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rfc:rfc6716

Internet Engineering Task Force (IETF) JM. Valin Request for Comments: 6716 Mozilla Corporation Category: Standards Track K. Vos ISSN: 2070-1721 Skype Technologies S.A.

                                                         T. Terriberry
                                                   Mozilla Corporation
                                                        September 2012
                 Definition of the Opus Audio Codec

Abstract

 This document defines the Opus interactive speech and audio codec.
 Opus is designed to handle a wide range of interactive audio
 applications, including Voice over IP, videoconferencing, in-game
 chat, and even live, distributed music performances.  It scales from
 low bitrate narrowband speech at 6 kbit/s to very high quality stereo
 music at 510 kbit/s.  Opus uses both Linear Prediction (LP) and the
 Modified Discrete Cosine Transform (MDCT) to achieve good compression
 of both speech and music.

Status of This Memo

 This is an Internet Standards Track document.
 This document is a product of the Internet Engineering Task Force
 (IETF).  It represents the consensus of the IETF community.  It has
 received public review and has been approved for publication by the
 Internet Engineering Steering Group (IESG).  Further information on
 Internet Standards is available in Section 2 of RFC 5741.
 Information about the current status of this document, any errata,
 and how to provide feedback on it may be obtained at
 http://www.rfc-editor.org/info/rfc6716.

Valin, et al. Standards Track [Page 1] RFC 6716 Interactive Audio Codec September 2012

Copyright Notice

 Copyright (c) 2012 IETF Trust and the persons identified as the
 document authors.  All rights reserved.
 This document is subject to BCP 78 and the IETF Trust's Legal
 Provisions Relating to IETF Documents
 (http://trustee.ietf.org/license-info) in effect on the date of
 publication of this document.  Please review these documents
 carefully, as they describe your rights and restrictions with respect
 to this document.  Code Components extracted from this document must
 include Simplified BSD License text as described in Section 4.e of
 the Trust Legal Provisions and are provided without warranty as
 described in the Simplified BSD License.
 The licenses granted by the IETF Trust to this RFC under Section 3.c
 of the Trust Legal Provisions shall also include the right to extract
 text from Sections 1 through 8 and Appendix A and Appendix B of this
 RFC and create derivative works from these extracts, and to copy,
 publish, display and distribute such derivative works in any medium
 and for any purpose, provided that no such derivative work shall be
 presented, displayed or published in a manner that states or implies
 that it is part of this RFC or any other IETF Document.

Table of Contents

 1. Introduction ....................................................5
    1.1. Notation and Conventions ...................................6
 2. Opus Codec Overview .............................................8
    2.1. Control Parameters ........................................10
         2.1.1. Bitrate ............................................10
         2.1.2. Number of Channels (Mono/Stereo) ...................11
         2.1.3. Audio Bandwidth ....................................11
         2.1.4. Frame Duration .....................................11
         2.1.5. Complexity .........................................11
         2.1.6. Packet Loss Resilience .............................12
         2.1.7. Forward Error Correction (FEC) .....................12
         2.1.8. Constant/Variable Bitrate ..........................12
         2.1.9. Discontinuous Transmission (DTX) ...................13
 3. Internal Framing ...............................................13
    3.1. The TOC Byte ..............................................13
    3.2. Frame Packing .............................................16
         3.2.1. Frame Length Coding ................................16
         3.2.2. Code 0: One Frame in the Packet ....................16
         3.2.3. Code 1: Two Frames in the Packet, Each with
                Equal Compressed Size ..............................17
         3.2.4. Code 2: Two Frames in the Packet, with
                Different Compressed Sizes .........................17

Valin, et al. Standards Track [Page 2] RFC 6716 Interactive Audio Codec September 2012

         3.2.5. Code 3: A Signaled Number of Frames in the Packet ..18
    3.3. Examples ..................................................21
    3.4. Receiving Malformed Packets ...............................22
 4. Opus Decoder ...................................................23
    4.1. Range Decoder .............................................23
         4.1.1. Range Decoder Initialization .......................25
         4.1.2. Decoding Symbols ...................................25
         4.1.3. Alternate Decoding Methods .........................27
         4.1.4. Decoding Raw Bits ..................................29
         4.1.5. Decoding Uniformly Distributed Integers ............29
         4.1.6. Current Bit Usage ..................................30
    4.2. SILK Decoder ..............................................32
         4.2.1. SILK Decoder Modules ...............................32
         4.2.2. LP Layer Organization ..............................33
         4.2.3. Header Bits ........................................35
         4.2.4. Per-Frame LBRR Flags ...............................36
         4.2.5. LBRR Frames ........................................36
         4.2.6. Regular SILK Frames ................................37
         4.2.7. SILK Frame Contents ................................37
                4.2.7.1. Stereo Prediction Weights .................40
                4.2.7.2. Mid-Only Flag .............................42
                4.2.7.3. Frame Type ................................43
                4.2.7.4. Subframe Gains ............................44
                4.2.7.5. Normalized Line Spectral Frequency
                         (LSF) and Linear Predictive Coding (LPC)
                         Coeffieients ..............................46
                4.2.7.6. Long-Term Prediction (LTP) Parameters .....74
                4.2.7.7. Linear Congruential Generator (LCG) Seed ..86
                4.2.7.8. Excitation ................................86
                4.2.7.9. SILK Frame Reconstruction .................98
         4.2.8. Stereo Unmixing ...................................102
         4.2.9. Resampling ........................................103
    4.3. CELT Decoder .............................................104
         4.3.1. Transient Decoding ................................108
         4.3.2. Energy Envelope Decoding ..........................108
         4.3.3. Bit Allocation ....................................110
         4.3.4. Shape Decoding ....................................116
         4.3.5. Anti-collapse Processing ..........................120
         4.3.6. Denormalization ...................................121
         4.3.7. Inverse MDCT ......................................121
    4.4. Packet Loss Concealment (PLC) ............................122
         4.4.1. Clock Drift Compensation ..........................122
    4.5. Configuration Switching ..................................123
         4.5.1. Transition Side Information (Redundancy) ..........124
         4.5.2. State Reset .......................................127
         4.5.3. Summary of Transitions ............................128
 5. Opus Encoder ..................................................131
    5.1. Range Encoder ............................................132

Valin, et al. Standards Track [Page 3] RFC 6716 Interactive Audio Codec September 2012

         5.1.1. Encoding Symbols ..................................133
         5.1.2. Alternate Encoding Methods ........................134
         5.1.3. Encoding Raw Bits .................................135
         5.1.4. Encoding Uniformly Distributed Integers ...........135
         5.1.5. Finalizing the Stream .............................135
         5.1.6. Current Bit Usage .................................136
    5.2. SILK Encoder .............................................136
         5.2.1. Sample Rate Conversion ............................137
         5.2.2. Stereo Mixing .....................................137
         5.2.3. SILK Core Encoder .................................138
    5.3. CELT Encoder .............................................150
         5.3.1. Pitch Pre-filter ..................................150
         5.3.2. Bands and Normalization ...........................151
         5.3.3. Energy Envelope Quantization ......................151
         5.3.4. Bit Allocation ....................................151
         5.3.5. Stereo Decisions ..................................152
         5.3.6. Time-Frequency Decision ...........................153
         5.3.7. Spreading Values Decision .........................153
         5.3.8. Spherical Vector Quantization .....................154
 6. Conformance ...................................................155
    6.1. Testing ..................................................155
    6.2. Opus Custom ..............................................156
 7. Security Considerations .......................................157
 8. Acknowledgements ..............................................158
 9. References ....................................................159
    9.1. Normative References .....................................159
    9.2. Informative References ...................................159
 Appendix A. Reference Implementation .............................163
    A.1. Extracting the Source ....................................164
    A.2. Up-to-Date Implementation ................................164
    A.3. Base64-Encoded Source Code ...............................164
    A.4. Test Vectors .............................................321
 Appendix B. Self-Delimiting Framing ..............................321

Valin, et al. Standards Track [Page 4] RFC 6716 Interactive Audio Codec September 2012

1. Introduction

 The Opus codec is a real-time interactive audio codec designed to
 meet the requirements described in [REQUIREMENTS].  It is composed of
 a layer based on Linear Prediction (LP) [LPC] and a layer based on
 the Modified Discrete Cosine Transform (MDCT) [MDCT].  The main idea
 behind using two layers is as follows: in speech, linear prediction
 techniques (such as Code-Excited Linear Prediction, or CELP) code low
 frequencies more efficiently than transform (e.g., MDCT) domain
 techniques, while the situation is reversed for music and higher
 speech frequencies.  Thus, a codec with both layers available can
 operate over a wider range than either one alone and can achieve
 better quality by combining them than by using either one
 individually.
 The primary normative part of this specification is provided by the
 source code in Appendix A.  Only the decoder portion of this software
 is normative, though a significant amount of code is shared by both
 the encoder and decoder.  Section 6 provides a decoder conformance
 test.  The decoder contains a great deal of integer and fixed-point
 arithmetic that needs to be performed exactly, including all rounding
 considerations, so any useful specification requires domain-specific
 symbolic language to adequately define these operations.
 Additionally, any conflict between the symbolic representation and
 the included reference implementation must be resolved.  For the
 practical reasons of compatibility and testability, it would be
 advantageous to give the reference implementation priority in any
 disagreement.  The C language is also one of the most widely
 understood, human-readable symbolic representations for machine
 behavior.  For these reasons, this RFC uses the reference
 implementation as the sole symbolic representation of the codec.
 While the symbolic representation is unambiguous and complete, it is
 not always the easiest way to understand the codec's operation.  For
 this reason, this document also describes significant parts of the
 codec in prose and takes the opportunity to explain the rationale
 behind many of the more surprising elements of the design.  These
 descriptions are intended to be accurate and informative, but the
 limitations of common English sometimes result in ambiguity, so it is
 expected that the reader will always read them alongside the symbolic
 representation.  Numerous references to the implementation are
 provided for this purpose.  The descriptions sometimes differ from
 the reference in ordering or through mathematical simplification
 wherever such deviation makes an explanation easier to understand.
 For example, the right shift and left shift operations in the
 reference implementation are often described using division and

Valin, et al. Standards Track [Page 5] RFC 6716 Interactive Audio Codec September 2012

 multiplication in the text.  In general, the text is focused on the
 "what" and "why" while the symbolic representation most clearly
 provides the "how".

1.1. Notation and Conventions

 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
 document are to be interpreted as described in RFC 2119 [RFC2119].
 Various operations in the codec require bit-exact fixed-point
 behavior, even when writing a floating point implementation.  The
 notation "Q<n>", where n is an integer, denotes the number of binary
 digits to the right of the decimal point in a fixed-point number.
 For example, a signed Q14 value in a 16-bit word can represent values
 from -2.0 to 1.99993896484375, inclusive.  This notation is for
 informational purposes only.  Arithmetic, when described, always
 operates on the underlying integer.  For example, the text will
 explicitly indicate any shifts required after a multiplication.
 Expressions, where included in the text, follow C operator rules and
 precedence, with the exception that the syntax "x**y" indicates x
 raised to the power y.  The text also makes use of the following
 functions.

1.1.1. min(x,y)

 The smallest of two values x and y.

1.1.2. max(x,y)

 The largest of two values x and y.

1.1.3. clamp(lo,x,hi)

                   clamp(lo,x,hi) = max(lo,min(x,hi))
 With this definition, if lo > hi, then lo is returned.

1.1.4. sign(x)

 The sign of x, i.e.,
                                  ( -1,  x < 0
                        sign(x) = <  0,  x == 0
                                  (  1,  x > 0

Valin, et al. Standards Track [Page 6] RFC 6716 Interactive Audio Codec September 2012

1.1.5. abs(x)

 The absolute value of x, i.e.,
                           abs(x) = sign(x)*x

1.1.6. floor(f)

 The largest integer z such that z <= f.

1.1.7. ceil(f)

 The smallest integer z such that z >= f.

1.1.8. round(f)

 The integer z nearest to f, with ties rounded towards negative
 infinity, i.e.,
                         round(f) = ceil(f - 0.5)

1.1.9. log2(f)

 The base-two logarithm of f.

1.1.10. ilog(n)

 The minimum number of bits required to store a positive integer n in
 binary, or 0 for a non-positive integer n.
                            ( 0,                 n <= 0
                  ilog(n) = <
                            ( floor(log2(n))+1,  n > 0
 Examples:
 o  ilog(-1) = 0
 o  ilog(0) = 0
 o  ilog(1) = 1
 o  ilog(2) = 2
 o  ilog(3) = 2
 o  ilog(4) = 3

Valin, et al. Standards Track [Page 7] RFC 6716 Interactive Audio Codec September 2012

 o  ilog(7) = 3

2. Opus Codec Overview

 The Opus codec scales from 6 kbit/s narrowband mono speech to
 510 kbit/s fullband stereo music, with algorithmic delays ranging
 from 5 ms to 65.2 ms.  At any given time, either the LP layer, the
 MDCT layer, or both, may be active.  It can seamlessly switch between
 all of its various operating modes, giving it a great deal of
 flexibility to adapt to varying content and network conditions
 without renegotiating the current session.  The codec allows input
 and output of various audio bandwidths, defined as follows:
 +----------------------+-----------------+-------------------------+
 | Abbreviation         | Audio Bandwidth | Sample Rate (Effective) |
 +----------------------+-----------------+-------------------------+
 | NB (narrowband)      |           4 kHz |                   8 kHz |
 |                      |                 |                         |
 | MB (medium-band)     |           6 kHz |                  12 kHz |
 |                      |                 |                         |
 | WB (wideband)        |           8 kHz |                  16 kHz |
 |                      |                 |                         |
 | SWB (super-wideband) |          12 kHz |                  24 kHz |
 |                      |                 |                         |
 | FB (fullband)        |      20 kHz (*) |                  48 kHz |
 +----------------------+-----------------+-------------------------+
                                Table 1
 (*) Although the sampling theorem allows a bandwidth as large as half
 the sampling rate, Opus never codes audio above 20 kHz, as that is
 the generally accepted upper limit of human hearing.
 Opus defines super-wideband (SWB) with an effective sample rate of
 24 kHz, unlike some other audio coding standards that use 32 kHz.
 This was chosen for a number of reasons.  The band layout in the MDCT
 layer naturally allows skipping coefficients for frequencies over
 12 kHz, but does not allow cleanly dropping just those frequencies
 over 16 kHz.  A sample rate of 24 kHz also makes resampling in the
 MDCT layer easier, as 24 evenly divides 48, and when 24 kHz is
 sufficient, it can save computation in other processing, such as
 Acoustic Echo Cancellation (AEC).  Experimental changes to the band
 layout to allow a 16 kHz cutoff (32 kHz effective sample rate) showed
 potential quality degradations at other sample rates, and, at typical
 bitrates, the number of bits saved by using such a cutoff instead of
 coding in fullband (FB) mode is very small.  Therefore, if an
 application wishes to process a signal sampled at 32 kHz, it should
 just use FB.

Valin, et al. Standards Track [Page 8] RFC 6716 Interactive Audio Codec September 2012

 The LP layer is based on the SILK codec [SILK].  It supports NB, MB,
 or WB audio and frame sizes from 10 ms to 60 ms, and requires an
 additional 5 ms look-ahead for noise shaping estimation.  A small
 additional delay (up to 1.5 ms) may be required for sampling rate
 conversion.  Like Vorbis [VORBIS-WEBSITE] and many other modern
 codecs, SILK is inherently designed for variable bitrate (VBR)
 coding, though the encoder can also produce constant bitrate (CBR)
 streams.  The version of SILK used in Opus is substantially modified
 from, and not compatible with, the stand-alone SILK codec previously
 deployed by Skype.  This document does not serve to define that
 format, but those interested in the original SILK codec should see
 [SILK] instead.
 The MDCT layer is based on the Constrained-Energy Lapped Transform
 (CELT) codec [CELT].  It supports NB, WB, SWB, or FB audio and frame
 sizes from 2.5 ms to 20 ms, and requires an additional 2.5 ms look-
 ahead due to the overlapping MDCT windows.  The CELT codec is
 inherently designed for CBR coding, but unlike many CBR codecs, it is
 not limited to a set of predetermined rates.  It internally allocates
 bits to exactly fill any given target budget, and an encoder can
 produce a VBR stream by varying the target on a per-frame basis.  The
 MDCT layer is not used for speech when the audio bandwidth is WB or
 less, as it is not useful there.  On the other hand, non-speech
 signals are not always adequately coded using linear prediction.
 Therefore, the MDCT layer should be used for music signals.
 A "Hybrid" mode allows the use of both layers simultaneously with a
 frame size of 10 or 20 ms and an SWB or FB audio bandwidth.  The LP
 layer codes the low frequencies by resampling the signal down to WB.
 The MDCT layer follows, coding the high frequency portion of the
 signal.  The cutoff between the two lies at 8 kHz, the maximum WB
 audio bandwidth.  In the MDCT layer, all bands below 8 kHz are
 discarded, so there is no coding redundancy between the two layers.
 The sample rate (in contrast to the actual audio bandwidth) can be
 chosen independently on the encoder and decoder side, e.g., a
 fullband signal can be decoded as wideband, or vice versa.  This
 approach ensures a sender and receiver can always interoperate,
 regardless of the capabilities of their actual audio hardware.
 Internally, the LP layer always operates at a sample rate of twice
 the audio bandwidth, up to a maximum of 16 kHz, which it continues to
 use for SWB and FB.  The decoder simply resamples its output to
 support different sample rates.  The MDCT layer always operates
 internally at a sample rate of 48 kHz.  Since all the supported
 sample rates evenly divide this rate, and since the decoder may
 easily zero out the high frequency portion of the spectrum in the
 frequency domain, it can simply decimate the MDCT layer output to
 achieve the other supported sample rates very cheaply.

Valin, et al. Standards Track [Page 9] RFC 6716 Interactive Audio Codec September 2012

 After conversion to the common, desired output sample rate, the
 decoder simply adds the output from the two layers together.  To
 compensate for the different look-ahead required by each layer, the
 CELT encoder input is delayed by an additional 2.7 ms.  This ensures
 that low frequencies and high frequencies arrive at the same time.
 This extra delay may be reduced by an encoder by using less look-
 ahead for noise shaping or using a simpler resampler in the LP layer,
 but this will reduce quality.  However, the base 2.5 ms look-ahead in
 the CELT layer cannot be reduced in the encoder because it is needed
 for the MDCT overlap, whose size is fixed by the decoder.
 Both layers use the same entropy coder, avoiding any waste from
 "padding bits" between them.  The hybrid approach makes it easy to
 support both CBR and VBR coding.  Although the LP layer is VBR, the
 bit allocation of the MDCT layer can produce a final stream that is
 CBR by using all the bits left unused by the LP layer.

2.1. Control Parameters

 The Opus codec includes a number of control parameters that can be
 changed dynamically during regular operation of the codec, without
 interrupting the audio stream from the encoder to the decoder.  These
 parameters only affect the encoder since any impact they have on the
 bitstream is signaled in-band such that a decoder can decode any Opus
 stream without any out-of-band signaling.  Any Opus implementation
 can add or modify these control parameters without affecting
 interoperability.  The most important encoder control parameters in
 the reference encoder are listed below.

2.1.1. Bitrate

 Opus supports all bitrates from 6 kbit/s to 510 kbit/s.  All other
 parameters being equal, higher bitrate results in higher quality.
 For a frame size of 20 ms, these are the bitrate "sweet spots" for
 Opus in various configurations:
 o  8-12 kbit/s for NB speech,
 o  16-20 kbit/s for WB speech,
 o  28-40 kbit/s for FB speech,
 o  48-64 kbit/s for FB mono music, and
 o  64-128 kbit/s for FB stereo music.

Valin, et al. Standards Track [Page 10] RFC 6716 Interactive Audio Codec September 2012

2.1.2. Number of Channels (Mono/Stereo)

 Opus can transmit either mono or stereo frames within a single
 stream.  When decoding a mono frame in a stereo decoder, the left and
 right channels are identical, and when decoding a stereo frame in a
 mono decoder, the mono output is the average of the left and right
 channels.  In some cases, it is desirable to encode a stereo input
 stream in mono (e.g., because the bitrate is too low to encode stereo
 with sufficient quality).  The number of channels encoded can be
 selected in real-time, but by default the reference encoder attempts
 to make the best decision possible given the current bitrate.

2.1.3. Audio Bandwidth

 The audio bandwidths supported by Opus are listed in Table 1.  Just
 like for the number of channels, any decoder can decode audio that is
 encoded at any bandwidth.  For example, any Opus decoder operating at
 8 kHz can decode an FB Opus frame, and any Opus decoder operating at
 48 kHz can decode an NB frame.  Similarly, the reference encoder can
 take a 48 kHz input signal and encode it as NB.  The higher the audio
 bandwidth, the higher the required bitrate to achieve acceptable
 quality.  The audio bandwidth can be explicitly specified in real-
 time, but, by default, the reference encoder attempts to make the
 best bandwidth decision possible given the current bitrate.

2.1.4. Frame Duration

 Opus can encode frames of 2.5, 5, 10, 20, 40, or 60 ms.  It can also
 combine multiple frames into packets of up to 120 ms.  For real-time
 applications, sending fewer packets per second reduces the bitrate,
 since it reduces the overhead from IP, UDP, and RTP headers.
 However, it increases latency and sensitivity to packet losses, as
 losing one packet constitutes a loss of a bigger chunk of audio.
 Increasing the frame duration also slightly improves coding
 efficiency, but the gain becomes small for frame sizes above 20 ms.
 For this reason, 20 ms frames are a good choice for most
 applications.

2.1.5. Complexity

 There are various aspects of the Opus encoding process where trade-
 offs can be made between CPU complexity and quality/bitrate.  In the
 reference encoder, the complexity is selected using an integer from 0
 to 10, where 0 is the lowest complexity and 10 is the highest.
 Examples of computations for which such trade-offs may occur are:
 o  The order of the pitch analysis whitening filter [WHITENING],

Valin, et al. Standards Track [Page 11] RFC 6716 Interactive Audio Codec September 2012

 o  The order of the short-term noise shaping filter,
 o  The number of states in delayed decision quantization of the
    residual signal, and
 o  The use of certain bitstream features such as variable time-
    frequency resolution and the pitch post-filter.

2.1.6. Packet Loss Resilience

 Audio codecs often exploit inter-frame correlations to reduce the
 bitrate at a cost in error propagation: after losing one packet,
 several packets need to be received before the decoder is able to
 accurately reconstruct the speech signal.  The extent to which Opus
 exploits inter-frame dependencies can be adjusted on the fly to
 choose a trade-off between bitrate and amount of error propagation.

2.1.7. Forward Error Correction (FEC)

 Another mechanism providing robustness against packet loss is the in-
 band Forward Error Correction (FEC).  Packets that are determined to
 contain perceptually important speech information, such as onsets or
 transients, are encoded again at a lower bitrate and this re-encoded
 information is added to a subsequent packet.

2.1.8. Constant/Variable Bitrate

 Opus is more efficient when operating with variable bitrate (VBR),
 which is the default.  When low-latency transmission is required over
 a relatively slow connection, then constrained VBR can also be used.
 This uses VBR in a way that simulates a "bit reservoir" and is
 equivalent to what MP3 (MPEG 1, Layer 3) and AAC (Advanced Audio
 Coding) call CBR (i.e., not true CBR due to the bit reservoir).  In
 some (rare) applications, constant bitrate (CBR) is required.  There
 are two main reasons to operate in CBR mode:
 o  When the transport only supports a fixed size for each compressed
    frame, or
 o  When encryption is used for an audio stream that is either highly
    constrained (e.g., yes/no, recorded prompts) or highly sensitive
    [SRTP-VBR].
 Bitrate may still be allowed to vary, even with sensitive data, as
 long as the variation is not driven by the input signal (for example,
 to match changing network conditions).  To achieve this, an
 application should still run Opus in CBR mode, but change the target
 rate before each packet.

Valin, et al. Standards Track [Page 12] RFC 6716 Interactive Audio Codec September 2012

2.1.9. Discontinuous Transmission (DTX)

 Discontinuous Transmission (DTX) reduces the bitrate during silence
 or background noise.  When DTX is enabled, only one frame is encoded
 every 400 milliseconds.

3. Internal Framing

 The Opus encoder produces "packets", which are each a contiguous set
 of bytes meant to be transmitted as a single unit.  The packets
 described here do not include such things as IP, UDP, or RTP headers,
 which are normally found in a transport-layer packet.  A single
 packet may contain multiple audio frames, so long as they share a
 common set of parameters, including the operating mode, audio
 bandwidth, frame size, and channel count (mono vs. stereo).  This
 section describes the possible combinations of these parameters and
 the internal framing used to pack multiple frames into a single
 packet.  This framing is not self-delimiting.  Instead, it assumes
 that a lower layer (such as UDP or RTP [RFC3550] or Ogg [RFC3533] or
 Matroska [MATROSKA-WEBSITE]) will communicate the length, in bytes,
 of the packet, and it uses this information to reduce the framing
 overhead in the packet itself.  A decoder implementation MUST support
 the framing described in this section.  An alternative, self-
 delimiting variant of the framing is described in Appendix B.
 Support for that variant is OPTIONAL.
 All bit diagrams in this document number the bits so that bit 0 is
 the most significant bit of the first byte, and bit 7 is the least
 significant.  Bit 8 is thus the most significant bit of the second
 byte, etc.  Well-formed Opus packets obey certain requirements,
 marked [R1] through [R7] below.  These are summarized in Section 3.4
 along with appropriate means of handling malformed packets.

3.1. The TOC Byte

 A well-formed Opus packet MUST contain at least one byte [R1].  This
 byte forms a table-of-contents (TOC) header that signals which of the
 various modes and configurations a given packet uses.  It is composed
 of a configuration number, "config", a stereo flag, "s", and a frame
 count code, "c", arranged as illustrated in Figure 1.  A description
 of each of these fields follows.

Valin, et al. Standards Track [Page 13] RFC 6716 Interactive Audio Codec September 2012

                            0
                            0 1 2 3 4 5 6 7
                           +-+-+-+-+-+-+-+-+
                           | config  |s| c |
                           +-+-+-+-+-+-+-+-+
                        Figure 1: The TOC Byte
 The top five bits of the TOC byte, labeled "config", encode one of 32
 possible configurations of operating mode, audio bandwidth, and frame
 size.  As described, the LP (SILK) layer and MDCT (CELT) layer can be
 combined in three possible operating modes:
 1.  A SILK-only mode for use in low bitrate connections with an audio
     bandwidth of WB or less,
 2.  A Hybrid (SILK+CELT) mode for SWB or FB speech at medium
     bitrates, and
 3.  A CELT-only mode for very low delay speech transmission as well
     as music transmission (NB to FB).
 The 32 possible configurations each identify which one of these
 operating modes the packet uses, as well as the audio bandwidth and
 the frame size.  Table 2 lists the parameters for each configuration.

Valin, et al. Standards Track [Page 14] RFC 6716 Interactive Audio Codec September 2012

 +-----------------------+-----------+-----------+-------------------+
 | Configuration         | Mode      | Bandwidth | Frame Sizes       |
 | Number(s)             |           |           |                   |
 +-----------------------+-----------+-----------+-------------------+
 | 0...3                 | SILK-only | NB        | 10, 20, 40, 60 ms |
 |                       |           |           |                   |
 | 4...7                 | SILK-only | MB        | 10, 20, 40, 60 ms |
 |                       |           |           |                   |
 | 8...11                | SILK-only | WB        | 10, 20, 40, 60 ms |
 |                       |           |           |                   |
 | 12...13               | Hybrid    | SWB       | 10, 20 ms         |
 |                       |           |           |                   |
 | 14...15               | Hybrid    | FB        | 10, 20 ms         |
 |                       |           |           |                   |
 | 16...19               | CELT-only | NB        | 2.5, 5, 10, 20 ms |
 |                       |           |           |                   |
 | 20...23               | CELT-only | WB        | 2.5, 5, 10, 20 ms |
 |                       |           |           |                   |
 | 24...27               | CELT-only | SWB       | 2.5, 5, 10, 20 ms |
 |                       |           |           |                   |
 | 28...31               | CELT-only | FB        | 2.5, 5, 10, 20 ms |
 +-----------------------+-----------+-----------+-------------------+
              Table 2: TOC Byte Configuration Parameters
 The configuration numbers in each range (e.g., 0...3 for NB SILK-
 only) correspond to the various choices of frame size, in the same
 order.  For example, configuration 0 has a 10 ms frame size and
 configuration 3 has a 60 ms frame size.
 One additional bit, labeled "s", signals mono vs. stereo, with 0
 indicating mono and 1 indicating stereo.
 The remaining two bits of the TOC byte, labeled "c", code the number
 of frames per packet (codes 0 to 3) as follows:
 o  0: 1 frame in the packet
 o  1: 2 frames in the packet, each with equal compressed size
 o  2: 2 frames in the packet, with different compressed sizes
 o  3: an arbitrary number of frames in the packet
 This document refers to a packet as a code 0 packet, code 1 packet,
 etc., based on the value of "c".

Valin, et al. Standards Track [Page 15] RFC 6716 Interactive Audio Codec September 2012

3.2. Frame Packing

 This section describes how frames are packed according to each
 possible value of "c" in the TOC byte.

3.2.1. Frame Length Coding

 When a packet contains multiple VBR frames (i.e., code 2 or 3), the
 compressed length of one or more of these frames is indicated with a
 one- or two-byte sequence, with the meaning of the first byte as
 follows:
 o  0: No frame (Discontinuous Transmission (DTX) or lost packet)
 o  1...251: Length of the frame in bytes
 o  252...255: A second byte is needed.  The total length is
    (second_byte*4)+first_byte
 The special length 0 indicates that no frame is available, either
 because it was dropped during transmission by some intermediary or
 because the encoder chose not to transmit it.  Any Opus frame in any
 mode MAY have a length of 0.
 The maximum representable length is 255*4+255=1275 bytes.  For 20 ms
 frames, this represents a bitrate of 510 kbit/s, which is
 approximately the highest useful rate for lossily compressed fullband
 stereo music.  Beyond this point, lossless codecs are more
 appropriate.  It is also roughly the maximum useful rate of the MDCT
 layer as, shortly thereafter, quality no longer improves with
 additional bits due to limitations on the codebook sizes.
 No length is transmitted for the last frame in a VBR packet, or for
 any of the frames in a CBR packet, as it can be inferred from the
 total size of the packet and the size of all other data in the
 packet.  However, the length of any individual frame MUST NOT exceed
 1275 bytes [R2] to allow for repacketization by gateways, conference
 bridges, or other software.

3.2.2. Code 0: One Frame in the Packet

 For code 0 packets, the TOC byte is immediately followed by N-1 bytes
 of compressed data for a single frame (where N is the size of the
 packet), as illustrated in Figure 2.

Valin, et al. Standards Track [Page 16] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|0|0|                                               |
   +-+-+-+-+-+-+-+-+                                               |
   |                    Compressed frame 1 (N-1 bytes)...          :
   :                                                               |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                       Figure 2: A Code 0 Packet

3.2.3. Code 1: Two Frames in the Packet, Each with Equal Compressed

      Size
 For code 1 packets, the TOC byte is immediately followed by the
 (N-1)/2 bytes of compressed data for the first frame, followed by
 (N-1)/2 bytes of compressed data for the second frame, as illustrated
 in Figure 3.  The number of payload bytes available for compressed
 data, N-1, MUST be even for all code 1 packets [R3].
    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|0|1|                                               |
   +-+-+-+-+-+-+-+-+                                               :
   |             Compressed frame 1 ((N-1)/2 bytes)...             |
   :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                               |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
   |             Compressed frame 2 ((N-1)/2 bytes)...             |
   :                                               +-+-+-+-+-+-+-+-+
   |                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                       Figure 3: A Code 1 Packet

3.2.4. Code 2: Two Frames in the Packet, with Different Compressed

      Sizes
 For code 2 packets, the TOC byte is followed by a one- or two-byte
 sequence indicating the length of the first frame (marked N1 in
 Figure 4), followed by N1 bytes of compressed data for the first
 frame.  The remaining N-N1-2 or N-N1-3 bytes are the compressed data
 for the second frame.  This is illustrated in Figure 4.  A code 2
 packet MUST contain enough bytes to represent a valid length.  For
 example, a 1-byte code 2 packet is always invalid, and a 2-byte code
 2 packet whose second byte is in the range 252...255 is also invalid.

Valin, et al. Standards Track [Page 17] RFC 6716 Interactive Audio Codec September 2012

 The length of the first frame, N1, MUST also be no larger than the
 size of the payload remaining after decoding that length for all code
 2 packets [R4].  This makes, for example, a 2-byte code 2 packet with
 a second byte in the range 1...251 invalid as well (the only valid
 2-byte code 2 packet is one where the length of both frames is zero).
    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|0| N1 (1-2 bytes):                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
   |               Compressed frame 1 (N1 bytes)...                |
   :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                               |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
   |                     Compressed frame 2...                     :
   :                                                               |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                       Figure 4: A Code 2 Packet

3.2.5. Code 3: A Signaled Number of Frames in the Packet

 Code 3 packets signal the number of frames, as well as additional
 padding, called "Opus padding" to indicate that this padding is added
 at the Opus layer rather than at the transport layer.  Code 3 packets
 MUST have at least 2 bytes [R6,R7].  The TOC byte is followed by a
 byte encoding the number of frames in the packet in bits 2 to 7
 (marked "M" in Figure 5), with bit 1 indicating whether or not Opus
 padding is inserted (marked "p" in Figure 5), and bit 0 indicating
 VBR (marked "v" in Figure 5).  M MUST NOT be zero, and the audio
 duration contained within a packet MUST NOT exceed 120 ms [R5].  This
 limits the maximum frame count for any frame size to 48 (for 2.5 ms
 frames), with lower limits for longer frame sizes.  Figure 5
 illustrates the layout of the frame count byte.
                            0
                            0 1 2 3 4 5 6 7
                           +-+-+-+-+-+-+-+-+
                           |v|p|     M     |
                           +-+-+-+-+-+-+-+-+
                    Figure 5: The frame count byte
 When Opus padding is used, the number of bytes of padding is encoded
 in the bytes following the frame count byte.  Values from 0...254
 indicate that 0...254 bytes of padding are included, in addition to

Valin, et al. Standards Track [Page 18] RFC 6716 Interactive Audio Codec September 2012

 the byte(s) used to indicate the size of the padding.  If the value
 is 255, then the size of the additional padding is 254 bytes, plus
 the padding value encoded in the next byte.  There MUST be at least
 one more byte in the packet in this case [R6,R7].  The additional
 padding bytes appear at the end of the packet and MUST be set to zero
 by the encoder to avoid creating a covert channel.  The decoder MUST
 accept any value for the padding bytes, however.
 Although this encoding provides multiple ways to indicate a given
 number of padding bytes, each uses a different number of bytes to
 indicate the padding size and thus will increase the total packet
 size by a different amount.  For example, to add 255 bytes to a
 packet, set the padding bit, p, to 1, insert a single byte after the
 frame count byte with a value of 254, and append 254 padding bytes
 with the value zero to the end of the packet.  To add 256 bytes to a
 packet, set the padding bit to 1, insert two bytes after the frame
 count byte with the values 255 and 0, respectively, and append 254
 padding bytes with the value zero to the end of the packet.  By using
 the value 255 multiple times, it is possible to create a packet of
 any specific, desired size.  Let P be the number of header bytes used
 to indicate the padding size plus the number of padding bytes
 themselves (i.e., P is the total number of bytes added to the
 packet).  Then, P MUST be no more than N-2 [R6,R7].
 In the CBR case, let R=N-2-P be the number of bytes remaining in the
 packet after subtracting the (optional) padding.  Then, the
 compressed length of each frame in bytes is equal to R/M.  The value
 R MUST be a non-negative integer multiple of M [R6].  The compressed
 data for all M frames follows, each of size R/M bytes, as illustrated
 in Figure 6.

Valin, et al. Standards Track [Page 19] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|1|0|p|     M     |  Padding length (Optional)    :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 1 (R/M bytes)...               :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 2 (R/M bytes)...               :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :                              ...                              :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame M (R/M bytes)...               :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :                  Opus Padding (Optional)...                   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                     Figure 6: A CBR Code 3 Packet
 In the VBR case, the (optional) padding length is followed by M-1
 frame lengths (indicated by "N1" to "N[M-1]" in Figure 7), each
 encoded in a one- or two-byte sequence as described above.  The
 packet MUST contain enough data for the M-1 lengths after removing
 the (optional) padding, and the sum of these lengths MUST be no
 larger than the number of bytes remaining in the packet after
 decoding them [R7].  The compressed data for all M frames follows,
 each frame consisting of the indicated number of bytes, with the
 final frame consuming any remaining bytes before the final padding,
 as illustrated in Figure 6.  The number of header bytes (TOC byte,
 frame count byte, padding length bytes, and frame length bytes), plus
 the signaled length of the first M-1 frames themselves, plus the
 signaled length of the padding MUST be no larger than N, the total
 size of the packet.

Valin, et al. Standards Track [Page 20] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|1|1|p|     M     | Padding length (Optional)     :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   : N1 (1-2 bytes): N2 (1-2 bytes):     ...       :     N[M-1]    |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 1 (N1 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 2 (N2 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :                              ...                              :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :                     Compressed frame M...                     :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :                  Opus Padding (Optional)...                   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                     Figure 7: A VBR Code 3 Packet

3.3. Examples

 Simplest case, one NB mono 20 ms SILK frame:
  0                   1                   2                   3
  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |    1    |0|0|0|               compressed data...              :
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                               Figure 8

Valin, et al. Standards Track [Page 21] RFC 6716 Interactive Audio Codec September 2012

 Two FB mono 5 ms CELT frames of the same compressed size:
  0                   1                   2                   3
  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |   29    |0|0|1|               compressed data...              :
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                               Figure 9
 Two FB mono 20 ms Hybrid frames of different compressed size:
  0                   1                   2                   3
  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |   15    |0|1|1|1|0|     2     |      N1       |               |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+               |
 |                       compressed data...                      :
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                               Figure 10
 Four FB stereo 20 ms CELT frames of the same compressed size:
  0                   1                   2                   3
  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |   31    |1|1|1|0|0|     4     |      compressed data...       :
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                               Figure 11

3.4. Receiving Malformed Packets

 A receiver MUST NOT process packets that violate any of the rules
 above as normal Opus packets.  They are reserved for future
 applications, such as in-band headers (containing metadata, etc.).
 Packets that violate these constraints may cause implementations of
 _this_ specification to treat them as malformed and discard them.
 These constraints are summarized here for reference:
 [R1]  Packets are at least one byte.
 [R2]  No implicit frame length is larger than 1275 bytes.
 [R3]  Code 1 packets have an odd total length, N, so that (N-1)/2 is
       an integer.

Valin, et al. Standards Track [Page 22] RFC 6716 Interactive Audio Codec September 2012

 [R4]  Code 2 packets have enough bytes after the TOC for a valid
       frame length, and that length is no larger than the number of
       bytes remaining in the packet.
 [R5]  Code 3 packets contain at least one frame, but no more than
       120 ms of audio total.
 [R6]  The length of a CBR code 3 packet, N, is at least two bytes,
       the number of bytes added to indicate the padding size plus the
       trailing padding bytes themselves, P, is no more than N-2, and
       the frame count, M, satisfies the constraint that (N-2-P) is a
       non-negative integer multiple of M.
 [R7]  VBR code 3 packets are large enough to contain all the header
       bytes (TOC byte, frame count byte, any padding length bytes,
       and any frame length bytes), plus the length of the first M-1
       frames, plus any trailing padding bytes.

4. Opus Decoder

 The Opus decoder consists of two main blocks: the SILK decoder and
 the CELT decoder.  At any given time, one or both of the SILK and
 CELT decoders may be active.  The output of the Opus decode is the
 sum of the outputs from the SILK and CELT decoders with proper sample
 rate conversion and delay compensation on the SILK side, and optional
 decimation (when decoding to sample rates less than 48 kHz) on the
 CELT side, as illustrated in the block diagram below.
                          +---------+    +------------+
                          |  SILK   |    |   Sample   |
                       +->| Decoder |--->|    Rate    |----+
 Bit-    +---------+   |  |         |    | Conversion |    v
 stream  |  Range  |---+  +---------+    +------------+  /---\  Audio
 ------->| Decoder |                                     | + |------>
         |         |---+  +---------+    +------------+  \---/
         +---------+   |  |  CELT   |    | Decimation |    ^
                       +->| Decoder |--->| (Optional) |----+
                          |         |    |            |
                          +---------+    +------------+

4.1. Range Decoder

 Opus uses an entropy coder based on range coding [RANGE-CODING]
 [MARTIN79], which is itself a rediscovery of the FIFO arithmetic code
 introduced by [CODING-THESIS].  It is very similar to arithmetic
 encoding, except that encoding is done with digits in any base

Valin, et al. Standards Track [Page 23] RFC 6716 Interactive Audio Codec September 2012

 instead of with bits, so it is faster when using larger bases (i.e.,
 a byte).  All of the calculations in the range coder must use bit-
 exact integer arithmetic.
 Symbols may also be coded as "raw bits" packed directly into the
 bitstream, bypassing the range coder.  These are packed backwards
 starting at the end of the frame, as illustrated in Figure 12.  This
 reduces complexity and makes the stream more resilient to bit errors,
 as corruption in the raw bits will not desynchronize the decoding
 process, unlike corruption in the input to the range decoder.  Raw
 bits are only used in the CELT layer.
    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | Range coder data (packed MSB to LSB) ->                       :
   +                                                               +
   :                                                               :
   +     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :     | <- Boundary occurs at an arbitrary bit position         :
   +-+-+-+                                                         +
   :                          <- Raw bits data (packed LSB to MSB) |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   Legend:
   LSB = Least Significant Bit
   MSB = Most Significant Bit
        Figure 12: Illustrative Example of Packing Range Coder
                           and Raw Bits Data
 Each symbol coded by the range coder is drawn from a finite alphabet
 and coded in a separate "context", which describes the size of the
 alphabet and the relative frequency of each symbol in that alphabet.
 Suppose there is a context with n symbols, identified with an index
 that ranges from 0 to n-1.  The parameters needed to encode or decode
 symbol k in this context are represented by a three-tuple
 (fl[k], fh[k], ft), all 16-bit unsigned integers, with
 0 <= fl[k] < fh[k] <= ft <= 65535.  The values of this tuple are
 derived from the probability model for the symbol, represented by
 traditional "frequency counts".  Because Opus uses static contexts,
 those are not updated as symbols are decoded.  Let f[i] be the
 frequency of symbol i.  Then, the three-tuple corresponding to symbol
 k is given by the following:

Valin, et al. Standards Track [Page 24] RFC 6716 Interactive Audio Codec September 2012

                 k-1                                   n-1
                 __                                    __
         fl[k] = \  f[i],  fh[k] = fl[k] + f[k],  ft = \  f[i]
                 /_                                    /_
                 i=0                                   i=0
 The range decoder extracts the symbols and integers encoded using the
 range encoder in Section 5.1.  The range decoder maintains an
 internal state vector composed of the two-tuple (val, rng), where val
 represents the difference between the high end of the current range
 and the actual coded value, minus one, and rng represents the size of
 the current range.  Both val and rng are 32-bit unsigned integer
 values.

4.1.1. Range Decoder Initialization

 Let b0 be an 8-bit unsigned integer containing first input byte (or
 containing zero if there are no bytes in this Opus frame).  The
 decoder initializes rng to 128 and initializes val to (127 -
  (b0>>1)), where (b0>>1) is the top 7 bits of the first input byte.
 It saves the remaining bit, (b0&1), for use in the renormalization
 procedure described in Section 4.1.2.1, which the decoder invokes
 immediately after initialization to read additional bits and
 establish the invariant that rng > 2**23.

4.1.2. Decoding Symbols

 Decoding a symbol is a two-step process.  The first step determines a
 16-bit unsigned value fs, which lies within the range of some symbol
 in the current context.  The second step updates the range decoder
 state with the three-tuple (fl[k], fh[k], ft) corresponding to that
 symbol.
 The first step is implemented by ec_decode() (entdec.c), which
 computes
                                    val
                     fs = ft - min(------ + 1, ft)
                                   rng/ft
 The divisions here are integer division.
 The decoder then identifies the symbol in the current context
 corresponding to fs; i.e., the value of k whose three-tuple
 (fl[k], fh[k], ft) satisfies fl[k] <= fs < fh[k].  It uses this tuple
 to update val according to

Valin, et al. Standards Track [Page 25] RFC 6716 Interactive Audio Codec September 2012

                                 rng
                     val = val - --- * (ft - fh[k])
                                 ft
 If fl[k] is greater than zero, then the decoder updates rng using
                            rng
                      rng = --- * (fh[k] - fl[k])
                            ft
 Otherwise, it updates rng using
                                 rng
                     rng = rng - --- * (ft - fh[k])
                                 ft
 Using a special case for the first symbol (rather than the last
 symbol, as is commonly done in other arithmetic coders) ensures that
 all the truncation error from the finite precision arithmetic
 accumulates in symbol 0.  This makes the cost of coding a 0 slightly
 smaller, on average, than its estimated probability indicates and
 makes the cost of coding any other symbol slightly larger.  When
 contexts are designed so that 0 is the most probable symbol, which is
 often the case, this strategy minimizes the inefficiency introduced
 by the finite precision.  It also makes some of the special-case
 decoding routines in Section 4.1.3 particularly simple.
 After the updates, implemented by ec_dec_update() (entdec.c), the
 decoder normalizes the range using the procedure in the next section,
 and returns the index k.

4.1.2.1. Renormalization

 To normalize the range, the decoder repeats the following process,
 implemented by ec_dec_normalize() (entdec.c), until rng > 2**23.  If
 rng is already greater than 2**23, the entire process is skipped.
 First, it sets rng to (rng<<8).  Then, it reads the next byte of the
 Opus frame and forms an 8-bit value sym, using the leftover bit
 buffered from the previous byte as the high bit and the top 7 bits of
 the byte just read as the other 7 bits of sym.  The remaining bit in
 the byte just read is buffered for use in the next iteration.  If no
 more input bytes remain, it uses zero bits instead.  See
 Section 4.1.1 for the initialization used to process the first byte.
 Then, it sets
               val = ((val<<8) + (255-sym)) & 0x7FFFFFFF

Valin, et al. Standards Track [Page 26] RFC 6716 Interactive Audio Codec September 2012

 It is normal and expected that the range decoder will read several
 bytes into the data of the raw bits (if any) at the end of the frame
 by the time the frame is completely decoded, as illustrated in
 Figure 13.  This same data MUST also be returned as raw bits when
 requested.  The encoder is expected to terminate the stream in such a
 way that the range decoder will decode the intended values regardless
 of the data contained in the raw bits.  Section 5.1.5 describes a
 procedure for doing this.  If the range decoder consumes all of the
 bytes belonging to the current frame, it MUST continue to use zero
 when any further input bytes are required, even if there is
 additional data in the current packet from padding or other frames.
    n              n+1             n+2             n+3
    0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :     | <----------- Overlap region ------------> |             :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
         ^                                           ^
         |   End of data buffered by the range coder |
   ...-----------------------------------------------+
         |
         | End of data consumed by raw bits
         +-------------------------------------------------------...
        Figure 13: Illustrative Example of Raw Bits Overlapping
                           Range Coder Data

4.1.3. Alternate Decoding Methods

 The reference implementation uses three additional decoding methods
 that are exactly equivalent to the above but make assumptions and
 simplifications that allow for a more efficient implementation.

4.1.3.1. ec_decode_bin()

 The first is ec_decode_bin() (entdec.c), defined using the parameter
 ftb instead of ft.  It is mathematically equivalent to calling
 ec_decode() with ft = (1<<ftb), but it avoids one of the divisions.

4.1.3.2. ec_dec_bit_logp()

 The next is ec_dec_bit_logp() (entdec.c), which decodes a single
 binary symbol, replacing both the ec_decode() and ec_dec_update()
 steps.  The context is described by a single parameter, logp, which
 is the absolute value of the base-2 logarithm of the probability of a
 "1".  It is mathematically equivalent to calling ec_decode() with
 ft = (1<<logp), followed by ec_dec_update() with the 3-tuple
 (fl[k] = 0, fh[k] = (1<<logp) - 1, ft = (1<<logp)) if the returned

Valin, et al. Standards Track [Page 27] RFC 6716 Interactive Audio Codec September 2012

 value of fs is less than (1<<logp) - 1 (a "0" was decoded), and with
 (fl[k] = (1<<logp) - 1, fh[k] = ft = (1<<logp)) otherwise (a "1" was
 decoded).  The implementation requires no multiplications or
 divisions.

4.1.3.3. ec_dec_icdf()

 The last is ec_dec_icdf() (entdec.c), which decodes a single symbol
 with a table-based context of up to 8 bits, also replacing both the
 ec_decode() and ec_dec_update() steps, as well as the search for the
 decoded symbol in between.  The context is described by two
 parameters, an icdf ("inverse" cumulative distribution function)
 table and ftb.  As with ec_decode_bin(), (1<<ftb) is equivalent to
 ft. idcf[k], on the other hand, stores (1<<ftb)-fh[k], which is equal
 to (1<<ftb) - fl[k+1]. fl[0] is assumed to be 0, and the table is
 terminated by a value of 0 (where fh[k] == ft).
 The function is mathematically equivalent to calling ec_decode() with
 ft = (1<<ftb), using the returned value fs to search the table for
 the first entry where fs < (1<<ftb)-icdf[k], and calling
 ec_dec_update() with fl[k] = (1<<ftb) - icdf[k-1] (or 0 if k == 0),
 fh[k] = (1<<ftb) - idcf[k], and ft = (1<<ftb).  Combining the search
 with the update allows the division to be replaced by a series of
 multiplications (which are usually much cheaper), and using an
 inverse CDF allows the use of an ftb as large as 8 in an 8-bit table
 without any special cases.  This is the primary interface with the
 range decoder in the SILK layer, though it is used in a few places in
 the CELT layer as well.
 Although icdf[k] is more convenient for the code, the frequency
 counts, f[k], are a more natural representation of the probability
 distribution function (PDF) for a given symbol.  Therefore, this
 document lists the latter, not the former, when describing the
 context in which a symbol is coded as a list, e.g., {4, 4, 4, 4}/16
 for a uniform context with four possible values and ft = 16.  The
 value of ft after the slash is always the sum of the entries in the
 PDF, but is included for convenience.  Contexts with identical
 probabilities, f[k]/ft, but different values of ft (or equivalently,
 ftb) are not the same, and cannot, in general, be used in place of
 one another.  An icdf table is also not capable of representing a PDF
 where the first symbol has 0 probability.  In such contexts,
 ec_dec_icdf() can decode the symbol by using a table that drops the
 entries for any initial zero-probability values and by adding the
 constant offset of the first value with a non-zero probability to its
 return value.

Valin, et al. Standards Track [Page 28] RFC 6716 Interactive Audio Codec September 2012

4.1.4. Decoding Raw Bits

 The raw bits used by the CELT layer are packed at the end of the
 frame, with the least significant bit of the first value packed in
 the least significant bit of the last byte, filling up to the most
 significant bit in the last byte, continuing on to the least
 significant bit of the penultimate byte, and so on.  The reference
 implementation reads them using ec_dec_bits() (entdec.c).  Because
 the range decoder must read several bytes ahead in the stream, as
 described in Section 4.1.2.1, the input consumed by the raw bits may
 overlap with the input consumed by the range coder, and a decoder
 MUST allow this.  The format should render it impossible to attempt
 to read more raw bits than there are actual bits in the frame, though
 a decoder may wish to check for this and report an error.

4.1.5. Decoding Uniformly Distributed Integers

 The function ec_dec_uint() (entdec.c) decodes one of ft equiprobable
 values in the range 0 to (ft - 1), inclusive, each with a frequency
 of 1, where ft may be as large as (2**32 - 1).  Because ec_decode()
 is limited to a total frequency of (2**16 - 1), it splits up the
 value into a range coded symbol representing up to 8 of the high
 bits, and, if necessary, raw bits representing the remainder of the
 value.  The limit of 8 bits in the range coded symbol is a trade-off
 between implementation complexity, modeling error (since the symbols
 no longer truly have equal coding cost), and rounding error
 introduced by the range coder itself (which gets larger as more bits
 are included).  Using raw bits reduces the maximum number of
 divisions required in the worst case, but means that it may be
 possible to decode a value outside the range 0 to (ft - 1),
 inclusive.
 ec_dec_uint() takes a single, positive parameter, ft, which is not
 necessarily a power of two, and returns an integer, t, whose value
 lies between 0 and (ft - 1), inclusive.  Let ftb = ilog(ft - 1),
 i.e., the number of bits required to store (ft - 1) in two's
 complement notation.  If ftb is 8 or less, then t is decoded with
 t = ec_decode(ft), and the range coder state is updated using the
 three-tuple (t, t + 1, ft).
 If ftb is greater than 8, then the top 8 bits of t are decoded using
               t = ec_decode(((ft - 1) >> (ftb - 8)) + 1)
 the decoder state is updated using the three-tuple (t, t + 1, ((ft -
  1) >> (ftb - 8)) + 1), and the remaining bits are decoded as raw
 bits, setting

Valin, et al. Standards Track [Page 29] RFC 6716 Interactive Audio Codec September 2012

              t = (t << (ftb - 8)) | ec_dec_bits(ftb - 8)
 If, at this point, t >= ft, then the current frame is corrupt.  In
 that case, the decoder should assume there has been an error in the
 coding, decoding, or transmission and SHOULD take measures to conceal
 the error (e.g., saturate to ft-1 or use the Packet Loss Concealment
 (PLC)) and/or report to the application that the error has occurred.

4.1.6. Current Bit Usage

 The bit allocation routines in the CELT decoder need a conservative
 upper bound on the number of bits that have been used from the
 current frame thus far, including both range coder bits and raw bits.
 This drives allocation decisions that must match those made in the
 encoder.  The upper bound is computed in the reference implementation
 to whole-bit precision by the function ec_tell() (entcode.h) and to
 fractional 1/8th bit precision by the function ec_tell_frac()
 (entcode.c).  Like all operations in the range coder, it must be
 implemented in a bit-exact manner, and it must produce exactly the
 same value returned by the same functions in the encoder after
 encoding the same symbols.
 ec_tell() is guaranteed to return ceil(ec_tell_frac()/8.0).  In
 various places, the codec will check to ensure there is enough room
 to contain a symbol before attempting to decode it.  In practice,
 although the number of bits used so far is an upper bound, decoding a
 symbol whose probability model suggests it has a worst-case cost of p
 1/8th bits may actually advance the return value of ec_tell_frac() by
 p-1, p, or p+1 1/8th bits, due to approximation error in that upper
 bound, truncation error in the range coder, and for large values of
 ft, modeling error in ec_dec_uint().
 However, this error is bounded, and periodic calls to ec_tell() or
 ec_tell_frac() at precisely defined points in the decoding process
 prevent it from accumulating.  For a range coder symbol that requires
 a whole number of bits (i.e., for which ft/(fh[k] - fl[k]) is a power
 of two), where there are at least p 1/8th bits available, decoding
 the symbol will never cause ec_tell() or ec_tell_frac() to exceed the
 size of the frame ("bust the budget").  In this case, the return
 value of ec_tell_frac() will only advance by more than p 1/8th bits
 if there were an additional, fractional number of bits remaining, and
 it will never advance beyond the next whole-bit boundary, which is
 safe, since frames always contain a whole number of bits.  However,
 when p is not a whole number of bits, an extra 1/8th bit is required
 to ensure that decoding the symbol will not bust the budget.

Valin, et al. Standards Track [Page 30] RFC 6716 Interactive Audio Codec September 2012

 The reference implementation keeps track of the total number of whole
 bits that have been processed by the decoder so far in the variable
 nbits_total, including the (possibly fractional) number of bits that
 are currently buffered, but not consumed, inside the range coder.
 nbits_total is initialized to 9 just before the initial range
 renormalization process completes (or equivalently, it can be
 initialized to 33 after the first renormalization).  The extra two
 bits over the actual amount buffered by the range coder guarantees
 that it is an upper bound and that there is enough room for the
 encoder to terminate the stream.  Each iteration through the range
 coder's renormalization loop increases nbits_total by 8.  Reading raw
 bits increases nbits_total by the number of raw bits read.

4.1.6.1. ec_tell()

 The whole number of bits buffered in rng may be estimated via
 lg = ilog(rng). ec_tell() then becomes a simple matter of removing
 these bits from the total.  It returns (nbits_total - lg).
 In a newly initialized decoder, before any symbols have been read,
 this reports that 1 bit has been used.  This is the bit reserved for
 termination of the encoder.

4.1.6.2. ec_tell_frac()

 ec_tell_frac() estimates the number of bits buffered in rng to
 fractional precision.  Since rng must be greater than 2**23 after
 renormalization, lg must be at least 24.  Let
                         r_Q15 = rng >> (lg-16)
 so that 32768 <= r_Q15 < 65536, an unsigned Q15 value representing
 the fractional part of rng.  Then, the following procedure can be
 used to add one bit of precision to lg.  First, update
                      r_Q15 = (r_Q15*r_Q15) >> 15
 Then, add the 16th bit of r_Q15 to lg via
                       lg = 2*lg + (r_Q15 >> 16)
 Finally, if this bit was a 1, reduce r_Q15 by a factor of two via
                           r_Q15 = r_Q15 >> 1
 so that it once again lies in the range 32768 <= r_Q15 < 65536.  This
 procedure is repeated three times to extend lg to 1/8th bit
 precision. ec_tell_frac() then returns (nbits_total*8 - lg).

Valin, et al. Standards Track [Page 31] RFC 6716 Interactive Audio Codec September 2012

4.2. SILK Decoder

 The decoder's LP layer uses a modified version of the SILK codec
 (herein simply called "SILK"), which runs a decoded excitation signal
 through adaptive long-term and short-term prediction synthesis
 filters.  It runs at NB, MB, and WB sample rates internally.  When
 used in a SWB or FB Hybrid frame, the LP layer itself still only runs
 in WB.

4.2.1. SILK Decoder Modules

 An overview of the decoder is given in Figure 14.
      +---------+    +------------+
   -->| Range   |--->| Decode     |---------------------------+
    1 | Decoder | 2  | Parameters |----------+       5        |
      +---------+    +------------+     4    |                |
                          3 |                |                |
                           \/               \/               \/
                     +------------+   +------------+   +------------+
                     | Generate   |-->| LTP        |-->| LPC        |
                     | Excitation |   | Synthesis  |   | Synthesis  |
                     +------------+   +------------+   +------------+
                                             ^                |
                                             |                |
                         +-------------------+----------------+
                         |                                      6
                         |   +------------+   +-------------+
                         +-->| Stereo     |-->| Sample Rate |-->
                             | Unmixing   | 7 | Conversion  | 8
                             +------------+   +-------------+
   1: Range encoded bitstream
   2: Coded parameters
   3: Pulses, LSBs, and signs
   4: Pitch lags, Long-Term Prediction (LTP) coefficients
   5: Linear Predictive Coding (LPC) coefficients and gains
   6: Decoded signal (mono or mid-side stereo)
   7: Unmixed signal (mono or left-right stereo)
   8: Resampled signal
                        Figure 14: SILK Decoder

Valin, et al. Standards Track [Page 32] RFC 6716 Interactive Audio Codec September 2012

 The decoder feeds the bitstream (1) to the range decoder from
 Section 4.1 and then decodes the parameters in it (2) using the
 procedures detailed in Sections 4.2.3 through 4.2.7.8.5.  These
 parameters (3, 4, 5) are used to generate an excitation signal (see
 Section 4.2.7.8.6), which is fed to an optional Long-Term Prediction
 (LTP) filter (voiced frames only, see Section 4.2.7.9.1) and then a
 short-term prediction filter (see Section 4.2.7.9.2), producing the
 decoded signal (6).  For stereo streams, the mid-side representation
 is converted to separate left and right channels (7).  The result is
 finally resampled to the desired output sample rate (e.g., 48 kHz) so
 that the resampled signal (8) can be mixed with the CELT layer.

4.2.2. LP Layer Organization

 Internally, the LP layer of a single Opus frame is composed of either
 a single 10 ms regular SILK frame or between one and three 20 ms
 regular SILK frames.  A stereo Opus frame may double the number of
 regular SILK frames (up to a total of six), since it includes
 separate frames for a mid channel and, optionally, a side channel.
 Optional Low Bit-Rate Redundancy (LBRR) frames, which are reduced-
 bitrate encodings of previous SILK frames, may be included to aid in
 recovery from packet loss.  If present, these appear before the
 regular SILK frames.  They are, in most respects, identical to
 regular, active SILK frames, except that they are usually encoded
 with a lower bitrate.  This document uses "SILK frame" to refer to
 either one and "regular SILK frame" if it needs to draw a distinction
 between the two.
 Logically, each SILK frame is, in turn, composed of either two or
 four 5 ms subframes.  Various parameters, such as the quantization
 gain of the excitation and the pitch lag and filter coefficients can
 vary on a subframe-by-subframe basis.  Physically, the parameters for
 each subframe are interleaved in the bitstream, as described in the
 relevant sections for each parameter.
 All of these frames and subframes are decoded from the same range
 coder, with no padding between them.  Thus, packing multiple SILK
 frames in a single Opus frame saves, on average, half a byte per SILK
 frame.  It also allows some parameters to be predicted from prior
 SILK frames in the same Opus frame, since this does not degrade
 packet loss robustness (beyond any penalty for merely using fewer,
 larger packets to store multiple frames).
 Stereo support in SILK uses a variant of mid-side coding, allowing a
 mono decoder to simply decode the mid channel.  However, the data for
 the two channels is interleaved, so a mono decoder must still unpack

Valin, et al. Standards Track [Page 33] RFC 6716 Interactive Audio Codec September 2012

 the data for the side channel.  It would be required to do so anyway
 for Hybrid Opus frames or to support decoding individual 20 ms
 frames.
 Table 3 summarizes the overall grouping of the contents of the LP
 layer.  Figures 15 and 16 illustrate the ordering of the various SILK
 frames for a 60 ms Opus frame, for both mono and stereo,
 respectively.
 +-----------------------------------+---------------+---------------+
 |             Symbol(s)             |     PDF(s)    |   Condition   |
 +-----------------------------------+---------------+---------------+
 |   Voice Activity Detection (VAD)  |    {1, 1}/2   |               |
 |               Flags               |               |               |
 |                                   |               |               |
 |             LBRR Flag             |    {1, 1}/2   |               |
 |                                   |               |               |
 |        Per-Frame LBRR Flags       |    Table 4    | Section 4.2.4 |
 |                                   |               |               |
 |           LBRR Frame(s)           | Section 4.2.7 | Section 4.2.4 |
 |                                   |               |               |
 |       Regular SILK Frame(s)       | Section 4.2.7 |               |
 +-----------------------------------+---------------+---------------+
       Table 3: Organization of the SILK layer of an Opus Frame
                  +---------------------------------+
                  |            VAD Flags            |
                  +---------------------------------+
                  |            LBRR Flag            |
                  +---------------------------------+
                  | Per-Frame LBRR Flags (Optional) |
                  +---------------------------------+
                  |     LBRR Frame 1 (Optional)     |
                  +---------------------------------+
                  |     LBRR Frame 2 (Optional)     |
                  +---------------------------------+
                  |     LBRR Frame 3 (Optional)     |
                  +---------------------------------+
                  |      Regular SILK Frame 1       |
                  +---------------------------------+
                  |      Regular SILK Frame 2       |
                  +---------------------------------+
                  |      Regular SILK Frame 3       |
                  +---------------------------------+
                     Figure 15: A 60 ms Mono Frame

Valin, et al. Standards Track [Page 34] RFC 6716 Interactive Audio Codec September 2012

               +---------------------------------------+
               |             Mid VAD Flags             |
               +---------------------------------------+
               |             Mid LBRR Flag             |
               +---------------------------------------+
               |             Side VAD Flags            |
               +---------------------------------------+
               |             Side LBRR Flag            |
               +---------------------------------------+
               |  Mid Per-Frame LBRR Flags (Optional)  |
               +---------------------------------------+
               | Side Per-Frame LBRR Flags (Optional)  |
               +---------------------------------------+
               |     Mid LBRR Frame 1 (Optional)       |
               +---------------------------------------+
               |     Side LBRR Frame 1 (Optional)      |
               +---------------------------------------+
               |     Mid LBRR Frame 2 (Optional)       |
               +---------------------------------------+
               |     Side LBRR Frame 2 (Optional)      |
               +---------------------------------------+
               |     Mid LBRR Frame 3 (Optional)       |
               +---------------------------------------+
               |     Side LBRR Frame 3 (Optional)      |
               +---------------------------------------+
               |      Mid Regular SILK Frame 1         |
               +---------------------------------------+
               | Side Regular SILK Frame 1 (Optional)  |
               +---------------------------------------+
               |      Mid Regular SILK Frame 2         |
               +---------------------------------------+
               | Side Regular SILK Frame 2 (Optional)  |
               +---------------------------------------+
               |      Mid Regular SILK Frame 3         |
               +---------------------------------------+
               | Side Regular SILK Frame 3 (Optional)  |
               +---------------------------------------+
                    Figure 16: A 60 ms Stereo Frame

4.2.3. Header Bits

 The LP layer begins with two to eight header bits, decoded in
 silk_Decode() (dec_API.c).  These consist of one Voice Activity
 Detection (VAD) bit per frame (up to 3), followed by a single flag
 indicating the presence of LBRR frames.  For a stereo packet, these
 first flags correspond to the mid channel, and a second set of flags
 is included for the side channel.

Valin, et al. Standards Track [Page 35] RFC 6716 Interactive Audio Codec September 2012

 Because these are the first symbols decoded by the range coder and
 because they are coded as binary values with uniform probability,
 they can be extracted directly from the most significant bits of the
 first byte of compressed data.  Thus, a receiver can determine if an
 Opus frame contains any active SILK frames without the overhead of
 using the range decoder.

4.2.4. Per-Frame LBRR Flags

 For Opus frames longer than 20 ms, a set of LBRR flags is decoded for
 each channel that has its LBRR flag set.  Each set contains one flag
 per 20 ms SILK frame. 40 ms Opus frames use the 2-frame LBRR flag PDF
 from Table 4, and 60 ms Opus frames use the 3-frame LBRR flag PDF.
 For each channel, the resulting 2- or 3-bit integer contains the
 corresponding LBRR flag for each frame, packed in order from the LSB
 to the MSB.
         +------------+-------------------------------------+
         | Frame Size | PDF                                 |
         +------------+-------------------------------------+
         | 40 ms      | {0, 53, 53, 150}/256                |
         |            |                                     |
         | 60 ms      | {0, 41, 20, 29, 41, 15, 28, 82}/256 |
         +------------+-------------------------------------+
                        Table 4: LBRR Flag PDFs
 A 10 or 20 ms Opus frame does not contain any per-frame LBRR flags,
 as there may be at most one LBRR frame per channel.  The global LBRR
 flag in the header bits (see Section 4.2.3) is already sufficient to
 indicate the presence of that single LBRR frame.

4.2.5. LBRR Frames

 The LBRR frames, if present, contain an encoded representation of the
 signal immediately prior to the current Opus frame as if it were
 encoded with the current mode, frame size, audio bandwidth, and
 channel count, even if those differ from the prior Opus frame.  When
 one of these parameters changes from one Opus frame to the next, this
 implies that the LBRR frames of the current Opus frame may not be
 simple drop-in replacements for the contents of the previous Opus
 frame.
 For example, when switching from 20 ms to 60 ms, the 60 ms Opus frame
 may contain LBRR frames covering up to three prior 20 ms Opus frames,
 even if those frames already contained LBRR frames covering some of
 the same time periods.  When switching from 20 ms to 10 ms, the 10 ms
 Opus frame can contain an LBRR frame covering at most half the prior

Valin, et al. Standards Track [Page 36] RFC 6716 Interactive Audio Codec September 2012

 20 ms Opus frame, potentially leaving a hole that needs to be
 concealed from even a single packet loss (see Section 4.4).  When
 switching from mono to stereo, the LBRR frames in the first stereo
 Opus frame MAY contain a non-trivial side channel.
 In order to properly produce LBRR frames under all conditions, an
 encoder might need to buffer up to 60 ms of audio and re-encode it
 during these transitions.  However, the reference implementation opts
 to disable LBRR frames at the transition point for simplicity.  Since
 transitions are relatively infrequent in normal usage, this does not
 have a significant impact on packet loss robustness.
 The LBRR frames immediately follow the LBRR flags, prior to any
 regular SILK frames.  Section 4.2.7 describes their exact contents.
 LBRR frames do not include their own separate VAD flags.  LBRR frames
 are only meant to be transmitted for active speech, thus all LBRR
 frames are treated as active.
 In a stereo Opus frame longer than 20 ms, although the per-frame LBRR
 flags for the mid channel are coded as a unit before the per-frame
 LBRR flags for the side channel, the LBRR frames themselves are
 interleaved.  The decoder parses an LBRR frame for the mid channel of
 a given 20 ms interval (if present) and then immediately parses the
 corresponding LBRR frame for the side channel (if present), before
 proceeding to the next 20 ms interval.

4.2.6. Regular SILK Frames

 The regular SILK frame(s) follow the LBRR frames (if any).
 Section 4.2.7 describes their contents, as well.  Unlike the LBRR
 frames, a regular SILK frame is coded for each time interval in an
 Opus frame, even if the corresponding VAD flags are unset.  For
 stereo Opus frames longer than 20 ms, the regular mid and side SILK
 frames for each 20 ms interval are interleaved, just as with the LBRR
 frames.  The side frame may be skipped by coding an appropriate flag,
 as detailed in Section 4.2.7.2.

4.2.7. SILK Frame Contents

 Each SILK frame includes a set of side information that encodes
 o  The frame type and quantization type (Section 4.2.7.3),
 o  Quantization gains (Section 4.2.7.4),
 o  Short-term prediction filter coefficients (Section 4.2.7.5),

Valin, et al. Standards Track [Page 37] RFC 6716 Interactive Audio Codec September 2012

 o  A Line Spectral Frequencies (LSFs) interpolation weight
    (Section 4.2.7.5.5),
 o  LTP filter lags and gains (Section 4.2.7.6), and
 o  A Linear Congruential Generator (LCG) seed (Section 4.2.7.7).
 The quantized excitation signal (see Section 4.2.7.8) follows these
 at the end of the frame.  Table 5 details the overall organization of
 a SILK frame.

Valin, et al. Standards Track [Page 38] RFC 6716 Interactive Audio Codec September 2012

 +---------------------------+-------------------+-------------------+
 |         Symbol(s)         |       PDF(s)      |     Condition     |
 +---------------------------+-------------------+-------------------+
 | Stereo Prediction Weights |      Table 6      |  Section 4.2.7.1  |
 |                           |                   |                   |
 |       Mid-only Flag       |      Table 8      |  Section 4.2.7.2  |
 |                           |                   |                   |
 |         Frame Type        |  Section 4.2.7.3  |                   |
 |                           |                   |                   |
 |       Subframe Gains      |  Section 4.2.7.4  |                   |
 |                           |                   |                   |
 |   Normalized LSF Stage-1  |      Table 14     |                   |
 |           Index           |                   |                   |
 |                           |                   |                   |
 |   Normalized LSF Stage-2  | Section 4.2.7.5.2 |                   |
 |          Residual         |                   |                   |
 |                           |                   |                   |
 |       Normalized LSF      |      Table 26     |    20 ms frame    |
 |    Interpolation Weight   |                   |                   |
 |                           |                   |                   |
 |     Primary Pitch Lag     | Section 4.2.7.6.1 |    Voiced frame   |
 |                           |                   |                   |
 |   Subframe Pitch Contour  |      Table 32     |    Voiced frame   |
 |                           |                   |                   |
 |     Periodicity Index     |      Table 37     |    Voiced frame   |
 |                           |                   |                   |
 |         LTP Filter        |      Table 38     |    Voiced frame   |
 |                           |                   |                   |
 |        LTP Scaling        |      Table 42     | Section 4.2.7.6.3 |
 |                           |                   |                   |
 |          LCG Seed         |      Table 43     |                   |
 |                           |                   |                   |
 |   Excitation Rate Level   |      Table 45     |                   |
 |                           |                   |                   |
 |  Excitation Pulse Counts  |      Table 46     |                   |
 |                           |                   |                   |
 |      Excitation Pulse     | Section 4.2.7.8.3 |   Non-zero pulse  |
 |         Locations         |                   |       count       |
 |                           |                   |                   |
 |      Excitation LSBs      |      Table 51     | Section 4.2.7.8.2 |
 |                           |                   |                   |
 |      Excitation Signs     |      Table 52     |                   |
 +---------------------------+-------------------+-------------------+
       Table 5: Order of the Symbols in an Individual SILK Frame

Valin, et al. Standards Track [Page 39] RFC 6716 Interactive Audio Codec September 2012

4.2.7.1. Stereo Prediction Weights

 A SILK frame corresponding to the mid channel of a stereo Opus frame
 begins with a pair of side channel prediction weights, designed such
 that zeros indicate normal mid-side coupling.  Since these weights
 can change on every frame, the first portion of each frame linearly
 interpolates between the previous weights and the current ones, using
 zeros for the previous weights if none are available.  These
 prediction weights are never included in a mono Opus frame, and the
 previous weights are reset to zeros on any transition from mono to
 stereo.  They are also not included in an LBRR frame for the side
 channel, even if the LBRR flags indicate the corresponding mid
 channel was not coded.  In that case, the previous weights are used,
 again substituting in zeros if no previous weights are available
 since the last decoder reset (see Section 4.5.2).
 To summarize, these weights are coded if and only if
 o  This is a stereo Opus frame (Section 3.1), and
 o  The current SILK frame corresponds to the mid channel.
 The prediction weights are coded in three separate pieces, which are
 decoded by silk_stereo_decode_pred() (stereo_decode_pred.c).  The
 first piece jointly codes the high-order part of a table index for
 both weights.  The second piece codes the low-order part of each
 table index.  The third piece codes an offset used to linearly
 interpolate between table indices.  The details are as follows.
 Let n be an index decoded with the 25-element stage-1 PDF in Table 6.
 Then, let i0 and i1 be indices decoded with the stage-2 and stage-3
 PDFs in Table 6, respectively, and let i2 and i3 be two more indices
 decoded with the stage-2 and stage-3 PDFs, all in that order.
 +-------+-----------------------------------------------------------+
 | Stage | PDF                                                       |
 +-------+-----------------------------------------------------------+
 | Stage | {7, 2, 1, 1, 1, 10, 24, 8, 1, 1, 3, 23, 92, 23, 3, 1, 1,  |
 | 1     | 8, 24, 10, 1, 1, 1, 2, 7}/256                             |
 |       |                                                           |
 | Stage | {85, 86, 85}/256                                          |
 | 2     |                                                           |
 |       |                                                           |
 | Stage | {51, 51, 52, 51, 51}/256                                  |
 | 3     |                                                           |
 +-------+-----------------------------------------------------------+
                      Table 6: Stereo Weight PDFs

Valin, et al. Standards Track [Page 40] RFC 6716 Interactive Audio Codec September 2012

 Then, use n, i0, and i2 to form two table indices, wi0 and wi1,
 according to
                           wi0 = i0 + 3*(n/5)
                           wi1 = i2 + 3*(n%5)
 where the division is integer division.  The range of these indices
 is 0 to 14, inclusive.  Let w_Q13[i] be the i'th weight from Table 7.
 Then, the two prediction weights, w0_Q13 and w1_Q13, are
    w1_Q13 = w_Q13[wi1]
             + (((w_Q13[wi1+1] - w_Q13[wi1])*6554) >> 16)*(2*i3 + 1)
    w0_Q13 = w_Q13[wi0]
             + (((w_Q13[wi0+1] - w_Q13[wi0])*6554) >> 16)*(2*i1 + 1)
             - w1_Q13
 N.B., w1_Q13 is computed first here, because w0_Q13 depends on it.
 The constant 6554 is approximately 0.1 in Q16.  Although wi0 and wi1
 only have 15 possible values, Table 7 contains 16 entries to allow
 interpolation between entry wi0 and (wi0 + 1) (and likewise for wi1).

Valin, et al. Standards Track [Page 41] RFC 6716 Interactive Audio Codec September 2012

                       +-------+--------------+
                       | Index | Weight (Q13) |
                       +-------+--------------+
                       | 0     |       -13732 |
                       |       |              |
                       | 1     |       -10050 |
                       |       |              |
                       | 2     |        -8266 |
                       |       |              |
                       | 3     |        -7526 |
                       |       |              |
                       | 4     |        -6500 |
                       |       |              |
                       | 5     |        -5000 |
                       |       |              |
                       | 6     |        -2950 |
                       |       |              |
                       | 7     |         -820 |
                       |       |              |
                       | 8     |          820 |
                       |       |              |
                       | 9     |         2950 |
                       |       |              |
                       | 10    |         5000 |
                       |       |              |
                       | 11    |         6500 |
                       |       |              |
                       | 12    |         7526 |
                       |       |              |
                       | 13    |         8266 |
                       |       |              |
                       | 14    |        10050 |
                       |       |              |
                       | 15    |        13732 |
                       +-------+--------------+
                     Table 7: Stereo Weight Table

4.2.7.2. Mid-Only Flag

 A flag appears after the stereo prediction weights that indicates if
 only the mid channel is coded for this time interval.  It appears
 only when
 o  This is a stereo Opus frame (see Section 3.1),
 o  The current SILK frame corresponds to the mid channel, and

Valin, et al. Standards Track [Page 42] RFC 6716 Interactive Audio Codec September 2012

 o  Either
  • This is a regular SILK frame where the VAD flags (see

Section 4.2.3) indicate that the corresponding side channel is

       not active.
  • This is an LBRR frame where the LBRR flags (see Sections 4.2.3

and 4.2.4) indicate that the corresponding side channel is not

       coded.
 It is omitted when there are no stereo weights, for all of the same
 reasons.  It is also omitted for a regular SILK frame when the VAD
 flag of the corresponding side channel frame is set (indicating it is
 active).  The side channel must be coded in this case, making the
 mid-only flag redundant.  It is also omitted for an LBRR frame when
 the corresponding LBRR flags indicate the side channel is coded.
 When the flag is present, the decoder reads a single value using the
 PDF in Table 8, as implemented in silk_stereo_decode_mid_only()
 (stereo_decode_pred.c).  If the flag is set, then there is no
 corresponding SILK frame for the side channel, the entire decoding
 process for the side channel is skipped, and zeros are fed to the
 stereo unmixing process (see Section 4.2.8) instead.  As stated
 above, LBRR frames still include this flag when the LBRR flag
 indicates that the side channel is not coded.  In that case, if this
 flag is zero (indicating that there should be a side channel), then
 Packet Loss Concealment (PLC, see Section 4.4) SHOULD be invoked to
 recover a side channel signal.  Otherwise, the stereo image will
 collapse.
                           +---------------+
                           | PDF           |
                           +---------------+
                           | {192, 64}/256 |
                           +---------------+
                      Table 8: Mid-only Flag PDF

4.2.7.3. Frame Type

 Each SILK frame contains a single "frame type" symbol that jointly
 codes the signal type and quantization offset type of the
 corresponding frame.  If the current frame is a regular SILK frame
 whose VAD bit was not set (an "inactive" frame), then the frame type
 symbol takes on a value of either 0 or 1 and is decoded using the
 first PDF in Table 9.  If the frame is an LBRR frame or a regular
 SILK frame whose VAD flag was set (an "active" frame), then the value
 of the symbol may range from 2 to 5, inclusive, and is decoded using

Valin, et al. Standards Track [Page 43] RFC 6716 Interactive Audio Codec September 2012

 the second PDF in Table 9.  Table 10 translates between the value of
 the frame type symbol and the corresponding signal type and
 quantization offset type.
              +----------+-----------------------------+
              | VAD Flag | PDF                         |
              +----------+-----------------------------+
              | Inactive | {26, 230, 0, 0, 0, 0}/256   |
              |          |                             |
              | Active   | {0, 0, 24, 74, 148, 10}/256 |
              +----------+-----------------------------+
                       Table 9: Frame Type PDFs
        +------------+-------------+--------------------------+
        | Frame Type | Signal Type | Quantization Offset Type |
        +------------+-------------+--------------------------+
        | 0          | Inactive    |                      Low |
        |            |             |                          |
        | 1          | Inactive    |                     High |
        |            |             |                          |
        | 2          | Unvoiced    |                      Low |
        |            |             |                          |
        | 3          | Unvoiced    |                     High |
        |            |             |                          |
        | 4          | Voiced      |                      Low |
        |            |             |                          |
        | 5          | Voiced      |                     High |
        +------------+-------------+--------------------------+
  Table 10: Signal Type and Quantization Offset Type from Frame Type

4.2.7.4. Subframe Gains

 A separate quantization gain is coded for each 5 ms subframe.  These
 gains control the step size between quantization levels of the
 excitation signal and, therefore, the quality of the reconstruction.
 They are independent of and unrelated to the pitch contours coded for
 voiced frames.  The quantization gains are themselves uniformly
 quantized to 6 bits on a log scale, giving them a resolution of
 approximately 1.369 dB and a range of approximately 1.94 dB to
 88.21 dB.
 The subframe gains are either coded independently, or relative to the
 gain from the most recent coded subframe in the same channel.
 Independent coding is used if and only if

Valin, et al. Standards Track [Page 44] RFC 6716 Interactive Audio Codec September 2012

 o  This is the first subframe in the current SILK frame, and
 o  Either
  • This is the first SILK frame of its type (LBRR or regular) for

this channel in the current Opus frame, or

  • The previous SILK frame of the same type (LBRR or regular) for

this channel in the same Opus frame was not coded.

 In an independently coded subframe gain, the 3 most significant bits
 of the quantization gain are decoded using a PDF selected from
 Table 11 based on the decoded signal type (see Section 4.2.7.3).
         +-------------+------------------------------------+
         | Signal Type | PDF                                |
         +-------------+------------------------------------+
         | Inactive    | {32, 112, 68, 29, 12, 1, 1, 1}/256 |
         |             |                                    |
         | Unvoiced    | {2, 17, 45, 60, 62, 47, 19, 4}/256 |
         |             |                                    |
         | Voiced      | {1, 3, 26, 71, 94, 50, 9, 2}/256   |
         +-------------+------------------------------------+
      Table 11: PDFs for Independent Quantization Gain MSB Coding
 The 3 least significant bits are decoded using a uniform PDF:
               +--------------------------------------+
               | PDF                                  |
               +--------------------------------------+
               | {32, 32, 32, 32, 32, 32, 32, 32}/256 |
               +--------------------------------------+
      Table 12: PDF for Independent Quantization Gain LSB Coding
 These 6 bits are combined to form a value, gain_index, between 0 and
 63.  When the gain for the previous subframe is available, then the
 current gain is limited as follows:
           log_gain = max(gain_index, previous_log_gain - 16)
 This may help some implementations limit the change in precision of
 their internal LTP history.  The indices to which this clamp applies
 cannot simply be removed from the codebook, because previous_log_gain
 will not be available after packet loss.  The clamping is skipped
 after a decoder reset, and in the side channel if the previous frame

Valin, et al. Standards Track [Page 45] RFC 6716 Interactive Audio Codec September 2012

 in the side channel was not coded, since there is no value for
 previous_log_gain available.  It MAY also be skipped after packet
 loss.
 For subframes that do not have an independent gain (including the
 first subframe of frames not listed as using independent coding
 above), the quantization gain is coded relative to the gain from the
 previous subframe (in the same channel).  The PDF in Table 13 yields
 a delta_gain_index value between 0 and 40, inclusive.
 +-------------------------------------------------------------------+
 | PDF                                                               |
 +-------------------------------------------------------------------+
 | {6, 5, 11, 31, 132, 21, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, |
 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,       |
 | 1}/256                                                            |
 +-------------------------------------------------------------------+
           Table 13: PDF for Delta Quantization Gain Coding
 The following formula translates this index into a quantization gain
 for the current subframe using the gain from the previous subframe:
   log_gain = clamp(0, max(2*delta_gain_index - 16,
                      previous_log_gain + delta_gain_index - 4), 63)
 silk_gains_dequant() (gain_quant.c) dequantizes log_gain for the k'th
 subframe and converts it into a linear Q16 scale factor via
       gain_Q16[k] = silk_log2lin((0x1D1C71*log_gain>>16) + 2090)
 The function silk_log2lin() (log2lin.c) computes an approximation of
 2**(inLog_Q7/128.0), where inLog_Q7 is its Q7 input.  Let i =
 inLog_Q7>>7 be the integer part of inLogQ7 and f = inLog_Q7&127 be
 the fractional part.  Then,
             (1<<i) + ((-174*f*(128-f)>>16)+f)*((1<<i)>>7)
 yields the approximate exponential.  The final Q16 gain values lies
 between 81920 and 1686110208, inclusive (representing scale factors
 of 1.25 to 25728, respectively).

4.2.7.5. Normalized Line Spectral Frequency (LSF) and Linear Predictive

        Coding (LPC) Coefficients
 A set of normalized Line Spectral Frequency (LSF) coefficients follow
 the quantization gains in the bitstream and represent the Linear
 Predictive Coding (LPC) coefficients for the current SILK frame.

Valin, et al. Standards Track [Page 46] RFC 6716 Interactive Audio Codec September 2012

 Once decoded, the normalized LSFs form an increasing list of Q15
 values between 0 and 1.  These represent the interleaved zeros on the
 upper half of the unit circle (between 0 and pi, hence "normalized")
 in the standard decomposition [SPECTRAL-PAIRS] of the LPC filter into
 a symmetric part and an anti-symmetric part (P and Q in
 Section 4.2.7.5.6).  Because of non-linear effects in the decoding
 process, an implementation SHOULD match the fixed-point arithmetic
 described in this section exactly.  An encoder SHOULD also use the
 same process.
 The normalized LSFs are coded using a two-stage vector quantizer (VQ)
 (Sections 4.2.7.5.1 and 4.2.7.5.2).  NB and MB frames use an order-10
 predictor, while WB frames use an order-16 predictor.  Thus, each of
 these two cases uses a different set of tables.  After reconstructing
 the normalized LSFs (Section 4.2.7.5.3), the decoder runs them
 through a stabilization process (Section 4.2.7.5.4), interpolates
 them between frames (Section 4.2.7.5.5), converts them back into LPC
 coefficients (Section 4.2.7.5.6), and then runs them through further
 processes to limit the range of the coefficients (Section 4.2.7.5.7)
 and the gain of the filter (Section 4.2.7.5.8).  All of this is
 necessary to ensure the reconstruction process is stable.

4.2.7.5.1. Normalized LSF Stage 1 Decoding

 The first VQ stage uses a 32-element codebook, coded with one of the
 PDFs in Table 14, depending on the audio bandwidth and the signal
 type of the current SILK frame.  This yields a single index, I1, for
 the entire frame, which
 1.  Indexes an element in a coarse codebook,
 2.  Selects the PDFs for the second stage of the VQ, and
 3.  Selects the prediction weights used to remove intra-frame
     redundancy from the second stage.
 The actual codebook elements are listed in Tables 23 and 24, but they
 are not needed until the last stages of reconstructing the LSF
 coefficients.

Valin, et al. Standards Track [Page 47] RFC 6716 Interactive Audio Codec September 2012

 +-----------+----------+--------------------------------------------+
 | Audio     | Signal   | PDF                                        |
 | Bandwidth | Type     |                                            |
 +-----------+----------+--------------------------------------------+
 | NB or MB  | Inactive | {44, 34, 30, 19, 21, 12, 11, 3, 3, 2, 16,  |
 |           | or       | 2, 2, 1, 5, 2, 1, 3, 3, 1, 1, 2, 2, 2, 3,  |
 |           | unvoiced | 1, 9, 9, 2, 7, 2, 1}/256                   |
 |           |          |                                            |
 | NB or MB  | Voiced   | {1, 10, 1, 8, 3, 8, 8, 14, 13, 14, 1, 14,  |
 |           |          | 12, 13, 11, 11, 12, 11, 10, 10, 11, 8, 9,  |
 |           |          | 8, 7, 8, 1, 1, 6, 1, 6, 5}/256             |
 |           |          |                                            |
 | WB        | Inactive | {31, 21, 3, 17, 1, 8, 17, 4, 1, 18, 16, 4, |
 |           | or       | 2, 3, 1, 10, 1, 3, 16, 11, 16, 2, 2, 3, 2, |
 |           | unvoiced | 11, 1, 4, 9, 8, 7, 3}/256                  |
 |           |          |                                            |
 | WB        | Voiced   | {1, 4, 16, 5, 18, 11, 5, 14, 15, 1, 3, 12, |
 |           |          | 13, 14, 14, 6, 14, 12, 2, 6, 1, 12, 12,    |
 |           |          | 11, 10, 3, 10, 5, 1, 1, 1, 3}/256          |
 +-----------+----------+--------------------------------------------+
       Table 14: PDFs for Normalized LSF Stage-1 Index Decoding

4.2.7.5.2. Normalized LSF Stage 2 Decoding

 A total of 16 PDFs are available for the LSF residual in the second
 stage: the 8 (a...h) for NB and MB frames given in Table 15, and the
 8 (i...p) for WB frames given in Table 16.  Which PDF is used for
 which coefficient is driven by the index, I1, decoded in the first
 stage.  Table 17 lists the letter of the corresponding PDF for each
 normalized LSF coefficient for NB and MB, and Table 18 lists the same
 information for WB.

Valin, et al. Standards Track [Page 48] RFC 6716 Interactive Audio Codec September 2012

          +----------+--------------------------------------+
          | Codebook | PDF                                  |
          +----------+--------------------------------------+
          | a        | {1, 1, 1, 15, 224, 11, 1, 1, 1}/256  |
          |          |                                      |
          | b        | {1, 1, 2, 34, 183, 32, 1, 1, 1}/256  |
          |          |                                      |
          | c        | {1, 1, 4, 42, 149, 55, 2, 1, 1}/256  |
          |          |                                      |
          | d        | {1, 1, 8, 52, 123, 61, 8, 1, 1}/256  |
          |          |                                      |
          | e        | {1, 3, 16, 53, 101, 74, 6, 1, 1}/256 |
          |          |                                      |
          | f        | {1, 3, 17, 55, 90, 73, 15, 1, 1}/256 |
          |          |                                      |
          | g        | {1, 7, 24, 53, 74, 67, 26, 3, 1}/256 |
          |          |                                      |
          | h        | {1, 1, 18, 63, 78, 58, 30, 6, 1}/256 |
          +----------+--------------------------------------+
    Table 15: PDFs for NB/MB Normalized LSF Stage-2 Index Decoding
         +----------+---------------------------------------+
         | Codebook | PDF                                   |
         +----------+---------------------------------------+
         | i        | {1, 1, 1, 9, 232, 9, 1, 1, 1}/256     |
         |          |                                       |
         | j        | {1, 1, 2, 28, 186, 35, 1, 1, 1}/256   |
         |          |                                       |
         | k        | {1, 1, 3, 42, 152, 53, 2, 1, 1}/256   |
         |          |                                       |
         | l        | {1, 1, 10, 49, 126, 65, 2, 1, 1}/256  |
         |          |                                       |
         | m        | {1, 4, 19, 48, 100, 77, 5, 1, 1}/256  |
         |          |                                       |
         | n        | {1, 1, 14, 54, 100, 72, 12, 1, 1}/256 |
         |          |                                       |
         | o        | {1, 1, 15, 61, 87, 61, 25, 4, 1}/256  |
         |          |                                       |
         | p        | {1, 7, 21, 50, 77, 81, 17, 1, 1}/256  |
         +----------+---------------------------------------+
      Table 16: PDFs for WB Normalized LSF Stage-2 Index Decoding

Valin, et al. Standards Track [Page 49] RFC 6716 Interactive Audio Codec September 2012

                     +----+---------------------+
                     | I1 | Coefficient         |
                     +----+---------------------+
                     |    | 0 1 2 3 4 5 6 7 8 9 |
                     | 0  | a a a a a a a a a a |
                     |    |                     |
                     | 1  | b d b c c b c b b b |
                     |    |                     |
                     | 2  | c b b b b b b b b b |
                     |    |                     |
                     | 3  | b c c c c b c b b b |
                     |    |                     |
                     | 4  | c d d d d c c c c c |
                     |    |                     |
                     | 5  | a f d d c c c c b b |
                     |    |                     |
                     | g  | a c c c c c c c c b |
                     |    |                     |
                     | 7  | c d g e e e f e f f |
                     |    |                     |
                     | 8  | c e f f e f e g e e |
                     |    |                     |
                     | 9  | c e e h e f e f f e |
                     |    |                     |
                     | 10 | e d d d c d c c c c |
                     |    |                     |
                     | 11 | b f f g e f e f f f |
                     |    |                     |
                     | 12 | c h e g f f f f f f |
                     |    |                     |
                     | 13 | c h f f f f f g f e |
                     |    |                     |
                     | 14 | d d f e e f e f e e |
                     |    |                     |
                     | 15 | c d d f f e e e e e |
                     |    |                     |
                     | 16 | c e e g e f e f f f |
                     |    |                     |
                     | 17 | c f e g f f f e f e |
                     |    |                     |
                     | 18 | c h e f e f e f f f |
                     |    |                     |
                     | 19 | c f e g h g f g f e |
                     |    |                     |
                     | 20 | d g h e g f f g e f |
                     |    |                     |
                     | 21 | c h g e e e f e f f |
                     |    |                     |

Valin, et al. Standards Track [Page 50] RFC 6716 Interactive Audio Codec September 2012

                     | 22 | e f f e g g f g f e |
                     |    |                     |
                     | 23 | c f f g f g e g e e |
                     |    |                     |
                     | 24 | e f f f d h e f f e |
                     |    |                     |
                     | 25 | c d e f f g e f f e |
                     |    |                     |
                     | 26 | c d c d d e c d d d |
                     |    |                     |
                     | 27 | b b c c c c c d c c |
                     |    |                     |
                     | 28 | e f f g g g f g e f |
                     |    |                     |
                     | 29 | d f f e e e e d d c |
                     |    |                     |
                     | 30 | c f d h f f e e f e |
                     |    |                     |
                     | 31 | e e f e f g f g f e |
                     +----+---------------------+
  Table 17: Codebook Selection for NB/MB Normalized LSF Stage-2 Index
                               Decoding
        +----+------------------------------------------------+
        | I1 | Coefficient                                    |
        +----+------------------------------------------------+
        |    | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 |
        |    |                                                |
        | 0  | i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i |
        |    |                                                |
        | 1  | k  l  l  l  l  l  k  k  k  k  k  j  j  j  i  l |
        |    |                                                |
        | 2  | k  n  n  l  p  m  m  n  k  n  m  n  n  m  l  l |
        |    |                                                |
        | 3  | i  k  j  k  k  j  j  j  j  j  i  i  i  i  i  j |
        |    |                                                |
        | 4  | i  o  n  m  o  m  p  n  m  m  m  n  n  m  m  l |
        |    |                                                |
        | 5  | i  l  n  n  m  l  l  n  l  l  l  l  l  l  k  m |
        |    |                                                |
        | 6  | i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i |
        |    |                                                |
        | 7  | i  k  o  l  p  k  n  l  m  n  n  m  l  l  k  l |
        |    |                                                |
        | 8  | i  o  k  o  o  m  n  m  o  n  m  m  n  l  l  l |
        |    |                                                |
        | 9  | k  j  i  i  i  i  i  i  i  i  i  i  i  i  i  i |

Valin, et al. Standards Track [Page 51] RFC 6716 Interactive Audio Codec September 2012

        |    |                                                |
        | 10 | i  j  i  i  i  i  i  i  i  i  i  i  i  i  i  j |
        |    |                                                |
        | 11 | k  k  l  m  n  l  l  l  l  l  l  l  k  k  j  l |
        |    |                                                |
        | 12 | k  k  l  l  m  l  l  l  l  l  l  l  l  k  j  l |
        |    |                                                |
        | 13 | l  m  m  m  o  m  m  n  l  n  m  m  n  m  l  m |
        |    |                                                |
        | 14 | i  o  m  n  m  p  n  k  o  n  p  m  m  l  n  l |
        |    |                                                |
        | 15 | i  j  i  j  j  j  j  j  j  j  i  i  i  i  j  i |
        |    |                                                |
        | 16 | j  o  n  p  n  m  n  l  m  n  m  m  m  l  l  m |
        |    |                                                |
        | 17 | j  l  l  m  m  l  l  n  k  l  l  n  n  n  l  m |
        |    |                                                |
        | 18 | k  l  l  k  k  k  l  k  j  k  j  k  j  j  j  m |
        |    |                                                |
        | 19 | i  k  l  n  l  l  k  k  k  j  j  i  i  i  i  i |
        |    |                                                |
        | 20 | l  m  l  n  l  l  k  k  j  j  j  j  j  k  k  m |
        |    |                                                |
        | 21 | k  o  l  p  p  m  n  m  n  l  n  l  l  k  l  l |
        |    |                                                |
        | 22 | k  l  n  o  o  l  n  l  m  m  l  l  l  l  k  m |
        |    |                                                |
        | 23 | j  l  l  m  m  m  m  l  n  n  n  l  j  j  j  j |
        |    |                                                |
        | 24 | k  n  l  o  o  m  p  m  m  n  l  m  m  l  l  l |
        |    |                                                |
        | 25 | i  o  j  j  i  i  i  i  i  i  i  i  i  i  i  i |
        |    |                                                |
        | 26 | i  o  o  l  n  k  n  n  l  m  m  p  p  m  m  m |
        |    |                                                |
        | 27 | l  l  p  l  n  m  l  l  l  k  k  l  l  l  k  l |
        |    |                                                |
        | 28 | i  i  j  i  i  i  k  j  k  j  j  k  k  k  j  j |
        |    |                                                |
        | 29 | i  l  k  n  l  l  k  l  k  j  i  i  j  i  i  j |
        |    |                                                |
        | 30 | l  n  n  m  p  n  l  l  k  l  k  k  j  i  j  i |
        |    |                                                |
        | 31 | k  l  n  l  m  l  l  l  k  j  k  o  m  i  i  i |
        +----+------------------------------------------------+
   Table 18: Codebook Selection for WB Normalized LSF Stage-2 Index
                               Decoding

Valin, et al. Standards Track [Page 52] RFC 6716 Interactive Audio Codec September 2012

 Decoding the second stage residual proceeds as follows.  For each
 coefficient, the decoder reads a symbol using the PDF corresponding
 to I1 from either Table 17 or Table 18, and subtracts 4 from the
 result to give an index in the range -4 to 4, inclusive.  If the
 index is either -4 or 4, it reads a second symbol using the PDF in
 Table 19, and adds the value of this second symbol to the index,
 using the same sign.  This gives the index, I2[k], a total range of
 -10 to 10, inclusive.
                   +-------------------------------+
                   | PDF                           |
                   +-------------------------------+
                   | {156, 60, 24, 9, 4, 2, 1}/256 |
                   +-------------------------------+
       Table 19: PDF for Normalized LSF Index Extension Decoding
 The decoded indices from both stages are translated back into
 normalized LSF coefficients in silk_NLSF_decode() (NLSF_decode.c).
 The stage-2 indices represent residuals after both the first stage of
 the VQ and a separate backwards-prediction step.  The backwards
 prediction process in the encoder subtracts a prediction from each
 residual formed by a multiple of the coefficient that follows it.
 The decoder must undo this process.  Table 20 contains lists of
 prediction weights for each coefficient.  There are two lists for NB
 and MB, and another two lists for WB, giving two possible prediction
 weights for each coefficient.

Valin, et al. Standards Track [Page 53] RFC 6716 Interactive Audio Codec September 2012

                +-------------+-----+-----+-----+-----+
                | Coefficient |   A |   B |   C |   D |
                +-------------+-----+-----+-----+-----+
                | 0           | 179 | 116 | 175 |  68 |
                |             |     |     |     |     |
                | 1           | 138 |  67 | 148 |  62 |
                |             |     |     |     |     |
                | 2           | 140 |  82 | 160 |  66 |
                |             |     |     |     |     |
                | 3           | 148 |  59 | 176 |  60 |
                |             |     |     |     |     |
                | 4           | 151 |  92 | 178 |  72 |
                |             |     |     |     |     |
                | 5           | 149 |  72 | 173 | 117 |
                |             |     |     |     |     |
                | 6           | 153 | 100 | 174 |  85 |
                |             |     |     |     |     |
                | 7           | 151 |  89 | 164 |  90 |
                |             |     |     |     |     |
                | 8           | 163 |  92 | 177 | 118 |
                |             |     |     |     |     |
                | 9           |     |     | 174 | 136 |
                |             |     |     |     |     |
                | 10          |     |     | 196 | 151 |
                |             |     |     |     |     |
                | 11          |     |     | 182 | 142 |
                |             |     |     |     |     |
                | 12          |     |     | 198 | 160 |
                |             |     |     |     |     |
                | 13          |     |     | 192 | 142 |
                |             |     |     |     |     |
                | 14          |     |     | 182 | 155 |
                +-------------+-----+-----+-----+-----+
       Table 20: Prediction Weights for Normalized LSF Decoding
 The prediction is undone using the procedure implemented in
 silk_NLSF_residual_dequant() (NLSF_decode.c), which is as follows.
 Each coefficient selects its prediction weight from one of the two
 lists based on the stage-1 index, I1.  Table 21 gives the selections
 for each coefficient for NB and MB, and Table 22 gives the selections
 for WB.  Let d_LPC be the order of the codebook, i.e., 10 for NB and
 MB, and 16 for WB, and let pred_Q8[k] be the weight for the k'th
 coefficient selected by this process for 0 <= k < d_LPC-1.  Then, the
 stage-2 residual for each coefficient is computed via
     res_Q10[k] = (k+1 < d_LPC ? (res_Q10[k+1]*pred_Q8[k])>>8 : 0)
                  + ((((I2[k]<<10) - sign(I2[k])*102)*qstep)>>16) ,

Valin, et al. Standards Track [Page 54] RFC 6716 Interactive Audio Codec September 2012

 where qstep is the Q16 quantization step size, which is 11796 for NB
 and MB and 9830 for WB (representing step sizes of approximately 0.18
 and 0.15, respectively).
                      +----+-------------------+
                      | I1 | Coefficient       |
                      +----+-------------------+
                      |    | 0 1 2 3 4 5 6 7 8 |
                      |    |                   |
                      | 0  | A B A A A A A A A |
                      |    |                   |
                      | 1  | B A A A A A A A A |
                      |    |                   |
                      | 2  | A A A A A A A A A |
                      |    |                   |
                      | 3  | B B B A A A A B A |
                      |    |                   |
                      | 4  | A B A A A A A A A |
                      |    |                   |
                      | 5  | A B A A A A A A A |
                      |    |                   |
                      | 6  | B A B B A A A B A |
                      |    |                   |
                      | 7  | A B B A A B B A A |
                      |    |                   |
                      | 8  | A A B B A B A B B |
                      |    |                   |
                      | 9  | A A B B A A B B B |
                      |    |                   |
                      | 10 | A A A A A A A A A |
                      |    |                   |
                      | 11 | A B A B B B B B A |
                      |    |                   |
                      | 12 | A B A B B B B B A |
                      |    |                   |
                      | 13 | A B B B B B B B A |
                      |    |                   |
                      | 14 | B A B B A B B B B |
                      |    |                   |
                      | 15 | A B B B B B A B A |
                      |    |                   |
                      | 16 | A A B B A B A B A |
                      |    |                   |
                      | 17 | A A B B B A B B B |
                      |    |                   |
                      | 18 | A B B A A B B B A |
                      |    |                   |
                      | 19 | A A A B B B A B A |

Valin, et al. Standards Track [Page 55] RFC 6716 Interactive Audio Codec September 2012

                      |    |                   |
                      | 20 | A B B A A B A B A |
                      |    |                   |
                      | 21 | A B B A A A B B A |
                      |    |                   |
                      | 22 | A A A A A B B B B |
                      |    |                   |
                      | 23 | A A B B A A A B B |
                      |    |                   |
                      | 24 | A A A B A B B B B |
                      |    |                   |
                      | 25 | A B B B B B B B A |
                      |    |                   |
                      | 26 | A A A A A A A A A |
                      |    |                   |
                      | 27 | A A A A A A A A A |
                      |    |                   |
                      | 28 | A A B A B B A B A |
                      |    |                   |
                      | 29 | B A A B A A A A A |
                      |    |                   |
                      | 30 | A A A B B A B A B |
                      |    |                   |
                      | 31 | B A B B A B B B B |
                      +----+-------------------+
    Table 21: Prediction Weight Selection for NB/MB Normalized LSF
                               Decoding
         +----+---------------------------------------------+
         | I1 | Coefficient                                 |
         +----+---------------------------------------------+
         |    | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 |
         |    |                                             |
         | 0  | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
         |    |                                             |
         | 1  | C  C  C  C  C  C  C  C  C  C  C  C  C  C  C |
         |    |                                             |
         | 2  | C  C  D  C  C  D  D  D  C  D  D  D  D  C  C |
         |    |                                             |
         | 3  | C  C  C  C  C  C  C  C  C  C  C  C  D  C  C |
         |    |                                             |
         | 4  | C  D  D  C  D  C  D  D  C  D  D  D  D  D  C |
         |    |                                             |
         | 5  | C  C  D  C  C  C  C  C  C  C  C  C  C  C  C |

Valin, et al. Standards Track [Page 56] RFC 6716 Interactive Audio Codec September 2012

         |    |                                             |
         | 6  | D  C  C  C  C  C  C  C  C  C  C  D  C  D  C |
         |    |                                             |
         | 7  | C  D  D  C  C  C  D  C  D  D  D  C  D  C  D |
         |    |                                             |
         | 8  | C  D  C  D  D  C  D  C  D  C  D  D  D  D  D |
         |    |                                             |
         | 9  | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
         |    |                                             |
         | 10 | C  D  C  C  C  C  C  C  C  C  C  C  C  C  C |
         |    |                                             |
         | 11 | C  C  D  C  D  D  D  D  D  D  D  C  D  C  C |
         |    |                                             |
         | 12 | C  C  D  C  C  D  C  D  C  D  C  C  D  C  C |
         |    |                                             |
         | 13 | C  C  C  C  D  D  C  D  C  D  D  D  D  C  C |
         |    |                                             |
         | 14 | C  D  C  C  C  D  D  C  D  D  D  C  D  D  D |
         |    |                                             |
         | 15 | C  C  D  D  C  C  C  C  C  C  C  C  D  D  C |
         |    |                                             |
         | 16 | C  D  D  C  D  C  D  D  D  D  D  C  D  C  C |
         |    |                                             |
         | 17 | C  C  D  C  C  C  C  D  C  C  D  D  D  C  C |
         |    |                                             |
         | 18 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
         |    |                                             |
         | 19 | C  C  C  C  C  C  C  C  C  C  C  C  D  C  C |
         |    |                                             |
         | 20 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  C |
         |    |                                             |
         | 21 | C  D  C  D  C  D  D  C  D  C  D  C  D  D  C |
         |    |                                             |
         | 22 | C  C  D  D  D  D  C  D  D  C  C  D  D  C  C |
         |    |                                             |
         | 23 | C  D  D  C  D  C  D  C  D  C  C  C  C  D  C |
         |    |                                             |
         | 24 | C  C  C  D  D  C  D  C  D  D  D  D  D  D  D |
         |    |                                             |
         | 25 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
         |    |                                             |
         | 26 | C  D  D  C  C  C  D  D  C  C  D  D  D  D  D |
         |    |                                             |
         | 27 | C  C  C  C  C  D  C  D  D  D  D  C  D  D  D |
         |    |                                             |
         | 28 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
         |    |                                             |
         | 29 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |

Valin, et al. Standards Track [Page 57] RFC 6716 Interactive Audio Codec September 2012

         |    |                                             |
         | 30 | D  C  C  C  C  C  C  C  C  C  C  D  C  C  C |
         |    |                                             |
         | 31 | C  C  D  C  C  D  D  D  C  C  D  C  C  D  C |
         +----+---------------------------------------------+
 Table 22: Prediction Weight Selection for WB Normalized LSF Decoding

4.2.7.5.3. Reconstructing the Normalized LSF Coefficients

 Once the stage-1 index I1 and the stage-2 residual res_Q10[] have
 been decoded, the final normalized LSF coefficients can be
 reconstructed.
 The spectral distortion introduced by the quantization of each LSF
 coefficient varies, so the stage-2 residual is weighted accordingly,
 using the low-complexity Inverse Harmonic Mean Weighting (IHMW)
 function proposed in [LAROIA-ICASSP].  The weights are derived
 directly from the stage-1 codebook vector.  Let cb1_Q8[k] be the k'th
 entry of the stage-1 codebook vector from Table 23 or Table 24.
 Then, for 0 <= k < d_LPC, the following expression computes the
 square of the weight as a Q18 value:
          w2_Q18[k] = (1024/(cb1_Q8[k] - cb1_Q8[k-1])
                       + 1024/(cb1_Q8[k+1] - cb1_Q8[k])) << 16
 where cb1_Q8[-1] = 0 and cb1_Q8[d_LPC] = 256, and the division is
 integer division.  This is reduced to an unsquared, Q9 value using
 the following square-root approximation:
               i = ilog(w2_Q18[k])
               f = (w2_Q18[k]>>(i-8)) & 127
               y = ((i&1) ? 32768 : 46214) >> ((32-i)>>1)
               w_Q9[k] = y + ((213*f*y)>>16)
 The constant 46214 here is approximately the square root of 2 in Q15.
 The cb1_Q8[] vector completely determines these weights, and they may
 be tabulated and stored as 13-bit unsigned values (with a range of
 1819 to 5227, inclusive) to avoid computing them when decoding.  The
 reference implementation already requires code to compute these
 weights on unquantized coefficients in the encoder, in
 silk_NLSF_VQ_weights_laroia() (NLSF_VQ_weights_laroia.c) and its
 callers, so it reuses that code in the decoder instead of using a
 pre-computed table to reduce the amount of ROM required.

Valin, et al. Standards Track [Page 58] RFC 6716 Interactive Audio Codec September 2012

            +----+----------------------------------------+
            | I1 | Codebook (Q8)                          |
            +----+----------------------------------------+
            |    |  0   1   2   3   4   5   6   7   8   9 |
            |    |                                        |
            | 0  | 12  35  60  83 108 132 157 180 206 228 |
            |    |                                        |
            | 1  | 15  32  55  77 101 125 151 175 201 225 |
            |    |                                        |
            | 2  | 19  42  66  89 114 137 162 184 209 230 |
            |    |                                        |
            | 3  | 12  25  50  72  97 120 147 172 200 223 |
            |    |                                        |
            | 4  | 26  44  69  90 114 135 159 180 205 225 |
            |    |                                        |
            | 5  | 13  22  53  80 106 130 156 180 205 228 |
            |    |                                        |
            | 6  | 15  25  44  64  90 115 142 168 196 222 |
            |    |                                        |
            | 7  | 19  24  62  82 100 120 145 168 190 214 |
            |    |                                        |
            | 8  | 22  31  50  79 103 120 151 170 203 227 |
            |    |                                        |
            | 9  | 21  29  45  65 106 124 150 171 196 224 |
            |    |                                        |
            | 10 | 30  49  75  97 121 142 165 186 209 229 |
            |    |                                        |
            | 11 | 19  25  52  70  93 116 143 166 192 219 |
            |    |                                        |
            | 12 | 26  34  62  75  97 118 145 167 194 217 |
            |    |                                        |
            | 13 | 25  33  56  70  91 113 143 165 196 223 |
            |    |                                        |
            | 14 | 21  34  51  72  97 117 145 171 196 222 |
            |    |                                        |
            | 15 | 20  29  50  67  90 117 144 168 197 221 |
            |    |                                        |
            | 16 | 22  31  48  66  95 117 146 168 196 222 |
            |    |                                        |
            | 17 | 24  33  51  77 116 134 158 180 200 224 |
            |    |                                        |
            | 18 | 21  28  70  87 106 124 149 170 194 217 |
            |    |                                        |
            | 19 | 26  33  53  64  83 117 152 173 204 225 |
            |    |                                        |
            | 20 | 27  34  65  95 108 129 155 174 210 225 |
            |    |                                        |
            | 21 | 20  26  72  99 113 131 154 176 200 219 |

Valin, et al. Standards Track [Page 59] RFC 6716 Interactive Audio Codec September 2012

            |    |                                        |
            | 22 | 34  43  61  78  93 114 155 177 205 229 |
            |    |                                        |
            | 23 | 23  29  54  97 124 138 163 179 209 229 |
            |    |                                        |
            | 24 | 30  38  56  89 118 129 158 178 200 231 |
            |    |                                        |
            | 25 | 21  29  49  63  85 111 142 163 193 222 |
            |    |                                        |
            | 26 | 27  48  77 103 133 158 179 196 215 232 |
            |    |                                        |
            | 27 | 29  47  74  99 124 151 176 198 220 237 |
            |    |                                        |
            | 28 | 33  42  61  76  93 121 155 174 207 225 |
            |    |                                        |
            | 29 | 29  53  87 112 136 154 170 188 208 227 |
            |    |                                        |
            | 30 | 24  30  52  84 131 150 166 186 203 229 |
            |    |                                        |
            | 31 | 37  48  64  84 104 118 156 177 201 230 |
            +----+----------------------------------------+
        Table 23: NB/MB Normalized LSF Stage-1 Codebook Vectors
  +----+------------------------------------------------------------+
  | I1 | Codebook (Q8)                                              |
  +----+------------------------------------------------------------+
  |    |  0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15 |
  |    |                                                            |
  | 0  |  7 23 38 54 69  85 100 116 131 147 162 178 193 208 223 239 |
  |    |                                                            |
  | 1  | 13 25 41 55 69  83  98 112 127 142 157 171 187 203 220 236 |
  |    |                                                            |
  | 2  | 15 21 34 51 61  78  92 106 126 136 152 167 185 205 225 240 |
  |    |                                                            |
  | 3  | 10 21 36 50 63  79  95 110 126 141 157 173 189 205 221 237 |
  |    |                                                            |
  | 4  | 17 20 37 51 59  78  89 107 123 134 150 164 184 205 224 240 |
  |    |                                                            |
  | 5  | 10 15 32 51 67  81  96 112 129 142 158 173 189 204 220 236 |
  |    |                                                            |
  | 6  |  8 21 37 51 65  79  98 113 126 138 155 168 179 192 209 218 |
  |    |                                                            |
  | 7  | 12 15 34 55 63  78  87 108 118 131 148 167 185 203 219 236 |
  |    |                                                            |
  | 8  | 16 19 32 36 56  79  91 108 118 136 154 171 186 204 220 237 |
  |    |                                                            |
  | 9  | 11 28 43 58 74  89 105 120 135 150 165 180 196 211 226 241 |

Valin, et al. Standards Track [Page 60] RFC 6716 Interactive Audio Codec September 2012

  |    |                                                            |
  | 10 |  6 16 33 46 60  75  92 107 123 137 156 169 185 199 214 225 |
  |    |                                                            |
  | 11 | 11 19 30 44 57  74  89 105 121 135 152 169 186 202 218 234 |
  |    |                                                            |
  | 12 | 12 19 29 46 57  71  88 100 120 132 148 165 182 199 216 233 |
  |    |                                                            |
  | 13 | 17 23 35 46 56  77  92 106 123 134 152 167 185 204 222 237 |
  |    |                                                            |
  | 14 | 14 17 45 53 63  75  89 107 115 132 151 171 188 206 221 240 |
  |    |                                                            |
  | 15 |  9 16 29 40 56  71  88 103 119 137 154 171 189 205 222 237 |
  |    |                                                            |
  | 16 | 16 19 36 48 57  76  87 105 118 132 150 167 185 202 218 236 |
  |    |                                                            |
  | 17 | 12 17 29 54 71  81  94 104 126 136 149 164 182 201 221 237 |
  |    |                                                            |
  | 18 | 15 28 47 62 79  97 115 129 142 155 168 180 194 208 223 238 |
  |    |                                                            |
  | 19 |  8 14 30 45 62  78  94 111 127 143 159 175 192 207 223 239 |
  |    |                                                            |
  | 20 | 17 30 49 62 79  92 107 119 132 145 160 174 190 204 220 235 |
  |    |                                                            |
  | 21 | 14 19 36 45 61  76  91 108 121 138 154 172 189 205 222 238 |
  |    |                                                            |
  | 22 | 12 18 31 45 60  76  91 107 123 138 154 171 187 204 221 236 |
  |    |                                                            |
  | 23 | 13 17 31 43 53  70  83 103 114 131 149 167 185 203 220 237 |
  |    |                                                            |
  | 24 | 17 22 35 42 58  78  93 110 125 139 155 170 188 206 224 240 |
  |    |                                                            |
  | 25 |  8 15 34 50 67  83  99 115 131 146 162 178 193 209 224 239 |
  |    |                                                            |
  | 26 | 13 16 41 66 73  86  95 111 128 137 150 163 183 206 225 241 |
  |    |                                                            |
  | 27 | 17 25 37 52 63  75  92 102 119 132 144 160 175 191 212 231 |
  |    |                                                            |
  | 28 | 19 31 49 65 83 100 117 133 147 161 174 187 200 213 227 242 |
  |    |                                                            |
  | 29 | 18 31 52 68 88 103 117 126 138 149 163 177 192 207 223 239 |
  |    |                                                            |
  | 30 | 16 29 47 61 76  90 106 119 133 147 161 176 193 209 224 240 |
  |    |                                                            |
  | 31 | 15 21 35 50 61  73  86  97 110 119 129 141 175 198 218 237 |
  +----+------------------------------------------------------------+
         Table 24: WB Normalized LSF Stage-1 Codebook Vectors

Valin, et al. Standards Track [Page 61] RFC 6716 Interactive Audio Codec September 2012

 Given the stage-1 codebook entry cb1_Q8[], the stage-2 residual
 res_Q10[], and their corresponding weights, w_Q9[], the reconstructed
 normalized LSF coefficients are
    NLSF_Q15[k] = clamp(0,
                   (cb1_Q8[k]<<7) + (res_Q10[k]<<14)/w_Q9[k], 32767)
 where the division is integer division.  However, nothing in either
 the reconstruction process or the quantization process in the encoder
 thus far guarantees that the coefficients are monotonically
 increasing and separated well enough to ensure a stable filter
 [KABAL86].  When using the reference encoder, roughly 2% of frames
 violate this constraint.  The next section describes a stabilization
 procedure used to make these guarantees.

4.2.7.5.4. Normalized LSF Stabilization

 The normalized LSF stabilization procedure is implemented in
 silk_NLSF_stabilize() (NLSF_stabilize.c).  This process ensures that
 consecutive values of the normalized LSF coefficients, NLSF_Q15[],
 are spaced some minimum distance apart (predetermined to be the 0.01
 percentile of a large training set).  Table 25 gives the minimum
 spacings for NB and MB and those for WB, where row k is the minimum
 allowed value of NLSF_Q15[k]-NLSF_Q15[k-1].  For the purposes of
 computing this spacing for the first and last coefficient,
 NLSF_Q15[-1] is taken to be 0 and NLSF_Q15[d_LPC] is taken to be
 32768.

Valin, et al. Standards Track [Page 62] RFC 6716 Interactive Audio Codec September 2012

                   +-------------+-----------+-----+
                   | Coefficient | NB and MB |  WB |
                   +-------------+-----------+-----+
                   | 0           |       250 | 100 |
                   |             |           |     |
                   | 1           |         3 |   3 |
                   |             |           |     |
                   | 2           |         6 |  40 |
                   |             |           |     |
                   | 3           |         3 |   3 |
                   |             |           |     |
                   | 4           |         3 |   3 |
                   |             |           |     |
                   | 5           |         3 |   3 |
                   |             |           |     |
                   | 6           |         4 |   5 |
                   |             |           |     |
                   | 7           |         3 |  14 |
                   |             |           |     |
                   | 8           |         3 |  14 |
                   |             |           |     |
                   | 9           |         3 |  10 |
                   |             |           |     |
                   | 10          |       461 |  11 |
                   |             |           |     |
                   | 11          |           |   3 |
                   |             |           |     |
                   | 12          |           |   8 |
                   |             |           |     |
                   | 13          |           |   9 |
                   |             |           |     |
                   | 14          |           |   7 |
                   |             |           |     |
                   | 15          |           |   3 |
                   |             |           |     |
                   | 16          |           | 347 |
                   +-------------+-----------+-----+
       Table 25: Minimum Spacing for Normalized LSF Coefficients
 The procedure starts off by trying to make small adjustments that
 attempt to minimize the amount of distortion introduced.  After 20
 such adjustments, it falls back to a more direct method that
 guarantees the constraints are enforced but may require large
 adjustments.

Valin, et al. Standards Track [Page 63] RFC 6716 Interactive Audio Codec September 2012

 Let NDeltaMin_Q15[k] be the minimum required spacing for the current
 audio bandwidth from Table 25.  First, the procedure finds the index
 i where NLSF_Q15[i] - NLSF_Q15[i-1] - NDeltaMin_Q15[i] is the
 smallest, breaking ties by using the lower value of i.  If this value
 is non-negative, then the stabilization stops; the coefficients
 satisfy all the constraints.  Otherwise, if i == 0, it sets
 NLSF_Q15[0] to NDeltaMin_Q15[0], and if i == d_LPC, it sets
 NLSF_Q15[d_LPC-1] to (32768 - NDeltaMin_Q15[d_LPC]).  For all other
 values of i, both NLSF_Q15[i-1] and NLSF_Q15[i] are updated as
 follows:
                                           i-1
                                           __
  min_center_Q15 = (NDeltaMin_Q15[i]>>1) + \  NDeltaMin_Q15[k]
                                           /_
                                           k=0
                                                  d_LPC
                                                   __
  max_center_Q15 = 32768 - (NDeltaMin_Q15[i]>>1) - \  NDeltaMin_Q15[k]
                                                   /_
                                                  k=i+1
 center_freq_Q15 = clamp(min_center_Q15[i],
                         (NLSF_Q15[i-1] + NLSF_Q15[i] + 1)>>1
                         max_center_Q15[i])
  NLSF_Q15[i-1] = center_freq_Q15 - (NDeltaMin_Q15[i]>>1)
    NLSF_Q15[i] = NLSF_Q15[i-1] + NDeltaMin_Q15[i]
 Then, the procedure repeats again, until it has either executed 20
 times or stopped because the coefficients satisfy all the
 constraints.
 After the 20th repetition of the above procedure, the following
 fallback procedure executes once.  First, the values of NLSF_Q15[k]
 for 0 <= k < d_LPC are sorted in ascending order.  Then, for each
 value of k from 0 to d_LPC-1, NLSF_Q15[k] is set to
           max(NLSF_Q15[k], NLSF_Q15[k-1] + NDeltaMin_Q15[k])
 Next, for each value of k from d_LPC-1 down to 0, NLSF_Q15[k] is set
 to
          min(NLSF_Q15[k], NLSF_Q15[k+1] - NDeltaMin_Q15[k+1])
 There is no need to check if the coefficients satisfy all the
 constraints before applying this fallback procedure.  If they do,
 then it will not change their values.

Valin, et al. Standards Track [Page 64] RFC 6716 Interactive Audio Codec September 2012

4.2.7.5.5. Normalized LSF Interpolation

 For 20 ms SILK frames, the first half of the frame (i.e., the first
 two subframes) may use normalized LSF coefficients that are
 interpolated between the decoded LSFs for the most recent coded frame
 (in the same channel) and the current frame.  A Q2 interpolation
 factor follows the LSF coefficient indices in the bitstream, which is
 decoded using the PDF in Table 26.  This happens in
 silk_decode_indices() (decode_indices.c).  After either
 o  An uncoded regular SILK frame in the side channel, or
 o  A decoder reset (see Section 4.5.2),
 the decoder still decodes this factor, but ignores its value and
 always uses 4 instead.  For 10 ms SILK frames, this factor is not
 stored at all.
                     +---------------------------+
                     | PDF                       |
                     +---------------------------+
                     | {13, 22, 29, 11, 181}/256 |
                     +---------------------------+
         Table 26: PDF for Normalized LSF Interpolation Index
 Let n2_Q15[k] be the normalized LSF coefficients decoded by the
 procedure in Section 4.2.7.5, n0_Q15[k] be the LSF coefficients
 decoded for the prior frame, and w_Q2 be the interpolation factor.
 Then, the normalized LSF coefficients used for the first half of a
 20 ms frame, n1_Q15[k], are
      n1_Q15[k] = n0_Q15[k] + (w_Q2*(n2_Q15[k] - n0_Q15[k]) >> 2)
 This interpolation is performed in silk_decode_parameters()
 (decode_parameters.c).

4.2.7.5.6. Converting Normalized LSFs to LPC Coefficients

 Any LPC filter A(z) can be split into a symmetric part P(z) and an
 anti-symmetric part Q(z) such that
                        d_LPC
                         __         -k   1
              A(z) = 1 - \  a[k] * z   = - * (P(z) + Q(z))
                         /_              2
                         k=1

Valin, et al. Standards Track [Page 65] RFC 6716 Interactive Audio Codec September 2012

 with
  1. d_LPC-1 -1

P(z) = A(z) + z * A(z )

  1. d_LPC-1 -1

Q(z) = A(z) - z * A(z )

 The even normalized LSF coefficients correspond to a pair of
 conjugate roots of P(z), while the odd coefficients correspond to a
 pair of conjugate roots of Q(z), all of which lie on the unit circle.
 In addition, P(z) has a root at pi and Q(z) has a root at 0.  Thus,
 they may be reconstructed mathematically from a set of normalized LSF
 coefficients, n[k], as
                        d_LPC/2-1
                    -1     ___                        -1    -2
       P(z) = (1 + z  ) *  | |  (1 - 2*cos(pi*n[2*k])*z  + z  )
                           k=0
                        d_LPC/2-1
                    -1     ___                          -1    -2
       Q(z) = (1 - z  ) *  | |  (1 - 2*cos(pi*n[2*k+1])*z  + z  )
                           k=0
 However, SILK performs this reconstruction using a fixed-point
 approximation so that all decoders can reproduce it in a bit-exact
 manner to avoid prediction drift.  The function silk_NLSF2A()
 (NLSF2A.c) implements this procedure.
 To start, it approximates cos(pi*n[k]) using a table lookup with
 linear interpolation.  The encoder SHOULD use the inverse of this
 piecewise linear approximation, rather than the true inverse of the
 cosine function, when deriving the normalized LSF coefficients.
 These values are also re-ordered to improve numerical accuracy when
 constructing the LPC polynomials.

Valin, et al. Standards Track [Page 66] RFC 6716 Interactive Audio Codec September 2012

                   +-------------+-----------+----+
                   | Coefficient | NB and MB | WB |
                   +-------------+-----------+----+
                   | 0           |         0 |  0 |
                   |             |           |    |
                   | 1           |         9 | 15 |
                   |             |           |    |
                   | 2           |         6 |  8 |
                   |             |           |    |
                   | 3           |         3 |  7 |
                   |             |           |    |
                   | 4           |         4 |  4 |
                   |             |           |    |
                   | 5           |         5 | 11 |
                   |             |           |    |
                   | 6           |         8 | 12 |
                   |             |           |    |
                   | 7           |         1 |  3 |
                   |             |           |    |
                   | 8           |         2 |  2 |
                   |             |           |    |
                   | 9           |         7 | 13 |
                   |             |           |    |
                   | 10          |           | 10 |
                   |             |           |    |
                   | 11          |           |  5 |
                   |             |           |    |
                   | 12          |           |  6 |
                   |             |           |    |
                   | 13          |           |  9 |
                   |             |           |    |
                   | 14          |           | 14 |
                   |             |           |    |
                   | 15          |           |  1 |
                   +-------------+-----------+----+
           Table 27: LSF Ordering for Polynomial Evaluation
 The top 7 bits of each normalized LSF coefficient index a value in
 the table, and the next 8 bits interpolate between it and the next
 value.  Let i = (n[k] >> 8) be the integer index and f = (n[k] & 255)
 be the fractional part of a given coefficient.  Then, the re-ordered,
 approximated cosine, c_Q17[ordering[k]], is
     c_Q17[ordering[k]] = (cos_Q12[i]*256
                           + (cos_Q12[i+1]-cos_Q12[i])*f + 4) >> 3

Valin, et al. Standards Track [Page 67] RFC 6716 Interactive Audio Codec September 2012

 where ordering[k] is the k'th entry of the column of Table 27
 corresponding to the current audio bandwidth and cos_Q12[i] is the
 i'th entry of Table 28.
                +-----+-------+-------+-------+-------+
                |   i |    +0 |    +1 |    +2 |    +3 |
                +-----+-------+-------+-------+-------+
                |   0 |  4096 |  4095 |  4091 |  4085 |
                |     |       |       |       |       |
                |   4 |  4076 |  4065 |  4052 |  4036 |
                |     |       |       |       |       |
                |   8 |  4017 |  3997 |  3973 |  3948 |
                |     |       |       |       |       |
                |  12 |  3920 |  3889 |  3857 |  3822 |
                |     |       |       |       |       |
                |  16 |  3784 |  3745 |  3703 |  3659 |
                |     |       |       |       |       |
                |  20 |  3613 |  3564 |  3513 |  3461 |
                |     |       |       |       |       |
                |  24 |  3406 |  3349 |  3290 |  3229 |
                |     |       |       |       |       |
                |  28 |  3166 |  3102 |  3035 |  2967 |
                |     |       |       |       |       |
                |  32 |  2896 |  2824 |  2751 |  2676 |
                |     |       |       |       |       |
                |  36 |  2599 |  2520 |  2440 |  2359 |
                |     |       |       |       |       |
                |  40 |  2276 |  2191 |  2106 |  2019 |
                |     |       |       |       |       |
                |  44 |  1931 |  1842 |  1751 |  1660 |
                |     |       |       |       |       |
                |  48 |  1568 |  1474 |  1380 |  1285 |
                |     |       |       |       |       |
                |  52 |  1189 |  1093 |   995 |   897 |
                |     |       |       |       |       |
                |  56 |   799 |   700 |   601 |   501 |
                |     |       |       |       |       |
                |  60 |   401 |   301 |   201 |   101 |
                |     |       |       |       |       |
                |  64 |     0 |  -101 |  -201 |  -301 |
                |     |       |       |       |       |
                |  68 |  -401 |  -501 |  -601 |  -700 |
                |     |       |       |       |       |
                |  72 |  -799 |  -897 |  -995 | -1093 |
                |     |       |       |       |       |
                |  76 | -1189 | -1285 | -1380 | -1474 |
                |     |       |       |       |       |
                |  80 | -1568 | -1660 | -1751 | -1842 |

Valin, et al. Standards Track [Page 68] RFC 6716 Interactive Audio Codec September 2012

                |     |       |       |       |       |
                |  84 | -1931 | -2019 | -2106 | -2191 |
                |     |       |       |       |       |
                |  88 | -2276 | -2359 | -2440 | -2520 |
                |     |       |       |       |       |
                |  92 | -2599 | -2676 | -2751 | -2824 |
                |     |       |       |       |       |
                |  96 | -2896 | -2967 | -3035 | -3102 |
                |     |       |       |       |       |
                | 100 | -3166 | -3229 | -3290 | -3349 |
                |     |       |       |       |       |
                | 104 | -3406 | -3461 | -3513 | -3564 |
                |     |       |       |       |       |
                | 108 | -3613 | -3659 | -3703 | -3745 |
                |     |       |       |       |       |
                | 112 | -3784 | -3822 | -3857 | -3889 |
                |     |       |       |       |       |
                | 116 | -3920 | -3948 | -3973 | -3997 |
                |     |       |       |       |       |
                | 120 | -4017 | -4036 | -4052 | -4065 |
                |     |       |       |       |       |
                | 124 | -4076 | -4085 | -4091 | -4095 |
                |     |       |       |       |       |
                | 128 | -4096 |       |       |       |
                +-----+-------+-------+-------+-------+
             Table 28: Q12 Cosine Table for LSF Conversion
 Given the list of cosine values, silk_NLSF2A_find_poly() (NLSF2A.c)
 computes the coefficients of P and Q, described here via a simple
 recurrence.  Let p_Q16[k][j] and q_Q16[k][j] be the coefficients of
 the products of the first (k+1) root pairs for P and Q, with j
 indexing the coefficient number.  Only the first (k+2) coefficients
 are needed, as the products are symmetric.  Let
 p_Q16[0][0] = q_Q16[0][0] = 1<<16, p_Q16[0][1] = -c_Q17[0],
 q_Q16[0][1] = -c_Q17[1], and d2 = d_LPC/2.  As boundary conditions,
 assume p_Q16[k][j] = q_Q16[k][j] = 0 for all j < 0.  Also, assume
 p_Q16[k][k+2] = p_Q16[k][k] and q_Q16[k][k+2] = q_Q16[k][k] (because
 of the symmetry).  Then, for 0 < k < d2 and 0 <= j <= k+1,
      p_Q16[k][j] = p_Q16[k-1][j] + p_Q16[k-1][j-2]
                    - ((c_Q17[2*k]*p_Q16[k-1][j-1] + 32768)>>16)
      q_Q16[k][j] = q_Q16[k-1][j] + q_Q16[k-1][j-2]
                    - ((c_Q17[2*k+1]*q_Q16[k-1][j-1] + 32768)>>16)

Valin, et al. Standards Track [Page 69] RFC 6716 Interactive Audio Codec September 2012

 The use of Q17 values for the cosine terms in an otherwise Q16
 expression implicitly scales them by a factor of 2.  The
 multiplications in this recurrence may require up to 48 bits of
 precision in the result to avoid overflow.  In practice, each row of
 the recurrence only depends on the previous row, so an implementation
 does not need to store all of them.
 silk_NLSF2A() uses the values from the last row of this recurrence to
 reconstruct a 32-bit version of the LPC filter (without the leading
 1.0 coefficient), a32_Q17[k], 0 <= k < d2:
      a32_Q17[k]         = -(q_Q16[d2-1][k+1] - q_Q16[d2-1][k])
                           - (p_Q16[d2-1][k+1] + p_Q16[d2-1][k]))
      a32_Q17[d_LPC-k-1] =  (q_Q16[d2-1][k+1] - q_Q16[d2-1][k])
                           - (p_Q16[d2-1][k+1] + p_Q16[d2-1][k]))
 The sum and difference of two terms from each of the p_Q16 and q_Q16
 coefficient lists reflect the (1 + z**-1) and (1 - z**-1) factors of
 P and Q, respectively.  The promotion of the expression from Q16 to
 Q17 implicitly scales the result by 1/2.

4.2.7.5.7. Limiting the Range of the LPC Coefficients

 The a32_Q17[] coefficients are too large to fit in a 16-bit value,
 which significantly increases the cost of applying this filter in
 fixed-point decoders.  Reducing them to Q12 precision doesn't incur
 any significant quality loss, but still does not guarantee they will
 fit. silk_NLSF2A() applies up to 10 rounds of bandwidth expansion to
 limit the dynamic range of these coefficients.  Even floating-point
 decoders SHOULD perform these steps, to avoid mismatch.
 For each round, the process first finds the index k such that
 abs(a32_Q17[k]) is largest, breaking ties by choosing the lowest
 value of k.  Then, it computes the corresponding Q12 precision value,
 maxabs_Q12, subject to an upper bound to avoid overflow in subsequent
 computations:
            maxabs_Q12 = min((maxabs_Q17 + 16) >> 5, 163838)
 If this is larger than 32767, the procedure derives the chirp factor,
 sc_Q16[0], to use in the bandwidth expansion as
                                 (maxabs_Q12 - 32767) << 14
             sc_Q16[0] = 65470 - --------------------------
                                 (maxabs_Q12 * (k+1)) >> 2

Valin, et al. Standards Track [Page 70] RFC 6716 Interactive Audio Codec September 2012

 where the division here is integer division.  This is an
 approximation of the chirp factor needed to reduce the target
 coefficient to 32767, though it is both less than 0.999 and, for
 k > 0 when maxabs_Q12 is much greater than 32767, still slightly too
 large.  The upper bound on maxabs_Q12, 163838, was chosen because it
 is equal to ((2**31 - 1) >> 14) + 32767, i.e., the largest value of
 maxabs_Q12 that would not overflow the numerator in the equation
 above when stored in a signed 32-bit integer.
 silk_bwexpander_32() (bwexpander_32.c) performs the bandwidth
 expansion (again, only when maxabs_Q12 is greater than 32767) using
 the following recurrence:
            a32_Q17[k] = (a32_Q17[k]*sc_Q16[k]) >> 16
           sc_Q16[k+1] = (sc_Q16[0]*sc_Q16[k] + 32768) >> 16
 The first multiply may require up to 48 bits of precision in the
 result to avoid overflow.  The second multiply must be unsigned to
 avoid overflow with only 32 bits of precision.  The reference
 implementation uses a slightly more complex formulation that avoids
 the 32-bit overflow using signed multiplication, but is otherwise
 equivalent.
 After 10 rounds of bandwidth expansion are performed, they are simply
 saturated to 16 bits:
     a32_Q17[k] = clamp(-32768, (a32_Q17[k] + 16) >> 5, 32767) << 5
 Because this performs the actual saturation in the Q12 domain, but
 converts the coefficients back to the Q17 domain for the purposes of
 prediction gain limiting, this step must be performed after the 10th
 round of bandwidth expansion, regardless of whether or not the Q12
 version of any coefficient still overflows a 16-bit integer.  This
 saturation is not performed if maxabs_Q12 drops to 32767 or less
 prior to the 10th round.

4.2.7.5.8. Limiting the Prediction Gain of the LPC Filter

 The prediction gain of an LPC synthesis filter is the square root of
 the output energy when the filter is excited by a unit-energy
 impulse.  Even if the Q12 coefficients would fit, the resulting
 filter may still have a significant gain (especially for voiced
 sounds), making the filter unstable. silk_NLSF2A() applies up to 16
 additional rounds of bandwidth expansion to limit the prediction
 gain.  Instead of controlling the amount of bandwidth expansion using
 the prediction gain itself (which may diverge to infinity for an
 unstable filter), silk_NLSF2A() uses silk_LPC_inverse_pred_gain_QA()

Valin, et al. Standards Track [Page 71] RFC 6716 Interactive Audio Codec September 2012

 (LPC_inv_pred_gain.c) to compute the reflection coefficients
 associated with the filter.  The filter is stable if and only if the
 magnitude of these coefficients is sufficiently less than one.  The
 reflection coefficients, rc[k], can be computed using a simple
 Levinson recurrence, initialized with the LPC coefficients a[d_LPC-
 1][n] = a[n], and then updated via
                    rc[k] = -a[k][k] ,
                            a[k][n] - a[k][k-n-1]*rc[k]
                a[k-1][n] = ---------------------------
                                             2
                                    1 - rc[k]
 However, silk_LPC_inverse_pred_gain_QA() approximates this using
 fixed-point arithmetic to guarantee reproducible results across
 platforms and implementations.  Since small changes in the
 coefficients can make a stable filter unstable, it takes the real Q12
 coefficients that will be used during reconstruction as input.  Thus,
 let
                  a32_Q12[n] = (a32_Q17[n] + 16) >> 5
 be the Q12 version of the LPC coefficients that will eventually be
 used.  As a simple initial check, the decoder computes the DC
 response as
                                d_PLC-1
                                  __
                        DC_resp = \   a32_Q12[n]
                                  /_
                                  n=0
 and if DC_resp > 4096, the filter is unstable.
 Increasing the precision of these Q12 coefficients to Q24 for
 intermediate computations allows more accurate computation of the
 reflection coefficients, so the decoder initializes the recurrence
 via
                 inv_gain_Q30[d_LPC] = 1 << 30
                 a32_Q24[d_LPC-1][n] = a32_Q12[n] << 12

Valin, et al. Standards Track [Page 72] RFC 6716 Interactive Audio Codec September 2012

 Then, for each k from d_LPC-1 down to 0, if
 abs(a32_Q24[k][k]) > 16773022, the filter is unstable and the
 recurrence stops.  The constant 16773022 here is approximately
 0.99975 in Q24.  Otherwise, the inverse of the prediction gain,
 inv_gain_Q30[k], is updated via
            rc_Q31[k] = -a32_Q24[k][k] << 7
           div_Q30[k] = (1<<30) - (rc_Q31[k]*rc_Q31[k] >> 32)
      inv_gain_Q30[k] = (inv_gain_Q30[k+1]*div_Q30[k] >> 32) << 2
 and if inv_gain_Q30[k] < 107374, the filter is unstable and the
 recurrence stops.  The constant 107374 here is approximately 1/10000
 in Q30.  If neither of these checks determine that the filter is
 unstable and k > 0, row k-1 of a32_Q24 is computed from row k as
            b1[k] = ilog(div_Q30[k])
            b2[k] = b1[k] - 16
                          (1<<29) - 1
       inv_Qb2[k] = -----------------------
                    div_Q30[k] >> (b2[k]+1)
       err_Q29[k] = (1<<29)
                    - ((div_Q30[k]<<(15-b2[k]))*inv_Qb2[k] >> 16)
      gain_Qb1[k] = ((inv_Qb2[k] << 16)
                     + (err_Q29[k]*inv_Qb2[k] >> 13))
  num_Q24[k-1][n] = a32_Q24[k][n]
                    - ((a32_Q24[k][k-n-1]*rc_Q31[k] + (1<<30)) >> 31)
  a32_Q24[k-1][n] = (num_Q24[k-1][n]*gain_Qb1[k]
                     + (1<<(b1[k]-1))) >> b1[k]
 where 0 <= n < k.  In the above, rc_Q31[k] are the reflection
 coefficients. div_Q30[k] is the denominator for each iteration, and
 gain_Qb1[k] is its multiplicative inverse (with b1[k] fractional
 bits, where b1[k] ranges from 20 to 31). inv_Qb2[k], which ranges
 from 16384 to 32767, is a low-precision version of that inverse (with
 b2[k] fractional bits). err_Q29[k] is the residual error, ranging
 from -32763 to 32392, which is used to improve the accuracy.  The
 values t_Q24[k-1][n] for each n are the numerators for the next row
 of coefficients in the recursion, and a32_Q24[k-1][n] is the final
 version of that row.  Every multiply in this procedure except the one
 used to compute gain_Qb1[k] requires more than 32 bits of precision,

Valin, et al. Standards Track [Page 73] RFC 6716 Interactive Audio Codec September 2012

 but otherwise all intermediate results fit in 32 bits or less.  In
 practice, because each row only depends on the next one, an
 implementation does not need to store them all.
 If abs(a32_Q24[k][k]) <= 16773022 and inv_gain_Q30[k] >= 107374 for
 0 <= k < d_LPC, then the filter is considered stable.  However, the
 problem of determining stability is ill-conditioned when the filter
 contains several reflection coefficients whose magnitude is very
 close to one.  This fixed-point algorithm is not mathematically
 guaranteed to correctly classify filters as stable or unstable in
 this case, though it does very well in practice.
 On round i, 0 <= i < 16, if the filter passes these stability checks,
 then this procedure stops, and the final LPC coefficients to use for
 reconstruction in Section 4.2.7.9.2 are
                   a_Q12[k] = (a32_Q17[k] + 16) >> 5
 Otherwise, a round of bandwidth expansion is applied using the same
 procedure as in Section 4.2.7.5.7, with
                       sc_Q16[0] = 65536 - (2<<i)
 During round 15, sc_Q16[0] becomes 0 in the above equation, so
 a_Q12[k] is set to 0 for all k, guaranteeing a stable filter.

4.2.7.6. Long-Term Prediction (LTP) Parameters

 After the normalized LSF indices and, for 20 ms frames, the LSF
 interpolation index, voiced frames (see Section 4.2.7.3) include
 additional LTP parameters.  There is one primary lag index for each
 SILK frame, but this is refined to produce a separate lag index per
 subframe using a vector quantizer.  Each subframe also gets its own
 prediction gain coefficient.

4.2.7.6.1. Pitch Lags

 The primary lag index is coded either relative to the primary lag of
 the prior frame in the same channel or as an absolute index.
 Absolute coding is used if and only if
 o  This is the first SILK frame of its type (LBRR or regular) for
    this channel in the current Opus frame,
 o  The previous SILK frame of the same type (LBRR or regular) for
    this channel in the same Opus frame was not coded, or

Valin, et al. Standards Track [Page 74] RFC 6716 Interactive Audio Codec September 2012

 o  That previous SILK frame was coded, but was not voiced (see
    Section 4.2.7.3).
 With absolute coding, the primary pitch lag may range from 2 ms
 (inclusive) up to 18 ms (exclusive), corresponding to pitches from
 500 Hz down to 55.6 Hz, respectively.  It is comprised of a high part
 and a low part, where the decoder first reads the high part using the
 32-entry codebook in Table 29 and then the low part using the
 codebook corresponding to the current audio bandwidth from Table 30.
 The final primary pitch lag is then
              lag = lag_high*lag_scale + lag_low + lag_min
 where lag_high is the high part, lag_low is the low part, and
 lag_scale and lag_min are the values from the "Scale" and "Minimum
 Lag" columns of Table 30, respectively.
 +-------------------------------------------------------------------+
 | PDF                                                               |
 +-------------------------------------------------------------------+
 | {3, 3, 6, 11, 21, 30, 32, 19, 11, 10, 12, 13, 13, 12, 11, 9, 8,   |
 | 7, 6, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256                  |
 +-------------------------------------------------------------------+
           Table 29: PDF for High Part of Primary Pitch Lag
 +------------+------------------------+-------+----------+----------+
 | Audio      | PDF                    | Scale | Minimum  | Maximum  |
 | Bandwidth  |                        |       | Lag      | Lag      |
 +------------+------------------------+-------+----------+----------+
 | NB         | {64, 64, 64, 64}/256   | 4     | 16       | 144      |
 |            |                        |       |          |          |
 | MB         | {43, 42, 43, 43, 42,   | 6     | 24       | 216      |
 |            | 43}/256                |       |          |          |
 |            |                        |       |          |          |
 | WB         | {32, 32, 32, 32, 32,   | 8     | 32       | 288      |
 |            | 32, 32, 32}/256        |       |          |          |
 +------------+------------------------+-------+----------+----------+
            Table 30: PDF for Low Part of Primary Pitch Lag
 All frames that do not use absolute coding for the primary lag index
 use relative coding instead.  The decoder reads a single delta value
 using the 21-entry PDF in Table 31.  If the resulting value is zero,
 it falls back to the absolute coding procedure from the prior
 paragraph.  Otherwise, the final primary pitch lag is then
               lag = previous_lag + (delta_lag_index - 9)

Valin, et al. Standards Track [Page 75] RFC 6716 Interactive Audio Codec September 2012

 where previous_lag is the primary pitch lag from the most recent
 frame in the same channel and delta_lag_index is the value just
 decoded.  This allows a per-frame change in the pitch lag of -8 to
 +11 samples.  The decoder does no clamping at this point, so this
 value can fall outside the range of 2 ms to 18 ms, and the decoder
 must use this unclamped value when using relative coding in the next
 SILK frame (if any).  However, because an Opus frame can use relative
 coding for at most two consecutive SILK frames, integer overflow
 should not be an issue.
 +-------------------------------------------------------------------+
 | PDF                                                               |
 +-------------------------------------------------------------------+
 | {46, 2, 2, 3, 4, 6, 10, 15, 26, 38, 30, 22, 15, 10, 7, 6, 4, 4,   |
 | 2, 2, 2}/256                                                      |
 +-------------------------------------------------------------------+
              Table 31: PDF for Primary Pitch Lag Change
 After the primary pitch lag, a "pitch contour", stored as a single
 entry from one of four small VQ codebooks, gives lag offsets for each
 subframe in the current SILK frame.  The codebook index is decoded
 using one of the PDFs in Table 32 depending on the current frame size
 and audio bandwidth.  Tables 33 through 36 give the corresponding
 offsets to apply to the primary pitch lag for each subframe given the
 decoded codebook index.
 +-----------+--------+----------+-----------------------------------+
 | Audio     | SILK   | Codebook | PDF                               |
 | Bandwidth | Frame  |     Size |                                   |
 |           | Size   |          |                                   |
 +-----------+--------+----------+-----------------------------------+
 | NB        | 10 ms  |        3 | {143, 50, 63}/256                 |
 |           |        |          |                                   |
 | NB        | 20 ms  |       11 | {68, 12, 21, 17, 19, 22, 30, 24,  |
 |           |        |          | 17, 16, 10}/256                   |
 |           |        |          |                                   |
 | MB or WB  | 10 ms  |       12 | {91, 46, 39, 19, 14, 12, 8, 7, 6, |
 |           |        |          | 5, 5, 4}/256                      |
 |           |        |          |                                   |
 | MB or WB  | 20 ms  |       34 | {33, 22, 18, 16, 15, 14, 14, 13,  |
 |           |        |          | 13, 10, 9, 9, 8, 6, 6, 6, 5, 4,   |
 |           |        |          | 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2,  |
 |           |        |          | 2, 1, 1, 1, 1}/256                |
 +-----------+--------+----------+-----------------------------------+
               Table 32: PDFs for Subframe Pitch Contour

Valin, et al. Standards Track [Page 76] RFC 6716 Interactive Audio Codec September 2012

                     +-------+------------------+
                     | Index | Subframe Offsets |
                     +-------+------------------+
                     | 0     |             0  0 |
                     |       |                  |
                     | 1     |             1  0 |
                     |       |                  |
                     | 2     |             0  1 |
                     +-------+------------------+
        Table 33: Codebook Vectors for Subframe Pitch Contour:
                           NB, 10 ms Frames
                     +-------+------------------+
                     | Index | Subframe Offsets |
                     +-------+------------------+
                     | 0     |       0  0  0  0 |
                     |       |                  |
                     | 1     |       2  1  0 -1 |
                     |       |                  |
                     | 2     |      -1  0  1  2 |
                     |       |                  |
                     | 3     |      -1  0  0  1 |
                     |       |                  |
                     | 4     |      -1  0  0  0 |
                     |       |                  |
                     | 5     |       0  0  0  1 |
                     |       |                  |
                     | 6     |       0  0  1  1 |
                     |       |                  |
                     | 7     |       1  1  0  0 |
                     |       |                  |
                     | 8     |       1  0  0  0 |
                     |       |                  |
                     | 9     |       0  0  0 -1 |
                     |       |                  |
                     | 10    |       1  0  0 -1 |
                     +-------+------------------+
        Table 34: Codebook Vectors for Subframe Pitch Contour:
                           NB, 20 ms Frames

Valin, et al. Standards Track [Page 77] RFC 6716 Interactive Audio Codec September 2012

                     +-------+------------------+
                     | Index | Subframe Offsets |
                     +-------+------------------+
                     | 0     |             0  0 |
                     |       |                  |
                     | 1     |             0  1 |
                     |       |                  |
                     | 2     |             1  0 |
                     |       |                  |
                     | 3     |            -1  1 |
                     |       |                  |
                     | 4     |             1 -1 |
                     |       |                  |
                     | 5     |            -1  2 |
                     |       |                  |
                     | 6     |             2 -1 |
                     |       |                  |
                     | 7     |            -2  2 |
                     |       |                  |
                     | 8     |             2 -2 |
                     |       |                  |
                     | 9     |            -2  3 |
                     |       |                  |
                     | 10    |             3 -2 |
                     |       |                  |
                     | 11    |            -3  3 |
                     +-------+------------------+
   Table 35: Codebook Vectors for Subframe Pitch Contour: MB or WB,
                             10 ms Frames
                     +-------+------------------+
                     | Index | Subframe Offsets |
                     +-------+------------------+
                     | 0     |       0  0  0  0 |
                     |       |                  |
                     | 1     |       0  0  1  1 |
                     |       |                  |
                     | 2     |       1  1  0  0 |
                     |       |                  |
                     | 3     |      -1  0  0  0 |
                     |       |                  |
                     | 4     |       0  0  0  1 |
                     |       |                  |
                     | 5     |       1  0  0  0 |
                     |       |                  |
                     | 6     |      -1  0  0  1 |
                     |       |                  |

Valin, et al. Standards Track [Page 78] RFC 6716 Interactive Audio Codec September 2012

                     | 7     |       0  0  0 -1 |
                     |       |                  |
                     | 8     |      -1  0  1  2 |
                     |       |                  |
                     | 9     |       1  0  0 -1 |
                     |       |                  |
                     | 10    |      -2 -1  1  2 |
                     |       |                  |
                     | 11    |       2  1  0 -1 |
                     |       |                  |
                     | 12    |      -2  0  0  2 |
                     |       |                  |
                     | 13    |      -2  0  1  3 |
                     |       |                  |
                     | 14    |       2  1 -1 -2 |
                     |       |                  |
                     | 15    |      -3 -1  1  3 |
                     |       |                  |
                     | 16    |       2  0  0 -2 |
                     |       |                  |
                     | 17    |       3  1  0 -2 |
                     |       |                  |
                     | 18    |      -3 -1  2  4 |
                     |       |                  |
                     | 19    |      -4 -1  1  4 |
                     |       |                  |
                     | 20    |       3  1 -1 -3 |
                     |       |                  |
                     | 21    |      -4 -1  2  5 |
                     |       |                  |
                     | 22    |       4  2 -1 -3 |
                     |       |                  |
                     | 23    |       4  1 -1 -4 |
                     |       |                  |
                     | 24    |      -5 -1  2  6 |
                     |       |                  |
                     | 25    |       5  2 -1 -4 |
                     |       |                  |
                     | 26    |      -6 -2  2  6 |
                     |       |                  |
                     | 27    |      -5 -2  2  5 |
                     |       |                  |
                     | 28    |       6  2 -1 -5 |
                     |       |                  |
                     | 29    |      -7 -2  3  8 |
                     |       |                  |
                     | 30    |       6  2 -2 -6 |
                     |       |                  |

Valin, et al. Standards Track [Page 79] RFC 6716 Interactive Audio Codec September 2012

                     | 31    |       5  2 -2 -5 |
                     |       |                  |
                     | 32    |       8  3 -2 -7 |
                     |       |                  |
                     | 33    |      -9 -3  3  9 |
                     +-------+------------------+
   Table 36: Codebook Vectors for Subframe Pitch Contour: MB or WB,
                             20 ms Frames
 The final pitch lag for each subframe is assembled in
 silk_decode_pitch() (decode_pitch.c).  Let lag be the primary pitch
 lag for the current SILK frame, contour_index be index of the VQ
 codebook, and lag_cb[contour_index][k] be the corresponding entry of
 the codebook from the appropriate table given above for the k'th
 subframe.  Then the final pitch lag for that subframe is
     pitch_lags[k] = clamp(lag_min, lag + lag_cb[contour_index][k],
                           lag_max)
 where lag_min and lag_max are the values from the "Minimum Lag" and
 "Maximum Lag" columns of Table 30, respectively.

4.2.7.6.2. LTP Filter Coefficients

 SILK uses a separate 5-tap pitch filter for each subframe, selected
 from one of three codebooks.  The three codebooks each represent
 different rate-distortion trade-offs, with average rates of
 1.61 bits/subframe, 3.68 bits/subframe, and 4.85 bits/subframe,
 respectively.
 The importance of the filter coefficients generally depends on two
 factors: the periodicity of the signal and relative energy between
 the current subframe and the signal from one period earlier.  Greater
 periodicity and decaying energy both lead to more important filter
 coefficients.  Thus, they should be coded with lower distortion and
 higher rate.  These properties are relatively stable over the
 duration of a single SILK frame.  Hence, all of the subframes in a
 SILK frame choose their filter from the same codebook.  This is
 signaled with an explicitly-coded "periodicity index".  This
 immediately follows the subframe pitch lags, and is coded using the
 3-entry PDF from Table 37.

Valin, et al. Standards Track [Page 80] RFC 6716 Interactive Audio Codec September 2012

                         +------------------+
                         | PDF              |
                         +------------------+
                         | {77, 80, 99}/256 |
                         +------------------+
                    Table 37: Periodicity Index PDF
 The indices of the filters for each subframe follow.  They are all
 coded using the PDF from Table 38 corresponding to the periodicity
 index.  Tables 39 through 41 contain the corresponding filter taps as
 signed Q7 integers.
 +-------------+----------+------------------------------------------+
 | Periodicity | Codebook | PDF                                      |
 | Index       |     Size |                                          |
 +-------------+----------+------------------------------------------+
 | 0           |        8 | {185, 15, 13, 13, 9, 9, 6, 6}/256        |
 |             |          |                                          |
 | 1           |       16 | {57, 34, 21, 20, 15, 13, 12, 13, 10, 10, |
 |             |          | 9, 10, 9, 8, 7, 8}/256                   |
 |             |          |                                          |
 | 2           |       32 | {15, 16, 14, 12, 12, 12, 11, 11, 11, 10, |
 |             |          | 9, 9, 9, 9, 8, 8, 8, 8, 7, 7, 6, 6, 5,   |
 |             |          | 4, 5, 4, 4, 4, 3, 4, 3, 2}/256           |
 +-------------+----------+------------------------------------------+
                       Table 38: LTP Filter PDFs

Valin, et al. Standards Track [Page 81] RFC 6716 Interactive Audio Codec September 2012

                    +-------+---------------------+
                    | Index |    Filter Taps (Q7) |
                    +-------+---------------------+
                    | 0     |   4   6  24   7   5 |
                    |       |                     |
                    | 1     |   0   0   2   0   0 |
                    |       |                     |
                    | 2     |  12  28  41  13  -4 |
                    |       |                     |
                    | 3     |  -9  15  42  25  14 |
                    |       |                     |
                    | 4     |   1  -2  62  41  -9 |
                    |       |                     |
                    | 5     | -10  37  65  -4   3 |
                    |       |                     |
                    | 6     |  -6   4  66   7  -8 |
                    |       |                     |
                    | 7     |  16  14  38  -3  33 |
                    +-------+---------------------+
    Table 39: Codebook Vectors for LTP Filter, Periodicity Index 0

Valin, et al. Standards Track [Page 82] RFC 6716 Interactive Audio Codec September 2012

                    +-------+---------------------+
                    | Index |    Filter Taps (Q7) |
                    +-------+---------------------+
                    | 0     |  13  22  39  23  12 |
                    |       |                     |
                    | 1     |  -1  36  64  27  -6 |
                    |       |                     |
                    | 2     |  -7  10  55  43  17 |
                    |       |                     |
                    | 3     |   1   1   8   1   1 |
                    |       |                     |
                    | 4     |   6 -11  74  53  -9 |
                    |       |                     |
                    | 5     | -12  55  76 -12   8 |
                    |       |                     |
                    | 6     |  -3   3  93  27  -4 |
                    |       |                     |
                    | 7     |  26  39  59   3  -8 |
                    |       |                     |
                    | 8     |   2   0  77  11   9 |
                    |       |                     |
                    | 9     |  -8  22  44  -6   7 |
                    |       |                     |
                    | 10    |  40   9  26   3   9 |
                    |       |                     |
                    | 11    |  -7  20 101  -7   4 |
                    |       |                     |
                    | 12    |   3  -8  42  26   0 |
                    |       |                     |
                    | 13    | -15  33  68   2  23 |
                    |       |                     |
                    | 14    |  -2  55  46  -2  15 |
                    |       |                     |
                    | 15    |   3  -1  21  16  41 |
                    +-------+---------------------+
    Table 40: Codebook Vectors for LTP Filter, Periodicity Index 1
                    +-------+---------------------+
                    | Index |    Filter Taps (Q7) |
                    +-------+---------------------+
                    | 0     |  -6  27  61  39   5 |
                    |       |                     |
                    | 1     | -11  42  88   4   1 |
                    |       |                     |
                    | 2     |  -2  60  65   6  -4 |
                    |       |                     |
                    | 3     |  -1  -5  73  56   1 |

Valin, et al. Standards Track [Page 83] RFC 6716 Interactive Audio Codec September 2012

                    | 4     |  -9  19  94  29  -9 |
                    |       |                     |
                    | 5     |   0  12  99   6   4 |
                    |       |                     |
                    | 6     |   8 -19 102  46 -13 |
                    |       |                     |
                    | 7     |   3   2  13   3   2 |
                    |       |                     |
                    | 8     |   9 -21  84  72 -18 |
                    |       |                     |
                    | 9     | -11  46 104 -22   8 |
                    |       |                     |
                    | 10    |  18  38  48  23   0 |
                    |       |                     |
                    | 11    | -16  70  83 -21  11 |
                    |       |                     |
                    | 12    |   5 -11 117  22  -8 |
                    |       |                     |
                    | 13    |  -6  23 117 -12   3 |
                    |       |                     |
                    | 14    |   3  -8  95  28   4 |
                    |       |                     |
                    | 15    | -10  15  77  60 -15 |
                    |       |                     |
                    | 16    |  -1   4 124   2  -4 |
                    |       |                     |
                    | 17    |   3  38  84  24 -25 |
                    |       |                     |
                    | 18    |   2  13  42  13  31 |
                    |       |                     |
                    | 19    |  21  -4  56  46  -1 |
                    |       |                     |
                    | 20    |  -1  35  79 -13  19 |
                    |       |                     |
                    | 21    |  -7  65  88  -9 -14 |
                    |       |                     |
                    | 22    |  20   4  81  49 -29 |
                    |       |                     |
                    | 23    |  20   0  75   3 -17 |
                    |       |                     |
                    | 24    |   5  -9  44  92  -8 |
                    |       |                     |
                    | 25    |   1  -3  22  69  31 |
                    |       |                     |
                    | 26    |  -6  95  41 -12   5 |
                    |       |                     |
                    | 27    |  39  67  16  -4   1 |
                    |       |                     |

Valin, et al. Standards Track [Page 84] RFC 6716 Interactive Audio Codec September 2012

                    | 28    |   0  -6 120  55 -36 |
                    |       |                     |
                    | 29    | -13  44 122   4 -24 |
                    |       |                     |
                    | 30    |  81   5  11   3   7 |
                    |       |                     |
                    | 31    |   2   0   9  10  88 |
                    +-------+---------------------+
    Table 41: Codebook Vectors for LTP Filter, Periodicity Index 2

4.2.7.6.3. LTP Scaling Parameter

 An LTP scaling parameter appears after the LTP filter coefficients if
 and only if
 o  This is a voiced frame (see Section 4.2.7.3), and
 o  Either
  • This SILK frame corresponds to the first time interval of the

current Opus frame for its type (LBRR or regular), or

  • This is an LBRR frame where the LBRR flags (see Section 4.2.4)

indicate the previous LBRR frame in the same channel is not

       coded.
 This allows the encoder to trade off the prediction gain between
 packets against the recovery time after packet loss.  Unlike
 absolute-coding for pitch lags, regular SILK frames that are not at
 the start of an Opus frame (i.e., that do not correspond to the first
 20 ms time interval in Opus frames of 40 or 60 ms) do not include
 this field, even if the prior frame was not voiced, or (in the case
 of the side channel) not even coded.  After an uncoded frame in the
 side channel, the LTP buffer (see Section 4.2.7.9.1) is cleared to
 zero, and is thus in a known state.  In contrast, LBRR frames do
 include this field when the prior frame was not coded, since the LTP
 buffer contains the output of the PLC, which is non-normative.
 If present, the decoder reads a value using the 3-entry PDF in
 Table 42.  The three possible values represent Q14 scale factors of
 15565, 12288, and 8192, respectively (corresponding to approximately
 0.95, 0.75, and 0.5).  Frames that do not code the scaling parameter
 use the default factor of 15565 (approximately 0.95).

Valin, et al. Standards Track [Page 85] RFC 6716 Interactive Audio Codec September 2012

                         +-------------------+
                         | PDF               |
                         +-------------------+
                         | {128, 64, 64}/256 |
                         +-------------------+
                Table 42: PDF for LTP Scaling Parameter

4.2.7.7. Linear Congruential Generator (LCG) Seed

 As described in Section 4.2.7.8.6, SILK uses a Linear Congruential
 Generator (LCG) to inject pseudorandom noise into the quantized
 excitation.  To ensure synchronization of this process between the
 encoder and decoder, each SILK frame stores a 2-bit seed after the
 LTP parameters (if any).  The encoder may consider the choice of seed
 during quantization, and the flexibility of this choice lets it
 reduce distortion, helping to pay for the bit cost required to signal
 it.  The decoder reads the seed using the uniform 4-entry PDF in
 Table 43, yielding a value between 0 and 3, inclusive.
                       +----------------------+
                       | PDF                  |
                       +----------------------+
                       | {64, 64, 64, 64}/256 |
                       +----------------------+
                      Table 43: PDF for LCG Seed

4.2.7.8. Excitation

 SILK codes the excitation using a modified version of the Pyramid
 Vector Quantizer (PVQ) codebook [PVQ].  The PVQ codebook is designed
 for Laplace-distributed values and consists of all sums of K signed,
 unit pulses in a vector of dimension N, where two pulses at the same
 position are required to have the same sign.  Thus, the codebook
 includes all integer codevectors y of dimension N that satisfy
                            N-1
                            __
                            \  abs(y[j]) = K
                            /_
                            j=0
 Unlike regular PVQ, SILK uses a variable-length, rather than fixed-
 length, encoding.  This encoding is better suited to the more
 Gaussian-like distribution of the coefficient magnitudes and the non-
 uniform distribution of their signs (caused by the quantization
 offset described below).  SILK also handles large codebooks by coding

Valin, et al. Standards Track [Page 86] RFC 6716 Interactive Audio Codec September 2012

 the least significant bits (LSBs) of each coefficient directly.  This
 adds a small coding efficiency loss, but greatly reduces the
 computation time and ROM size required for decoding, as implemented
 in silk_decode_pulses() (decode_pulses.c).
 SILK fixes the dimension of the codebook to N = 16.  The excitation
 is made up of a number of "shell blocks", each 16 samples in size.
 Table 44 lists the number of shell blocks required for a SILK frame
 for each possible audio bandwidth and frame size. 10 ms MB frames
 nominally contain 120 samples (10 ms at 12 kHz), which is not a
 multiple of 16.  This is handled by coding 8 shell blocks (128
 samples) and discarding the final 8 samples of the last block.  The
 decoder contains no special case that prevents an encoder from
 placing pulses in these samples, and they must be correctly parsed
 from the bitstream if present, but they are otherwise ignored.
       +-----------------+------------+------------------------+
       | Audio Bandwidth | Frame Size | Number of Shell Blocks |
       +-----------------+------------+------------------------+
       | NB              | 10 ms      |                      5 |
       |                 |            |                        |
       | MB              | 10 ms      |                      8 |
       |                 |            |                        |
       | WB              | 10 ms      |                     10 |
       |                 |            |                        |
       | NB              | 20 ms      |                     10 |
       |                 |            |                        |
       | MB              | 20 ms      |                     15 |
       |                 |            |                        |
       | WB              | 20 ms      |                     20 |
       +-----------------+------------+------------------------+
            Table 44: Number of Shell Blocks Per SILK Frame

4.2.7.8.1. Rate Level

 The first symbol in the excitation is a "rate level", which is an
 index from 0 to 8, inclusive, coded using the PDF in Table 45
 corresponding to the signal type of the current frame (from
 Section 4.2.7.3).  The rate level selects the PDF used to decode the
 number of pulses in the individual shell blocks.  It does not
 directly convey any information about the bitrate or the number of
 pulses itself, but merely changes the probability of the symbols in
 Section 4.2.7.8.2.  Level 0 provides a more efficient encoding at low
 rates generally, and level 8 provides a more efficient encoding at
 high rates generally, though the most efficient level for a

Valin, et al. Standards Track [Page 87] RFC 6716 Interactive Audio Codec September 2012

 particular SILK frame may depend on the exact distribution of the
 coded symbols.  An encoder should, but is not required to, use the
 most efficient rate level.
  +----------------------+------------------------------------------+
  | Signal Type          | PDF                                      |
  +----------------------+------------------------------------------+
  | Inactive or Unvoiced | {15, 51, 12, 46, 45, 13, 33, 27, 14}/256 |
  |                      |                                          |
  | Voiced               | {33, 30, 36, 17, 34, 49, 18, 21, 18}/256 |
  +----------------------+------------------------------------------+
                   Table 45: PDFs for the Rate Level

4.2.7.8.2. Pulses per Shell Block

 The total number of pulses in each of the shell blocks follows the
 rate level.  The pulse counts for all of the shell blocks are coded
 consecutively, before the content of any of the blocks.  Each block
 may have anywhere from 0 to 16 pulses, inclusive, coded using the 18-
 entry PDF in Table 46 corresponding to the rate level from
 Section 4.2.7.8.1.  The special value 17 indicates that this block
 has one or more additional LSBs to decode for each coefficient.  If
 the decoder encounters this value, it decodes another value for the
 actual pulse count of the block, but uses the PDF corresponding to
 the special rate level 9 instead of the normal rate level.  This
 process repeats until the decoder reads a value less than 17, and it
 then sets the number of extra LSBs used to the number of 17's decoded
 for that block.  If it reads the value 17 ten times, then the next
 iteration uses the special rate level 10 instead of 9.  The
 probability of decoding a 17 when using the PDF for rate level 10 is
 zero, ensuring that the number of LSBs for a block will not exceed
 10.  The cumulative distribution for rate level 10 is just a shifted
 version of that for 9 and thus does not require any additional
 storage.

Valin, et al. Standards Track [Page 88] RFC 6716 Interactive Audio Codec September 2012

 +----------+--------------------------------------------------------+
 | Rate     | PDF                                                    |
 | Level    |                                                        |
 +----------+--------------------------------------------------------+
 | 0        | {131, 74, 25, 8, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,   |
 |          | 1, 1}/256                                              |
 |          |                                                        |
 | 1        | {58, 93, 60, 23, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,   |
 |          | 1, 1}/256                                              |
 |          |                                                        |
 | 2        | {43, 51, 46, 33, 24, 16, 11, 8, 6, 3, 3, 3, 2, 1, 1,   |
 |          | 2, 1, 2}/256                                           |
 |          |                                                        |
 | 3        | {17, 52, 71, 57, 31, 12, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, |
 |          | 1, 1}/256                                              |
 |          |                                                        |
 | 4        | {6, 21, 41, 53, 49, 35, 21, 11, 6, 3, 2, 2, 1, 1, 1,   |
 |          | 1, 1, 1}/256                                           |
 |          |                                                        |
 | 5        | {7, 14, 22, 28, 29, 28, 25, 20, 17, 13, 11, 9, 7, 5,   |
 |          | 4, 4, 3, 10}/256                                       |
 |          |                                                        |
 | 6        | {2, 5, 14, 29, 42, 46, 41, 31, 19, 11, 6, 3, 2, 1, 1,  |
 |          | 1, 1, 1}/256                                           |
 |          |                                                        |
 | 7        | {1, 2, 4, 10, 19, 29, 35, 37, 34, 28, 20, 14, 8, 5, 4, |
 |          | 2, 2, 2}/256                                           |
 |          |                                                        |
 | 8        | {1, 2, 2, 5, 9, 14, 20, 24, 27, 28, 26, 23, 20, 15,    |
 |          | 11, 8, 6, 15}/256                                      |
 |          |                                                        |
 | 9        | {1, 1, 1, 6, 27, 58, 56, 39, 25, 14, 10, 6, 3, 3, 2,   |
 |          | 1, 1, 2}/256                                           |
 |          |                                                        |
 | 10       | {2, 1, 6, 27, 58, 56, 39, 25, 14, 10, 6, 3, 3, 2, 1,   |
 |          | 1, 2, 0}/256                                           |
 +----------+--------------------------------------------------------+
                  Table 46: PDFs for the Pulse Count

4.2.7.8.3. Pulse Location Decoding

 The locations of the pulses in each shell block follow the pulse
 counts, as decoded by silk_shell_decoder() (shell_coder.c).  As with
 the pulse counts, these locations are coded for all the shell blocks
 before any of the remaining information for each block.  Unlike many
 other codecs, SILK places no restriction on the distribution of

Valin, et al. Standards Track [Page 89] RFC 6716 Interactive Audio Codec September 2012

 pulses within a shell block.  All of the pulses may be placed in a
 single location, or each one in a unique location, or anything in
 between.
 The location of pulses is coded by recursively partitioning each
 block into halves, and coding how many pulses fall on the left side
 of the split.  All remaining pulses must fall on the right side of
 the split.  The process then recurses into the left half, and after
 that returns, the right half (preorder traversal).  The PDF to use is
 chosen by the size of the current partition (16, 8, 4, or 2) and the
 number of pulses in the partition (1 to 16, inclusive).  Tables 47
 through 50 list the PDFs used for each partition size and pulse
 count.  This process skips partitions without any pulses, i.e., where
 the initial pulse count from Section 4.2.7.8.2 was zero, or where the
 split in the prior level indicated that all of the pulses fell on the
 other side.  These partitions have nothing to code, so they require
 no PDF.

Valin, et al. Standards Track [Page 90] RFC 6716 Interactive Audio Codec September 2012

 +------------+------------------------------------------------------+
 | Pulse      | PDF                                                  |
 | Count      |                                                      |
 +------------+------------------------------------------------------+
 | 1          | {126, 130}/256                                       |
 |            |                                                      |
 | 2          | {56, 142, 58}/256                                    |
 |            |                                                      |
 | 3          | {25, 101, 104, 26}/256                               |
 |            |                                                      |
 | 4          | {12, 60, 108, 64, 12}/256                            |
 |            |                                                      |
 | 5          | {7, 35, 84, 87, 37, 6}/256                           |
 |            |                                                      |
 | 6          | {4, 20, 59, 86, 63, 21, 3}/256                       |
 |            |                                                      |
 | 7          | {3, 12, 38, 72, 75, 42, 12, 2}/256                   |
 |            |                                                      |
 | 8          | {2, 8, 25, 54, 73, 59, 27, 7, 1}/256                 |
 |            |                                                      |
 | 9          | {2, 5, 17, 39, 63, 65, 42, 18, 4, 1}/256             |
 |            |                                                      |
 | 10         | {1, 4, 12, 28, 49, 63, 54, 30, 11, 3, 1}/256         |
 |            |                                                      |
 | 11         | {1, 4, 8, 20, 37, 55, 57, 41, 22, 8, 2, 1}/256       |
 |            |                                                      |
 | 12         | {1, 3, 7, 15, 28, 44, 53, 48, 33, 16, 6, 1, 1}/256   |
 |            |                                                      |
 | 13         | {1, 2, 6, 12, 21, 35, 47, 48, 40, 25, 12, 5, 1,      |
 |            | 1}/256                                               |
 |            |                                                      |
 | 14         | {1, 1, 4, 10, 17, 27, 37, 47, 43, 33, 21, 9, 4, 1,   |
 |            | 1}/256                                               |
 |            |                                                      |
 | 15         | {1, 1, 1, 8, 14, 22, 33, 40, 43, 38, 28, 16, 8, 1,   |
 |            | 1, 1}/256                                            |
 |            |                                                      |
 | 16         | {1, 1, 1, 1, 13, 18, 27, 36, 41, 41, 34, 24, 14, 1,  |
 |            | 1, 1, 1}/256                                         |
 +------------+------------------------------------------------------+
      Table 47: PDFs for Pulse Count Split, 16 Sample Partitions

Valin, et al. Standards Track [Page 91] RFC 6716 Interactive Audio Codec September 2012

 +------------+------------------------------------------------------+
 | Pulse      | PDF                                                  |
 | Count      |                                                      |
 +------------+------------------------------------------------------+
 | 1          | {127, 129}/256                                       |
 |            |                                                      |
 | 2          | {53, 149, 54}/256                                    |
 |            |                                                      |
 | 3          | {22, 105, 106, 23}/256                               |
 |            |                                                      |
 | 4          | {11, 61, 111, 63, 10}/256                            |
 |            |                                                      |
 | 5          | {6, 35, 86, 88, 36, 5}/256                           |
 |            |                                                      |
 | 6          | {4, 20, 59, 87, 62, 21, 3}/256                       |
 |            |                                                      |
 | 7          | {3, 13, 40, 71, 73, 41, 13, 2}/256                   |
 |            |                                                      |
 | 8          | {3, 9, 27, 53, 70, 56, 28, 9, 1}/256                 |
 |            |                                                      |
 | 9          | {3, 8, 19, 37, 57, 61, 44, 20, 6, 1}/256             |
 |            |                                                      |
 | 10         | {3, 7, 15, 28, 44, 54, 49, 33, 17, 5, 1}/256         |
 |            |                                                      |
 | 11         | {1, 7, 13, 22, 34, 46, 48, 38, 28, 14, 4, 1}/256     |
 |            |                                                      |
 | 12         | {1, 1, 11, 22, 27, 35, 42, 47, 33, 25, 10, 1, 1}/256 |
 |            |                                                      |
 | 13         | {1, 1, 6, 14, 26, 37, 43, 43, 37, 26, 14, 6, 1,      |
 |            | 1}/256                                               |
 |            |                                                      |
 | 14         | {1, 1, 4, 10, 20, 31, 40, 42, 40, 31, 20, 10, 4, 1,  |
 |            | 1}/256                                               |
 |            |                                                      |
 | 15         | {1, 1, 3, 8, 16, 26, 35, 38, 38, 35, 26, 16, 8, 3,   |
 |            | 1, 1}/256                                            |
 |            |                                                      |
 | 16         | {1, 1, 2, 6, 12, 21, 30, 36, 38, 36, 30, 21, 12, 6,  |
 |            | 2, 1, 1}/256                                         |
 +------------+------------------------------------------------------+
       Table 48: PDFs for Pulse Count Split, 8 Sample Partitions

Valin, et al. Standards Track [Page 92] RFC 6716 Interactive Audio Codec September 2012

 +------------+------------------------------------------------------+
 | Pulse      | PDF                                                  |
 | Count      |                                                      |
 +------------+------------------------------------------------------+
 | 1          | {127, 129}/256                                       |
 |            |                                                      |
 | 2          | {49, 157, 50}/256                                    |
 |            |                                                      |
 | 3          | {20, 107, 109, 20}/256                               |
 |            |                                                      |
 | 4          | {11, 60, 113, 62, 10}/256                            |
 |            |                                                      |
 | 5          | {7, 36, 84, 87, 36, 6}/256                           |
 |            |                                                      |
 | 6          | {6, 24, 57, 82, 60, 23, 4}/256                       |
 |            |                                                      |
 | 7          | {5, 18, 39, 64, 68, 42, 16, 4}/256                   |
 |            |                                                      |
 | 8          | {6, 14, 29, 47, 61, 52, 30, 14, 3}/256               |
 |            |                                                      |
 | 9          | {1, 15, 23, 35, 51, 50, 40, 30, 10, 1}/256           |
 |            |                                                      |
 | 10         | {1, 1, 21, 32, 42, 52, 46, 41, 18, 1, 1}/256         |
 |            |                                                      |
 | 11         | {1, 6, 16, 27, 36, 42, 42, 36, 27, 16, 6, 1}/256     |
 |            |                                                      |
 | 12         | {1, 5, 12, 21, 31, 38, 40, 38, 31, 21, 12, 5, 1}/256 |
 |            |                                                      |
 | 13         | {1, 3, 9, 17, 26, 34, 38, 38, 34, 26, 17, 9, 3,      |
 |            | 1}/256                                               |
 |            |                                                      |
 | 14         | {1, 3, 7, 14, 22, 29, 34, 36, 34, 29, 22, 14, 7, 3,  |
 |            | 1}/256                                               |
 |            |                                                      |
 | 15         | {1, 2, 5, 11, 18, 25, 31, 35, 35, 31, 25, 18, 11, 5, |
 |            | 2, 1}/256                                            |
 |            |                                                      |
 | 16         | {1, 1, 4, 9, 15, 21, 28, 32, 34, 32, 28, 21, 15, 9,  |
 |            | 4, 1, 1}/256                                         |
 +------------+------------------------------------------------------+
       Table 49: PDFs for Pulse Count Split, 4 Sample Partitions

Valin, et al. Standards Track [Page 93] RFC 6716 Interactive Audio Codec September 2012

 +------------+------------------------------------------------------+
 | Pulse      | PDF                                                  |
 | Count      |                                                      |
 +------------+------------------------------------------------------+
 | 1          | {128, 128}/256                                       |
 |            |                                                      |
 | 2          | {42, 172, 42}/256                                    |
 |            |                                                      |
 | 3          | {21, 107, 107, 21}/256                               |
 |            |                                                      |
 | 4          | {12, 60, 112, 61, 11}/256                            |
 |            |                                                      |
 | 5          | {8, 34, 86, 86, 35, 7}/256                           |
 |            |                                                      |
 | 6          | {8, 23, 55, 90, 55, 20, 5}/256                       |
 |            |                                                      |
 | 7          | {5, 15, 38, 72, 72, 36, 15, 3}/256                   |
 |            |                                                      |
 | 8          | {6, 12, 27, 52, 77, 47, 20, 10, 5}/256               |
 |            |                                                      |
 | 9          | {6, 19, 28, 35, 40, 40, 35, 28, 19, 6}/256           |
 |            |                                                      |
 | 10         | {4, 14, 22, 31, 37, 40, 37, 31, 22, 14, 4}/256       |
 |            |                                                      |
 | 11         | {3, 10, 18, 26, 33, 38, 38, 33, 26, 18, 10, 3}/256   |
 |            |                                                      |
 | 12         | {2, 8, 13, 21, 29, 36, 38, 36, 29, 21, 13, 8, 2}/256 |
 |            |                                                      |
 | 13         | {1, 5, 10, 17, 25, 32, 38, 38, 32, 25, 17, 10, 5,    |
 |            | 1}/256                                               |
 |            |                                                      |
 | 14         | {1, 4, 7, 13, 21, 29, 35, 36, 35, 29, 21, 13, 7, 4,  |
 |            | 1}/256                                               |
 |            |                                                      |
 | 15         | {1, 2, 5, 10, 17, 25, 32, 36, 36, 32, 25, 17, 10, 5, |
 |            | 2, 1}/256                                            |
 |            |                                                      |
 | 16         | {1, 2, 4, 7, 13, 21, 28, 34, 36, 34, 28, 21, 13, 7,  |
 |            | 4, 2, 1}/256                                         |
 +------------+------------------------------------------------------+
       Table 50: PDFs for Pulse Count Split, 2 Sample Partitions

4.2.7.8.4. LSB Decoding

 After the decoder reads the pulse locations for all blocks, it reads
 the LSBs (if any) for each block in turn.  Inside each block, it
 reads all the LSBs for each coefficient in turn, even those where no

Valin, et al. Standards Track [Page 94] RFC 6716 Interactive Audio Codec September 2012

 pulses were allocated, before proceeding to the next one.  For 10 ms
 MB frames, it reads LSBs even for the extra 8 samples in the last
 block.  The LSBs are coded from most significant to least
 significant, and they all use the PDF in Table 51.
                          +----------------+
                          | PDF            |
                          +----------------+
                          | {136, 120}/256 |
                          +----------------+
                   Table 51: PDF for Excitation LSBs
 The number of LSBs read for each coefficient in a block is determined
 in Section 4.2.7.8.2.  The magnitude of the coefficient is initially
 equal to the number of pulses placed at that location in
 Section 4.2.7.8.3.  As each LSB is decoded, the magnitude is doubled,
 and then the value of the LSB added to it, to obtain an updated
 magnitude.

4.2.7.8.5. Sign Decoding

 After decoding the pulse locations and the LSBs, the decoder knows
 the magnitude of each coefficient in the excitation.  It then decodes
 a sign for all coefficients with a non-zero magnitude, using one of
 the PDFs from Table 52.  If the value decoded is 0, then the
 coefficient magnitude is negated.  Otherwise, it remains positive.
 The decoder chooses the PDF for the sign based on the signal type and
 quantization offset type (from Section 4.2.7.3) and the number of
 pulses in the block (from Section 4.2.7.8.2).  The number of pulses
 in the block does not take into account any LSBs.  Most PDFs are
 skewed towards negative signs because of the quantization offset, but
 the PDFs for zero pulses are highly skewed towards positive signs.
 If a block contains many positive coefficients, it is sometimes
 beneficial to code it solely using LSBs (i.e., with zero pulses),
 since the encoder may be able to save enough bits on the signs to
 justify the less efficient coefficient magnitude encoding.
 +-------------+-----------------------+-------------+---------------+
 | Signal Type | Quantization Offset   | Pulse Count | PDF           |
 |             | Type                  |             |               |
 +-------------+-----------------------+-------------+---------------+
 | Inactive    | Low                   | 0           | {2, 254}/256  |
 |             |                       |             |               |
 | Inactive    | Low                   | 1           | {207, 49}/256 |
 |             |                       |             |               |
 | Inactive    | Low                   | 2           | {189, 67}/256 |

Valin, et al. Standards Track [Page 95] RFC 6716 Interactive Audio Codec September 2012

 | Inactive    | Low                   | 3           | {179, 77}/256 |
 |             |                       |             |               |
 | Inactive    | Low                   | 4           | {174, 82}/256 |
 |             |                       |             |               |
 | Inactive    | Low                   | 5           | {163, 93}/256 |
 |             |                       |             |               |
 | Inactive    | Low                   | 6 or more   | {157, 99}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 0           | {58, 198}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 1           | {245, 11}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 2           | {238, 18}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 3           | {232, 24}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 4           | {225, 31}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 5           | {220, 36}/256 |
 |             |                       |             |               |
 | Inactive    | High                  | 6 or more   | {211, 45}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 0           | {1, 255}/256  |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 1           | {210, 46}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 2           | {190, 66}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 3           | {178, 78}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 4           | {169, 87}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 5           | {162, 94}/256 |
 |             |                       |             |               |
 | Unvoiced    | Low                   | 6 or more   | {152,         |
 |             |                       |             | 104}/256      |
 |             |                       |             |               |
 | Unvoiced    | High                  | 0           | {48, 208}/256 |
 |             |                       |             |               |
 | Unvoiced    | High                  | 1           | {242, 14}/256 |
 |             |                       |             |               |
 | Unvoiced    | High                  | 2           | {235, 21}/256 |
 |             |                       |             |               |
 | Unvoiced    | High                  | 3           | {224, 32}/256 |
 |             |                       |             |               |
 | Unvoiced    | High                  | 4           | {214, 42}/256 |
 |             |                       |             |               |
 | Unvoiced    | High                  | 5           | {205, 51}/256 |

Valin, et al. Standards Track [Page 96] RFC 6716 Interactive Audio Codec September 2012

 | Unvoiced    | High                  | 6 or more   | {190, 66}/256 |
 |             |                       |             |               |
 | Voiced      | Low                   | 0           | {1, 255}/256  |
 |             |                       |             |               |
 | Voiced      | Low                   | 1           | {162, 94}/256 |
 |             |                       |             |               |
 | Voiced      | Low                   | 2           | {152,         |
 |             |                       |             | 104}/256      |
 |             |                       |             |               |
 | Voiced      | Low                   | 3           | {147,         |
 |             |                       |             | 109}/256      |
 |             |                       |             |               |
 | Voiced      | Low                   | 4           | {144,         |
 |             |                       |             | 112}/256      |
 |             |                       |             |               |
 | Voiced      | Low                   | 5           | {141,         |
 |             |                       |             | 115}/256      |
 |             |                       |             |               |
 | Voiced      | Low                   | 6 or more   | {138,         |
 |             |                       |             | 118}/256      |
 |             |                       |             |               |
 | Voiced      | High                  | 0           | {8, 248}/256  |
 |             |                       |             |               |
 | Voiced      | High                  | 1           | {203, 53}/256 |
 |             |                       |             |               |
 | Voiced      | High                  | 2           | {187, 69}/256 |
 |             |                       |             |               |
 | Voiced      | High                  | 3           | {176, 80}/256 |
 |             |                       |             |               |
 | Voiced      | High                  | 4           | {168, 88}/256 |
 |             |                       |             |               |
 | Voiced      | High                  | 5           | {161, 95}/256 |
 |             |                       |             |               |
 | Voiced      | High                  | 6 or more   | {154,         |
 |             |                       |             | 102}/256      |
 +-------------+-----------------------+-------------+---------------+
                  Table 52: PDFs for Excitation Signs

4.2.7.8.6. Reconstructing the Excitation

 After the signs have been read, there is enough information to
 reconstruct the complete excitation signal.  This requires adding a
 constant quantization offset to each non-zero sample and then
 pseudorandomly inverting and offsetting every sample.  The constant
 quantization offset varies depending on the signal type and
 quantization offset type (see Section 4.2.7.3).

Valin, et al. Standards Track [Page 97] RFC 6716 Interactive Audio Codec September 2012

 +-------------+--------------------------+--------------------------+
 | Signal Type | Quantization Offset Type |      Quantization Offset |
 |             |                          |                    (Q23) |
 +-------------+--------------------------+--------------------------+
 | Inactive    | Low                      |                       25 |
 |             |                          |                          |
 | Inactive    | High                     |                       60 |
 |             |                          |                          |
 | Unvoiced    | Low                      |                       25 |
 |             |                          |                          |
 | Unvoiced    | High                     |                       60 |
 |             |                          |                          |
 | Voiced      | Low                      |                        8 |
 |             |                          |                          |
 | Voiced      | High                     |                       25 |
 +-------------+--------------------------+--------------------------+
               Table 53: Excitation Quantization Offsets
 Let e_raw[i] be the raw excitation value at position i, with a
 magnitude composed of the pulses at that location (see
 Section 4.2.7.8.3) combined with any additional LSBs (see
 Section 4.2.7.8.4), and with the corresponding sign decoded in
 Section 4.2.7.8.5.  Additionally, let seed be the current
 pseudorandom seed, which is initialized to the value decoded from
 Section 4.2.7.7 for the first sample in the current SILK frame, and
 updated for each subsequent sample according to the procedure below.
 Finally, let offset_Q23 be the quantization offset from Table 53.
 Then the following procedure produces the final reconstructed
 excitation value, e_Q23[i]:
      e_Q23[i] = (e_raw[i] << 8) - sign(e_raw[i])*20 + offset_Q23;
          seed = (196314165*seed + 907633515) & 0xFFFFFFFF;
      e_Q23[i] = (seed & 0x80000000) ? -e_Q23[i] : e_Q23[i];
          seed = (seed + e_raw[i]) & 0xFFFFFFFF;
 When e_raw[i] is zero, sign() returns 0 by the definition in
 Section 1.1.4, so the factor of 20 does not get added.  The final
 e_Q23[i] value may require more than 16 bits per sample, but it will
 not require more than 23, including the sign.

4.2.7.9. SILK Frame Reconstruction

 The remainder of the reconstruction process for the frame does not
 need to be bit-exact, as small errors should only introduce
 proportionally small distortions.  Although the reference
 implementation only includes a fixed-point version of the remaining

Valin, et al. Standards Track [Page 98] RFC 6716 Interactive Audio Codec September 2012

 steps, this section describes them in terms of a floating-point
 version for simplicity.  This produces a signal with a nominal range
 of -1.0 to 1.0.
 silk_decode_core() (decode_core.c) contains the code for the main
 reconstruction process.  It proceeds subframe-by-subframe, since
 quantization gains, LTP parameters, and (in 20 ms SILK frames) LPC
 coefficients can vary from one to the next.
 Let a_Q12[k] be the LPC coefficients for the current subframe.  If
 this is the first or second subframe of a 20 ms SILK frame and the
 LSF interpolation factor, w_Q2 (see Section 4.2.7.5.5), is less than
 4, then these correspond to the final LPC coefficients produced by
 Section 4.2.7.5.8 from the interpolated LSF coefficients, n1_Q15[k]
 (computed in Section 4.2.7.5.5).  Otherwise, they correspond to the
 final LPC coefficients produced from the uninterpolated LSF
 coefficients for the current frame, n2_Q15[k].
 Also, let n be the number of samples in a subframe (40 for NB, 60 for
 MB, and 80 for WB), s be the index of the current subframe in this
 SILK frame (0 or 1 for 10 ms frames, or 0 to 3 for 20 ms frames), and
 j be the index of the first sample in the residual corresponding to
 the current subframe.

4.2.7.9.1. LTP Synthesis

 For unvoiced frames (see Section 4.2.7.3), the LPC residual for i
 such that j <= i < (j + n) is simply a normalized copy of the
 excitation signal, i.e.,
                                     e_Q23[i]
                           res[i] = ---------
                                     2.0**23
 Voiced SILK frames, on the other hand, pass the excitation through an
 LTP filter using the parameters decoded in Section 4.2.7.6 to produce
 an LPC residual.  The LTP filter requires LPC residual values from
 before the current subframe as input.  However, since the LPC
 coefficients may have changed, it obtains this residual by
 "rewhitening" the corresponding output signal using the LPC
 coefficients from the current subframe.  Let out[i] for i such that
 (j - pitch_lags[s] - d_LPC - 2) <= i < j be the fully reconstructed
 output signal from the last (pitch_lags[s] + d_LPC + 2) samples of
 previous subframes (see Section 4.2.7.9.2), where pitch_lags[s] is
 the pitch lag for the current subframe from Section 4.2.7.6.1.
 Additionally, let lpc[i] for i such that (j - s*n - d_LPC) <= i < j
 be the fully reconstructed output signal from the last (s*n + d_LPC)

Valin, et al. Standards Track [Page 99] RFC 6716 Interactive Audio Codec September 2012

 samples of previous subframes before clamping (see
 Section 4.2.7.9.2).  During reconstruction of the first subframe for
 this channel after either
 o  An uncoded regular SILK frame (if this is the side channel), or
 o  A decoder reset (see Section 4.5.2),
 out[i] and lpc[i] are initially cleared to all zeros.  If this is the
 third or fourth subframe of a 20 ms SILK frame and the LSF
 interpolation factor, w_Q2 (see Section 4.2.7.5.5), is less than 4,
 then let out_end be set to (j - (s-2)*n) and let LTP_scale_Q14 be set
 to 16384.  Otherwise, set out_end to (j - s*n) and set LTP_scale_Q14
 to the Q14 LTP scaling value from Section 4.2.7.6.3.  Then, for i
 such that (j - pitch_lags[s] - 2) <= i < out_end, out[i] is
 rewhitened into an LPC residual, res[i], via
           4.0*LTP_scale_Q14
  res[i] = ----------------- * clamp(-1.0,
              gain_Q16[s]
                                     d_LPC-1
                                       __              a_Q12[k]
                              out[i] - \  out[i-k-1] * --------, 1.0)
                                       /_               4096.0
                                       k=0
 This requires storage to buffer up to 306 values of out[i] from
 previous subframes.  This corresponds to WB with a maximum pitch lag
 of 18 ms * 16 kHz samples, plus 16 samples for d_LPC, plus 2 samples
 for the width of the LTP filter.  Then, for i such that
 out_end <= i < j, lpc[i] is rewhitened into an LPC residual, res[i],
 via
                                      d_LPC-1
                  65536.0               __              a_Q12[k]
       res[i] = ----------- * (lpc[i] - \  lpc[i-k-1] * --------)
                gain_Q16[s]             /_               4096.0
                                        k=0
 This requires storage to buffer up to 256 values of lpc[i] from
 previous subframes (240 from the current SILK frame and 16 from the
 previous SILK frame).  This corresponds to WB with up to three
 previous subframes in the current SILK frame, plus 16 samples for
 d_LPC.  The astute reader will notice that, given the definition of
 lpc[i] in Section 4.2.7.9.2, the output of this latter equation is
 merely a scaled version of the values of res[i] from previous
 subframes.

Valin, et al. Standards Track [Page 100] RFC 6716 Interactive Audio Codec September 2012

 Let e_Q23[i] for j <= i < (j + n) be the excitation for the current
 subframe, and b_Q7[k] for 0 <= k < 5 be the coefficients of the LTP
 filter taken from the codebook entry in one of Tables 39 through 41
 corresponding to the index decoded for the current subframe in
 Section 4.2.7.6.2.  Then for i such that j <= i < (j + n), the LPC
 residual is
                          4
              e_Q23[i]   __                                  b_Q7[k]
    res[i] = --------- + \  res[i - pitch_lags[s] + 2 - k] * -------
              2.0**23    /_                                   128.0
                         k=0

4.2.7.9.2. LPC Synthesis

 LPC synthesis uses the short-term LPC filter to predict the next
 output coefficient.  For i such that (j - d_LPC) <= i < j, let lpc[i]
 be the result of LPC synthesis from the last d_LPC samples of the
 previous subframe or zeros in the first subframe for this channel
 after either
 o  An uncoded regular SILK frame (if this is the side channel), or
 o  A decoder reset (see Section 4.5.2).
 Then, for i such that j <= i < (j + n), the result of LPC synthesis
 for the current subframe is
                                      d_LPC-1
                 gain_Q16[i]            __              a_Q12[k]
        lpc[i] = ----------- * res[i] + \  lpc[i-k-1] * --------
                   65536.0              /_               4096.0
                                        k=0
 The decoder saves the final d_LPC values, i.e., lpc[i] such that
 (j + n - d_LPC) <= i < (j + n), to feed into the LPC synthesis of the
 next subframe.  This requires storage for up to 16 values of lpc[i]
 (for WB frames).
 Then, the signal is clamped into the final nominal range:
                   out[i] = clamp(-1.0, lpc[i], 1.0)
 This clamping occurs entirely after the LPC synthesis filter has run.
 The decoder saves the unclamped values, lpc[i], to feed into the LPC
 filter for the next subframe, but saves the clamped values, out[i],
 for rewhitening in voiced frames.

Valin, et al. Standards Track [Page 101] RFC 6716 Interactive Audio Codec September 2012

4.2.8. Stereo Unmixing

 For stereo streams, after decoding a frame from each channel, the
 decoder must convert the mid-side (MS) representation into a left-
 right (LR) representation.  The function silk_stereo_MS_to_LR
 (stereo_MS_to_LR.c) implements this process.  In it, the decoder
 predicts the side channel using a) a simple low-passed version of the
 mid channel, and b) the unfiltered mid channel, using the prediction
 weights decoded in Section 4.2.7.1.  This simple low-pass filter
 imposes a one-sample delay, and the unfiltered mid channel is also
 delayed by one sample.  In order to allow seamless switching between
 stereo and mono, mono streams must also impose the same one-sample
 delay.  The encoder requires an additional one-sample delay for both
 mono and stereo streams, though an encoder may omit the delay for
 mono if it knows it will never switch to stereo.
 The unmixing process operates in two phases.  The first phase lasts
 for 8 ms, during which it interpolates the prediction weights from
 the previous frame, prev_w0_Q13 and prev_w1_Q13, to the values for
 the current frame, w0_Q13 and w1_Q13.  The second phase simply uses
 these weights for the remainder of the frame.
 Let mid[i] and side[i] be the contents of out[i] (from
 Section 4.2.7.9.2) for the current mid and side channels,
 respectively, and let left[i] and right[i] be the corresponding
 stereo output channels.  If the side channel is not coded (see
 Section 4.2.7.2), then side[i] is set to zero.  Also, let j be
 defined as in Section 4.2.7.9, n1 be the number of samples in phase 1
 (64 for NB, 96 for MB, and 128 for WB), and n2 be the total number of
 samples in the frame.  Then, for i such that j <= i < (j + n2), the
 left and right channel output is
                 prev_w0_Q13                  (w0_Q13 - prev_w0_Q13)
           w0 =  ----------- + min(i - j, n1)*----------------------
                   8192.0                           8192.0*n1
                 prev_w1_Q13                  (w1_Q13 - prev_w1_Q13)
           w1 =  ----------- + min(i - j, n1)*----------------------
                   8192.0                            8192.0*n1
                mid[i-2] + 2*mid[i-1] + mid[i]
           p0 = ------------------------------
                             4.0
    left[i] = clamp(-1.0, (1 + w1)*mid[i-1] + side[i-1] + w0*p0, 1.0)
   right[i] = clamp(-1.0, (1 - w1)*mid[i-1] - side[i-1] - w0*p0, 1.0)

Valin, et al. Standards Track [Page 102] RFC 6716 Interactive Audio Codec September 2012

 These formulas require two samples prior to index j, the start of the
 frame, for the mid channel, and one prior sample for the side
 channel.  For the first frame after a decoder reset, zeros are used
 instead.

4.2.9. Resampling

 After stereo unmixing (if any), the decoder applies resampling to
 convert the decoded SILK output to the sample rate desired by the
 application.  This is necessary when decoding a Hybrid frame at SWB
 or FB sample rates, or whenever the decoder wants the output at a
 different sample rate than the internal SILK sampling rate (e.g., to
 allow a constant sample rate when the audio bandwidth changes, or to
 allow mixing with audio from other applications).  The resampler
 itself is non-normative, and a decoder can use any method it wants to
 perform the resampling.
 However, a minimum amount of delay is imposed to allow the resampler
 to operate, and this delay is normative, so that the corresponding
 delay can be applied to the MDCT layer in the encoder.  A decoder is
 always free to use a resampler that requires more delay than allowed
 for here (e.g., to improve quality), but it must then delay the
 output of the MDCT layer by this extra amount.  Keeping as much delay
 as possible on the encoder side allows an encoder that knows it will
 never use any of the SILK or Hybrid modes to skip this delay.  By
 contrast, if it were all applied by the decoder, then a decoder that
 processes audio in fixed-size blocks would be forced to delay the
 output of CELT frames just in case of a later switch to a SILK or
 Hybrid mode.
 Table 54 gives the maximum resampler delay in samples at 48 kHz for
 each SILK audio bandwidth.  Because the actual output rate may not be
 48 kHz, it may not be possible to achieve exactly these delays while
 using a whole number of input or output samples.  The reference
 implementation is able to resample to any of the supported output
 sampling rates (8, 12, 16, 24, or 48 kHz) within or near this delay
 constraint.  Some resampling filters (including those used by the
 reference implementation) may add a delay that is not an exact
 integer, or is not linear-phase, and so cannot be represented by a
 single delay at all frequencies.  However, such deviations are
 unlikely to be perceptible, and the comparison tool described in
 Section 6 is designed to be relatively insensitive to them.  The
 delays listed here are the ones that should be targeted by the
 encoder.

Valin, et al. Standards Track [Page 103] RFC 6716 Interactive Audio Codec September 2012

              +-----------------+-----------------------+
              | Audio Bandwidth | Delay in Milliseconds |
              +-----------------+-----------------------+
              | NB              | 0.538                 |
              |                 |                       |
              | MB              | 0.692                 |
              |                 |                       |
              | WB              | 0.706                 |
              +-----------------+-----------------------+
              Table 54: SILK Resampler Delay Allocations
 NB is given a smaller decoder delay allocation than MB and WB to
 allow a higher-order filter when resampling to 8 kHz in both the
 encoder and decoder.  This implies that the audio content of two SILK
 frames operating at different bandwidths is not perfectly aligned in
 time.  This is not an issue for any transitions described in
 Section 4.5, because they all involve a SILK decoder reset.  When the
 decoder is reset, any samples remaining in the resampling buffer are
 discarded, and the resampler is re-initialized with silence.

4.3. CELT Decoder

 The CELT layer of Opus is based on the Modified Discrete Cosine
 Transform [MDCT] with partially overlapping windows of 5 to 22.5 ms.
 The main principle behind CELT is that the MDCT spectrum is divided
 into bands that (roughly) follow the Bark scale, i.e., the scale of
 the ear's critical bands [ZWICKER61].  The normal CELT layer uses 21
 of those bands, though Opus Custom (see Section 6.2) may use a
 different number of bands.  In Hybrid mode, the first 17 bands (up to
 8 kHz) are not coded.  A band can contain as little as one MDCT bin
 per channel, and as many as 176 bins per channel, as detailed in
 Table 55.  In each band, the gain (energy) is coded separately from
 the shape of the spectrum.  Coding the gain explicitly makes it easy
 to preserve the spectral envelope of the signal.  The remaining unit-
 norm shape vector is encoded using a Pyramid Vector Quantizer
 (PVQ) Section 4.3.4.
 +--------+--------+------+-------+-------+-------------+------------+
 | Frame  | 2.5 ms | 5 ms | 10 ms | 20 ms |       Start |       Stop |
 | Size:  |        |      |       |       |   Frequency |  Frequency |
 +--------+--------+------+-------+-------+-------------+------------+
 | Band   |  Bins: |      |       |       |             |            |
 |        |        |      |       |       |             |            |
 | 0      |      1 |    2 |     4 |     8 |        0 Hz |     200 Hz |
 |        |        |      |       |       |             |            |
 | 1      |      1 |    2 |     4 |     8 |      200 Hz |     400 Hz |
 |        |        |      |       |       |             |            |

Valin, et al. Standards Track [Page 104] RFC 6716 Interactive Audio Codec September 2012

 | 2      |      1 |    2 |     4 |     8 |      400 Hz |     600 Hz |
 |        |        |      |       |       |             |            |
 | 3      |      1 |    2 |     4 |     8 |      600 Hz |     800 Hz |
 |        |        |      |       |       |             |            |
 | 4      |      1 |    2 |     4 |     8 |      800 Hz |    1000 Hz |
 |        |        |      |       |       |             |            |
 | 5      |      1 |    2 |     4 |     8 |     1000 Hz |    1200 Hz |
 |        |        |      |       |       |             |            |
 | 6      |      1 |    2 |     4 |     8 |     1200 Hz |    1400 Hz |
 |        |        |      |       |       |             |            |
 | 7      |      1 |    2 |     4 |     8 |     1400 Hz |    1600 Hz |
 |        |        |      |       |       |             |            |
 | 8      |      2 |    4 |     8 |    16 |     1600 Hz |    2000 Hz |
 |        |        |      |       |       |             |            |
 | 9      |      2 |    4 |     8 |    16 |     2000 Hz |    2400 Hz |
 |        |        |      |       |       |             |            |
 | 10     |      2 |    4 |     8 |    16 |     2400 Hz |    2800 Hz |
 |        |        |      |       |       |             |            |
 | 11     |      2 |    4 |     8 |    16 |     2800 Hz |    3200 Hz |
 |        |        |      |       |       |             |            |
 | 12     |      4 |    8 |    16 |    32 |     3200 Hz |    4000 Hz |
 |        |        |      |       |       |             |            |
 | 13     |      4 |    8 |    16 |    32 |     4000 Hz |    4800 Hz |
 |        |        |      |       |       |             |            |
 | 14     |      4 |    8 |    16 |    32 |     4800 Hz |    5600 Hz |
 |        |        |      |       |       |             |            |
 | 15     |      6 |   12 |    24 |    48 |     5600 Hz |    6800 Hz |
 |        |        |      |       |       |             |            |
 | 16     |      6 |   12 |    24 |    48 |     6800 Hz |    8000 Hz |
 |        |        |      |       |       |             |            |
 | 17     |      8 |   16 |    32 |    64 |     8000 Hz |    9600 Hz |
 |        |        |      |       |       |             |            |
 | 18     |     12 |   24 |    48 |    96 |     9600 Hz |   12000 Hz |
 |        |        |      |       |       |             |            |
 | 19     |     18 |   36 |    72 |   144 |    12000 Hz |   15600 Hz |
 |        |        |      |       |       |             |            |
 | 20     |     22 |   44 |    88 |   176 |    15600 Hz |   20000 Hz |
 +--------+--------+------+-------+-------+-------------+------------+
     Table 55: MDCT Bins per Channel per Band for Each Frame Size
 Transients are notoriously difficult for transform codecs to code.
 CELT uses two different strategies for them:
 1.  Using multiple smaller MDCTs instead of a single large MDCT, and
 2.  Dynamic time-frequency resolution changes (See Section 4.3.4.5).

Valin, et al. Standards Track [Page 105] RFC 6716 Interactive Audio Codec September 2012

 To improve quality on highly tonal and periodic signals, CELT
 includes a pre-filter/post-filter combination.  The pre-filter on the
 encoder side attenuates the signal's harmonics.  The post-filter on
 the decoder side restores the original gain of the harmonics, while
 shaping the coding noise to roughly follow the harmonics.  Such noise
 shaping reduces the perception of the noise.
 When coding a stereo signal, three coding methods are available:
 o  mid-side stereo: encodes the mean and the difference of the left
    and right channels,
 o  intensity stereo: only encodes the mean of the left and right
    channels (discards the difference),
 o  dual stereo: encodes the left and right channels separately.
 An overview of the decoder is given in Figure 17.
                     +---------+
                     | Coarse  |
                  +->| decoder |----+
                  |  +---------+    |
                  |                 |
                  |  +---------+    v
                  |  |  Fine   |  +---+
                  +->| decoder |->| + |
                  |  +---------+  +---+
                  |       ^         |
      +---------+ |       |         |
      |  Range  | | +----------+    v
      | Decoder |-+ |   Bit    | +------+
      +---------+ | |Allocation| | 2**x |
                  | +----------+ +------+
                  |       |         |
                  |       v         v               +--------+
                  |  +---------+  +---+  +-------+  | pitch  |
                  +->|   PVQ   |->| * |->| IMDCT |->| post-  |--->
                  |  | decoder |  +---+  +-------+  | filter |
                  |  +---------+                    +--------+
                  |                                      ^
                  +--------------------------------------+
      Legend: IMDCT = Inverse MDCT
               Figure 17: Structure of the CELT decoder
 The decoder is based on the following symbols and sets of symbols:

Valin, et al. Standards Track [Page 106] RFC 6716 Interactive Audio Codec September 2012

        +---------------+---------------------+---------------+
        |   Symbol(s)   |         PDF         |   Condition   |
        +---------------+---------------------+---------------+
        |    silence    |   {32767, 1}/32768  |               |
        |               |                     |               |
        |  post-filter  |       {1, 1}/2      |               |
        |               |                     |               |
        |     octave    |     uniform (6)     |  post-filter  |
        |               |                     |               |
        |     period    | raw bits (4+octave) |  post-filter  |
        |               |                     |               |
        |      gain     |     raw bits (3)    |  post-filter  |
        |               |                     |               |
        |     tapset    |     {2, 1, 1}/4     |  post-filter  |
        |               |                     |               |
        |   transient   |       {7, 1}/8      |               |
        |               |                     |               |
        |     intra     |       {7, 1}/8      |               |
        |               |                     |               |
        | coarse energy |    Section 4.3.2    |               |
        |               |                     |               |
        |   tf_change   |    Section 4.3.1    |               |
        |               |                     |               |
        |   tf_select   |       {1, 1}/2      | Section 4.3.1 |
        |               |                     |               |
        |     spread    |   {7, 2, 21, 2}/32  |               |
        |               |                     |               |
        |  dyn. alloc.  |    Section 4.3.3    |               |
        |               |                     |               |
        |  alloc. trim  |       Table 58      |               |
        |               |                     |               |
        |      skip     |       {1, 1}/2      | Section 4.3.3 |
        |               |                     |               |
        |   intensity   |       uniform       | Section 4.3.3 |
        |               |                     |               |
        |      dual     |       {1, 1}/2      |               |
        |               |                     |               |
        |  fine energy  |    Section 4.3.2    |               |
        |               |                     |               |
        |    residual   |    Section 4.3.4    |               |
        |               |                     |               |
        | anti-collapse |       {1, 1}/2      | Section 4.3.5 |
        |               |                     |               |
        |    finalize   |    Section 4.3.2    |               |
        +---------------+---------------------+---------------+
  Table 56: Order of the Symbols in the CELT Section of the Bitstream

Valin, et al. Standards Track [Page 107] RFC 6716 Interactive Audio Codec September 2012

 The decoder extracts information from the range-coded bitstream in
 the order described in Table 56.  In some circumstances, it is
 possible for a decoded value to be out of range due to a very small
 amount of redundancy in the encoding of large integers by the range
 coder.  In that case, the decoder should assume there has been an
 error in the coding, decoding, or transmission and SHOULD take
 measures to conceal the error and/or report to the application that a
 problem has occurred.  Such out of range errors cannot occur in the
 SILK layer.

4.3.1. Transient Decoding

 The "transient" flag indicates whether the frame uses a single long
 MDCT or several short MDCTs.  When it is set, then the MDCT
 coefficients represent multiple short MDCTs in the frame.  When not
 set, the coefficients represent a single long MDCT for the frame.
 The flag is encoded in the bitstream with a probability of 1/8.  In
 addition to the global transient flag is a per-band binary flag to
 change the time-frequency (tf) resolution independently in each band.
 The change in tf resolution is defined in tf_select_table[][] in
 celt.c and depends on the frame size, whether the transient flag is
 set, and the value of tf_select.  The tf_select flag uses a 1/2
 probability, but is only decoded if it can have an impact on the
 result knowing the value of all per-band tf_change flags.

4.3.2. Energy Envelope Decoding

 It is important to quantize the energy with sufficient resolution
 because any energy quantization error cannot be compensated for at a
 later stage.  Regardless of the resolution used for encoding the
 spectral shape of a band, it is perceptually important to preserve
 the energy in each band.  CELT uses a three-step coarse-fine-fine
 strategy for encoding the energy in the base-2 log domain, as
 implemented in quant_bands.c.

4.3.2.1. Coarse Energy Decoding

 Coarse quantization of the energy uses a fixed resolution of 6 dB
 (integer part of base-2 log).  To minimize the bitrate, prediction is
 applied both in time (using the previous frame) and in frequency
 (using the previous bands).  The part of the prediction that is based
 on the previous frame can be disabled, creating an "intra" frame
 where the energy is coded without reference to prior frames.  The
 decoder first reads the intra flag to determine what prediction is
 used.  The 2-D z-transform [Z-TRANSFORM] of the prediction filter is

Valin, et al. Standards Track [Page 108] RFC 6716 Interactive Audio Codec September 2012

  1. 1 -1

(1 - alpha*z_l )*(1 - z_b )

              A(z_l, z_b) = -----------------------------
                                               -1
                                   1 - beta*z_b
 where b is the band index and l is the frame index.  The prediction
 coefficients applied depend on the frame size in use when not using
 intra energy and are alpha=0, beta=4915/32768 when using intra
 energy.  The time-domain prediction is based on the final fine
 quantization of the previous frame, while the frequency domain
 (within the current frame) prediction is based on coarse quantization
 only (because the fine quantization has not been computed yet).  The
 prediction is clamped internally so that fixed-point implementations
 with limited dynamic range always remain in the same state as
 floating point implementations.  We approximate the ideal probability
 distribution of the prediction error using a Laplace distribution
 with separate parameters for each frame size in intra- and inter-
 frame modes.  These parameters are held in the e_prob_model table in
 quant_bands.c.  The coarse energy decoding is performed by
 unquant_coarse_energy() (quant_bands.c).  The decoding of the
 Laplace-distributed values is implemented in ec_laplace_decode()
 (laplace.c).

4.3.2.2. Fine Energy Quantization

 The number of bits assigned to fine energy quantization in each band
 is determined by the bit allocation computation described in
 Section 4.3.3.  Let B_i be the number of fine energy bits for band i;
 the refinement is an integer f in the range [0,2**B_i-1].  The
 mapping between f and the correction applied to the coarse energy is
 equal to (f+1/2)/2**B_i - 1/2.  Fine energy quantization is
 implemented in quant_fine_energy() (quant_bands.c).
 When some bits are left "unused" after all other flags have been
 decoded, these bits are assigned to a "final" step of fine
 allocation.  In effect, these bits are used to add one extra fine
 energy bit per band per channel.  The allocation process determines
 two "priorities" for the final fine bits.  Any remaining bits are
 first assigned only to bands of priority 0, starting from band 0 and
 going up.  If all bands of priority 0 have received one bit per
 channel, then bands of priority 1 are assigned an extra bit per
 channel, starting from band 0.  If any bits are left after this, they
 are left unused.  This is implemented in unquant_energy_finalise()
 (quant_bands.c).

Valin, et al. Standards Track [Page 109] RFC 6716 Interactive Audio Codec September 2012

4.3.3. Bit Allocation

 Because the bit allocation drives the decoding of the range-coder
 stream, it MUST be recovered exactly so that identical coding
 decisions are made in the encoder and decoder.  Any deviation from
 the reference's resulting bit allocation will result in corrupted
 output, though implementers are free to implement the procedure in
 any way that produces identical results.
 The per-band gain-shape structure of the CELT layer ensures that
 using the same number of bits for the spectral shape of a band in
 every frame will result in a roughly constant signal-to-noise ratio
 in that band.  This results in coding noise that has the same
 spectral envelope as the signal.  The masking curve produced by a
 standard psychoacoustic model also closely follows the spectral
 envelope of the signal.  This structure means that the ideal
 allocation is more consistent from frame to frame than it is for
 other codecs without an equivalent structure and that a fixed
 allocation provides fairly consistent perceptual
 performance [VALIN2010].
 Many codecs transmit significant amounts of side information to
 control the bit allocation within a frame.  Often this control is
 only indirect, and it must be exercised carefully to achieve the
 desired rate constraints.  The CELT layer, however, can adapt over a
 very wide range of rates, so it has a large number of codebook sizes
 to choose from for each band.  Explicitly signaling the size of each
 of these codebooks would impose considerable overhead, even though
 the allocation is relatively static from frame to frame.  This is
 because all of the information required to compute these codebook
 sizes must be derived from a single frame by itself, in order to
 retain robustness to packet loss, so the signaling cannot take
 advantage of knowledge of the allocation in neighboring frames.  This
 problem is exacerbated in low-latency (small frame size)
 applications, which would include this overhead in every frame.
 For this reason, in the MDCT mode, Opus uses a primarily implicit bit
 allocation.  The available bitstream capacity is known in advance to
 both the encoder and decoder without additional signaling, ultimately
 from the packet sizes expressed by a higher-level protocol.  Using
 this information, the codec interpolates an allocation from a hard-
 coded table.
 While the band-energy structure effectively models intra-band
 masking, it ignores the weaker inter-band masking, band-temporal
 masking, and other less significant perceptual effects.  While these
 effects can often be ignored, they can become significant for
 particular samples.  One mechanism available to encoders would be to

Valin, et al. Standards Track [Page 110] RFC 6716 Interactive Audio Codec September 2012

 simply increase the overall rate for these frames, but this is not
 possible in a constant rate mode and can be fairly inefficient.  As a
 result three explicitly signaled mechanisms are provided to alter the
 implicit allocation:
 o  Band boost
 o  Allocation trim
 o  Band skipping
 The first of these mechanisms, band boost, allows an encoder to boost
 the allocation in specific bands.  The second, allocation trim, works
 by biasing the overall allocation towards higher or lower frequency
 bands.  The third, band skipping, selects which low-precision high
 frequency bands will be allocated no shape bits at all.
 In stereo mode, there are two additional parameters potentially coded
 as part of the allocation procedure: a parameter to allow the
 selective elimination of allocation for the 'side' (i.e., intensity
 stereo) in jointly coded bands, and a flag to deactivate joint coding
 (i.e., dual stereo).  These values are not signaled if they would be
 meaningless in the overall context of the allocation.
 Because every signaled adjustment increases overhead and
 implementation complexity, none were included speculatively: the
 reference encoder makes use of all of these mechanisms.  While the
 decision logic in the reference was found to be effective enough to
 justify the overhead and complexity, further analysis techniques may
 be discovered that increase the effectiveness of these parameters.
 As with other signaled parameters, an encoder is free to choose the
 values in any manner, but, unless a technique is known to deliver
 superior perceptual results, the methods used by the reference
 implementation should be used.
 The allocation process consists of the following steps: determining
 the per-band maximum allocation vector, decoding the boosts, decoding
 the tilt, determining the remaining capacity of the frame, searching
 the mode table for the entry nearest but not exceeding the available
 space (subject to the tilt, boosts, band maximums, and band
 minimums), linear interpolation, reallocation of unused bits with
 concurrent skip decoding, determination of the fine-energy vs. shape
 split, and final reallocation.  This process results in a per-band
 shape allocation (in 1/8th-bit units), a per-band fine-energy
 allocation (in 1 bit per channel units), a set of band priorities for
 controlling the use of remaining bits at the end of the frame, and a
 remaining balance of unallocated space, which is usually zero except
 at very high rates.

Valin, et al. Standards Track [Page 111] RFC 6716 Interactive Audio Codec September 2012

 The "static" bit allocation (in 1/8 bits) for a quality q, excluding
 the minimums, maximums, tilt and boosts, is equal to
 channels*N*alloc[band][q]<<LM>>2, where alloc[][] is given in
 Table 57 and LM=log2(frame_size/120).  The allocation is obtained by
 linearly interpolating between two values of q (in steps of 1/64) to
 find the highest allocation that does not exceed the number of bits
 remaining.
  Rows indicate the MDCT bands, columns are the different quality (q)
           parameters.  The units are 1/32 bit per MDCT bin.
   +---+----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   | 0 |  1 |   2 |   3 |   4 |   5 |   6 |   7 |   8 |   9 |  10 |
   +---+----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   | 0 | 90 | 110 | 118 | 126 | 134 | 144 | 152 | 162 | 172 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 80 | 100 | 110 | 119 | 127 | 137 | 145 | 155 | 165 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 75 |  90 | 103 | 112 | 120 | 130 | 138 | 148 | 158 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 69 |  84 |  93 | 104 | 114 | 124 | 132 | 142 | 152 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 63 |  78 |  86 |  95 | 103 | 113 | 123 | 133 | 143 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 56 |  71 |  80 |  89 |  97 | 107 | 117 | 127 | 137 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 49 |  65 |  75 |  83 |  91 | 101 | 111 | 121 | 131 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 40 |  58 |  70 |  78 |  85 |  95 | 105 | 115 | 125 | 200 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 34 |  51 |  65 |  72 |  78 |  88 |  98 | 108 | 118 | 198 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 29 |  45 |  59 |  66 |  72 |  82 |  92 | 102 | 112 | 193 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 20 |  39 |  53 |  60 |  66 |  76 |  86 |  96 | 106 | 188 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 18 |  32 |  47 |  54 |  60 |  70 |  80 |  90 | 100 | 183 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 | 10 |  26 |  40 |  47 |  54 |  64 |  74 |  84 |  94 | 178 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |  20 |  31 |  39 |  47 |  57 |  67 |  77 |  87 | 173 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |  12 |  23 |  32 |  41 |  51 |  61 |  71 |  81 | 168 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |   0 |  15 |  25 |  35 |  45 |  55 |  65 |  75 | 163 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |   0 |   4 |  17 |  29 |  39 |  49 |  59 |  69 | 158 |
   |   |    |     |     |     |     |     |     |     |     |     |

Valin, et al. Standards Track [Page 112] RFC 6716 Interactive Audio Codec September 2012

   | 0 |  0 |   0 |   0 |  12 |  23 |  33 |  43 |  53 |  63 | 153 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |   0 |   0 |   1 |  16 |  26 |  36 |  46 |  56 | 148 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |   0 |   0 |   0 |  10 |  15 |  20 |  30 |  45 | 129 |
   |   |    |     |     |     |     |     |     |     |     |     |
   | 0 |  0 |   0 |   0 |   0 |   1 |   1 |   1 |   1 |  20 | 104 |
   +---+----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
                Table 57: CELT Static Allocation Table
 The maximum allocation vector is an approximation of the maximum
 space that can be used by each band for a given mode.  The value is
 approximate because the shape encoding is variable rate (due to
 entropy coding of splitting parameters).  Setting the maximum too low
 reduces the maximum achievable quality in a band while setting it too
 high may result in waste: bitstream capacity available at the end of
 the frame that cannot be put to any use.  The maximums specified by
 the codec reflect the average maximum.  In the reference
 implementation, the maximums in bits/sample are precomputed in a
 static table (see cache_caps50[] in static_modes_float.h) for each
 band, for each value of LM, and for both mono and stereo.
 Implementations are expected to simply use the same table data, but
 the procedure for generating this table is included in rate.c as part
 of compute_pulse_cache().
 To convert the values in cache.caps into the actual maximums: first,
 set nbBands to the maximum number of bands for this mode, and stereo
 to zero if stereo is not in use and one otherwise.  For each band,
 set N to the number of MDCT bins covered by the band (for one
 channel), set LM to the shift value for the frame size.  Then, set i
 to nbBands*(2*LM+stereo).  Next, set the maximum for the band to the
 i-th index of cache.caps + 64 and multiply by the number of channels
 in the current frame (one or two) and by N, then divide the result by
 4 using integer division.  The resulting vector will be called cap[].
 The elements fit in signed 16-bit integers but do not fit in 8 bits.
 This procedure is implemented in the reference in the function
 init_caps() in celt.c.
 The band boosts are represented by a series of binary symbols that
 are entropy coded with very low probability.  Each band can
 potentially be boosted multiple times, subject to the frame actually
 having enough room to obey the boost and having enough room to code
 the boost symbol.  The default coding cost for a boost starts out at
 six bits (probability p=1/64), but subsequent boosts in a band cost
 only a single bit and every time a band is boosted the initial cost
 is reduced (down to a minimum of two bits, or p=1/4).  Since the

Valin, et al. Standards Track [Page 113] RFC 6716 Interactive Audio Codec September 2012

 initial cost of coding a boost is 6 bits, the coding cost of the
 boost symbols when completely unused is 0.48 bits/frame for a 21 band
 mode (21*-log2(1-1/2**6)).
 To decode the band boosts: First, set 'dynalloc_logp' to 6, the
 initial amount of storage required to signal a boost in bits,
 'total_bits' to the size of the frame in 8th bits, 'total_boost' to
 zero, and 'tell' to the total number of 8th bits decoded so far.  For
 each band from the coding start (0 normally, but 17 in Hybrid mode)
 to the coding end (which changes depending on the signaled
 bandwidth), the boost quanta in units of 1/8 bit is calculated as
 quanta = min(8*N, max(48, N)).  This represents a boost step size of
 six bits, subject to a lower limit of 1/8th bit/sample and an upper
 limit of 1 bit/sample.  Set 'boost' to zero and 'dynalloc_loop_logp'
 to dynalloc_logp.  While dynalloc_loop_log (the current worst case
 symbol cost) in 8th bits plus tell is less than total_bits plus
 total_boost and boost is less than cap[] for this band: Decode a bit
 from the bitstream with dynalloc_loop_logp as the cost of a one and
 update tell to reflect the current used capacity.  If the decoded
 value is zero break the loop.  Otherwise, add quanta to boost and
 total_boost, subtract quanta from total_bits, and set
 dynalloc_loop_log to 1.  When the loop finishes 'boost' contains the
 bit allocation boost for this band.  If boost is non-zero and
 dynalloc_logp is greater than 2, decrease dynalloc_logp.  Once this
 process has been executed on all bands, the band boosts have been
 decoded.  This procedure is implemented around line 2474 of celt.c.
 At very low rates, it is possible that there won't be enough
 available space to execute the inner loop even once.  In these cases,
 band boost is not possible, but its overhead is completely
 eliminated.  Because of the high cost of band boost when activated, a
 reasonable encoder should not be using it at very low rates.  The
 reference implements its dynalloc decision logic around line 1304 of
 celt.c.
 The allocation trim is an integer value from 0-10.  The default value
 of 5 indicates no trim.  The trim parameter is entropy coded in order
 to lower the coding cost of less extreme adjustments.  Values lower
 than 5 bias the allocation towards lower frequencies and values above
 5 bias it towards higher frequencies.  Like other signaled
 parameters, signaling of the trim is gated so that it is not included
 if there is insufficient space available in the bitstream.  To decode
 the trim, first set the trim value to 5, then if and only if the
 count of decoded 8th bits so far (ec_tell_frac) plus 48 (6 bits) is
 less than or equal to the total frame size in 8th bits minus
 total_boost (a product of the above band boost procedure), decode the
 trim value using the PDF in Table 58.

Valin, et al. Standards Track [Page 114] RFC 6716 Interactive Audio Codec September 2012

            +--------------------------------------------+
            | PDF                                        |
            +--------------------------------------------+
            | {2, 2, 5, 10, 22, 46, 22, 10, 5, 2, 2}/128 |
            +--------------------------------------------+
                      Table 58: PDF for the Trim
 For 10 ms and 20 ms frames using short blocks and that have at least
 LM+2 bits left prior to the allocation process, one anti-collapse bit
 is reserved in the allocation process so it can be decoded later.
 Following the anti-collapse reservation, one bit is reserved for skip
 if available.
 For stereo frames, bits are reserved for intensity stereo and for
 dual stereo.  Intensity stereo requires ilog2(end-start) bits.  Those
 bits are reserved if there are enough bits left.  Following this, one
 bit is reserved for dual stereo if available.
 The allocation computation begins by setting up some initial
 conditions. 'total' is set to the remaining available 8th bits,
 computed by taking the size of the coded frame times 8 and
 subtracting ec_tell_frac().  From this value, one (8th bit) is
 subtracted to ensure that the resulting allocation will be
 conservative. 'anti_collapse_rsv' is set to 8 (8th bits) if and only
 if the frame is a transient, LM is greater than 1, and total is
 greater than or equal to (LM+2) * 8.  Total is then decremented by
 anti_collapse_rsv and clamped to be equal to or greater than zero.
 'skip_rsv' is set to 8 (8th bits) if total is greater than 8,
 otherwise it is zero.  Total is then decremented by skip_rsv.  This
 reserves space for the final skipping flag.
 If the current frame is stereo, intensity_rsv is set to the
 conservative log2 in 8th bits of the number of coded bands for this
 frame (given by the table LOG2_FRAC_TABLE in rate.c).  If
 intensity_rsv is greater than total, then intensity_rsv is set to
 zero.  Otherwise, total is decremented by intensity_rsv, and if total
 is still greater than 8, dual_stereo_rsv is set to 8 and total is
 decremented by dual_stereo_rsv.
 The allocation process then computes a vector representing the hard
 minimum amounts allocation any band will receive for shape.  This
 minimum is higher than the technical limit of the PVQ process, but
 very low rate allocations produce an excessively sparse spectrum and
 these bands are better served by having no allocation at all.  For
 each coded band, set thresh[band] to 24 times the number of MDCT bins
 in the band and divide by 16.  If 8 times the number of channels is
 greater, use that instead.  This sets the minimum allocation to one

Valin, et al. Standards Track [Page 115] RFC 6716 Interactive Audio Codec September 2012

 bit per channel or 48 128th bits per MDCT bin, whichever is greater.
 The band-size dependent part of this value is not scaled by the
 channel count, because at the very low rates where this limit is
 applicable there will usually be no bits allocated to the side.
 The previously decoded allocation trim is used to derive a vector of
 per-band adjustments, 'trim_offsets[]'.  For each coded band take the
 alloc_trim and subtract 5 and LM.  Then, multiply the result by the
 number of channels, the number of MDCT bins in the shortest frame
 size for this mode, the number of remaining bands, 2**LM, and 8.
 Next, divide this value by 64.  Finally, if the number of MDCT bins
 in the band per channel is only one, 8 times the number of channels
 is subtracted in order to diminish the allocation by one bit, because
 width 1 bands receive greater benefit from the coarse energy coding.

4.3.4. Shape Decoding

 In each band, the normalized "shape" is encoded using Pyramid Vector
 Quantizer.
 In the simplest case, the number of bits allocated in Section 4.3.3
 is converted to a number of pulses as described by Section 4.3.4.1.
 Knowing the number of pulses and the number of samples in the band,
 the decoder calculates the size of the codebook as detailed in
 Section 4.3.4.2.  The size is used to decode an unsigned integer
 (uniform probability model), which is the codeword index.  This index
 is converted into the corresponding vector as explained in
 Section 4.3.4.2.  This vector is then scaled to unit norm.

4.3.4.1. Bits to Pulses

 Although the allocation is performed in 1/8th bit units, the
 quantization requires an integer number of pulses K.  To do this, the
 encoder searches for the value of K that produces the number of bits
 nearest to the allocated value (rounding down if exactly halfway
 between two values), not to exceed the total number of bits
 available.  For efficiency reasons, the search is performed against a
 precomputed allocation table that only permits some K values for each
 N.  The number of codebook entries can be computed as explained in
 Section 4.3.4.2.  The difference between the number of bits allocated
 and the number of bits used is accumulated to a "balance"
 (initialized to zero) that helps adjust the allocation for the next
 bands.  One third of the balance is applied to the bit allocation of
 each band to help achieve the target allocation.  The only exceptions
 are the band before the last and the last band, for which half the
 balance and the whole balance are applied, respectively.

Valin, et al. Standards Track [Page 116] RFC 6716 Interactive Audio Codec September 2012

4.3.4.2. PVQ Decoding

 Decoding of PVQ vectors is implemented in decode_pulses() (cwrs.c).
 The unique codeword index is decoded as a uniformly distributed
 integer value between 0 and V(N,K)-1, where V(N,K) is the number of
 possible combinations of K pulses in N samples.  The index is then
 converted to a vector in the same way specified in [PVQ].  The
 indexing is based on the calculation of V(N,K) (denoted N(L,K) in
 [PVQ]).
 The number of combinations can be computed recursively as V(N,K) =
 V(N-1,K) + V(N,K-1) + V(N-1,K-1), with V(N,0) = 1 and V(0,K) = 0, K
 != 0.  There are many different ways to compute V(N,K), including
 precomputed tables and direct use of the recursive formulation.  The
 reference implementation applies the recursive formulation one line
 (or column) at a time to save on memory use, along with an alternate,
 univariate recurrence to initialize an arbitrary line, and direct
 polynomial solutions for small N.  All of these methods are
 equivalent, and have different trade-offs in speed, memory usage, and
 code size.  Implementations MAY use any methods they like, as long as
 they are equivalent to the mathematical definition.
 The decoded vector X is recovered as follows.  Let i be the index
 decoded with the procedure in Section 4.1.5 with ft = V(N,K), so that
 0 <= i < V(N,K).  Let k = K.  Then, for j = 0 to (N - 1), inclusive,
 do:
 1.  Let p = (V(N-j-1,k) + V(N-j,k))/2.
 2.  If i < p, then let sgn = 1, else let sgn = -1 and set i = i - p.
 3.  Let k0 = k and set p = p - V(N-j-1,k).
 4.  While p > i, set k = k - 1 and p = p - V(N-j-1,k).
 5.  Set X[j] = sgn*(k0 - k) and i = i - p.
 The decoded vector X is then normalized such that its L2-norm equals
 one.

4.3.4.3. Spreading

 The normalized vector decoded in Section 4.3.4.2 is then rotated for
 the purpose of avoiding tonal artifacts.  The rotation gain is equal
 to
                         g_r = N / (N + f_r*K)

Valin, et al. Standards Track [Page 117] RFC 6716 Interactive Audio Codec September 2012

 where N is the number of dimensions, K is the number of pulses, and
 f_r depends on the value of the "spread" parameter in the bitstream.
               +--------------+------------------------+
               | Spread value | f_r                    |
               +--------------+------------------------+
               | 0            | infinite (no rotation) |
               |              |                        |
               | 1            | 15                     |
               |              |                        |
               | 2            | 10                     |
               |              |                        |
               | 3            | 5                      |
               +--------------+------------------------+
                      Table 59: Spreading Values
 The rotation angle is then calculated as
                                            2
                                   pi *  g_r
                           theta = ----------
                                       4
 A 2-D rotation R(i,j) between points x_i and x_j is defined as:
                x_i' =  cos(theta)*x_i + sin(theta)*x_j
                x_j' = -sin(theta)*x_i + cos(theta)*x_j
 An N-D rotation is then achieved by applying a series of 2-D
 rotations back and forth, in the following order: R(x_1, x_2), R(x_2,
 x_3), ..., R(x_N-2, X_N-1), R(x_N-1, X_N), R(x_N-2, X_N-1), ...,
 R(x_1, x_2).
 If the decoded vector represents more than one time block, then this
 spreading process is applied separately on each time block.  Also, if
 each block represents 8 samples or more, then another N-D rotation,
 by (pi/2-theta), is applied _before_ the rotation described above.
 This extra rotation is applied in an interleaved manner with a stride
 equal to round(sqrt(N/nb_blocks)), i.e., it is applied independently
 for each set of sample S_k = {stride*n + k}, n=0..N/stride-1.

4.3.4.4. Split Decoding

 To avoid the need for multi-precision calculations when decoding PVQ
 codevectors, the maximum size allowed for codebooks is 32 bits.  When
 larger codebooks are needed, the vector is instead split in two sub-
 vectors of size N/2.  A quantized gain parameter with precision

Valin, et al. Standards Track [Page 118] RFC 6716 Interactive Audio Codec September 2012

 derived from the current allocation is entropy coded to represent the
 relative gains of each side of the split, and the entire decoding
 process is recursively applied.  Multiple levels of splitting may be
 applied up to a limit of LM+1 splits.  The same recursive mechanism
 is applied for the joint coding of stereo audio.

4.3.4.5. Time-Frequency Change

 The time-frequency (TF) parameters are used to control the time-
 frequency resolution trade-off in each coded band.  For each band,
 there are two possible TF choices.  For the first band coded, the PDF
 is {3, 1}/4 for frames marked as transient and {15, 1}/16 for the
 other frames.  For subsequent bands, the TF choice is coded relative
 to the previous TF choice with probability {15, 1}/16 for transient
 frames and {31, 1}/32 otherwise.  The mapping between the decoded TF
 choices and the adjustment in TF resolution is shown in the tables
 below.
                     +-----------------+---+----+
                     | Frame size (ms) | 0 |  1 |
                     +-----------------+---+----+
                     |       2.5       | 0 | -1 |
                     |                 |   |    |
                     |        5        | 0 | -1 |
                     |                 |   |    |
                     |        10       | 0 | -2 |
                     |                 |   |    |
                     |        20       | 0 | -2 |
                     +-----------------+---+----+
   Table 60: TF Adjustments for Non-transient Frames and tf_select=0
                     +-----------------+---+----+
                     | Frame size (ms) | 0 |  1 |
                     +-----------------+---+----+
                     |       2.5       | 0 | -1 |
                     |                 |   |    |
                     |        5        | 0 | -2 |
                     |                 |   |    |
                     |        10       | 0 | -3 |
                     |                 |   |    |
                     |        20       | 0 | -3 |
                     +-----------------+---+----+
   Table 61: TF Adjustments for Non-transient Frames and tf_select=1

Valin, et al. Standards Track [Page 119] RFC 6716 Interactive Audio Codec September 2012

                     +-----------------+---+----+
                     | Frame size (ms) | 0 |  1 |
                     +-----------------+---+----+
                     |       2.5       | 0 | -1 |
                     |                 |   |    |
                     |        5        | 1 |  0 |
                     |                 |   |    |
                     |        10       | 2 |  0 |
                     |                 |   |    |
                     |        20       | 3 |  0 |
                     +-----------------+---+----+
     Table 62: TF Adjustments for Transient Frames and tf_select=0
                     +-----------------+---+----+
                     | Frame size (ms) | 0 |  1 |
                     +-----------------+---+----+
                     |       2.5       | 0 | -1 |
                     |                 |   |    |
                     |        5        | 1 | -1 |
                     |                 |   |    |
                     |        10       | 1 | -1 |
                     |                 |   |    |
                     |        20       | 1 | -1 |
                     +-----------------+---+----+
     Table 63: TF Adjustments for Transient Frames and tf_select=1
 A negative TF adjustment means that the temporal resolution is
 increased, while a positive TF adjustment means that the frequency
 resolution is increased.  Changes in TF resolution are implemented
 using the Hadamard transform [HADAMARD].  To increase the time
 resolution by N, N "levels" of the Hadamard transform are applied to
 the decoded vector for each interleaved MDCT vector.  To increase the
 frequency resolution (assumes a transient frame), then N levels of
 the Hadamard transform are applied _across_ the interleaved MDCT
 vector.  In the case of increased time resolution, the decoder uses
 the "sequency order" because the input vector is sorted in time.

4.3.5. Anti-collapse Processing

 The anti-collapse feature is designed to avoid the situation where
 the use of multiple short MDCTs causes the energy in one or more of
 the MDCTs to be zero for some bands, causing unpleasant artifacts.
 When the frame has the transient bit set, an anti-collapse bit is
 decoded.  When anti-collapse is set, the energy in each small MDCT is
 prevented from collapsing to zero.  For each band of each MDCT where
 a collapse is detected, a pseudo-random signal is inserted with an

Valin, et al. Standards Track [Page 120] RFC 6716 Interactive Audio Codec September 2012

 energy corresponding to the minimum energy over the two previous
 frames.  A renormalization step is then required to ensure that the
 anti-collapse step did not alter the energy preservation property.

4.3.6. Denormalization

 Just as each band was normalized in the encoder, the last step of the
 decoder before the inverse MDCT is to denormalize the bands.  Each
 decoded normalized band is multiplied by the square root of the
 decoded energy.  This is done by denormalise_bands() (bands.c).

4.3.7. Inverse MDCT

 The inverse MDCT implementation has no special characteristics.  The
 input is N frequency-domain samples and the output is 2*N time-domain
 samples, while scaling by 1/2.  A "low-overlap" window reduces the
 algorithmic delay.  It is derived from a basic (full-overlap) 240-
 sample version of the window used by the Vorbis codec:
                                                       2
                        /   /pi      /pi   n + 1/2\ \ \
                 W(n) = |sin|-- * sin|-- * -------| | |
                        \   \2       \2       L   / / /
 The low-overlap window is created by zero-padding the basic window
 and inserting ones in the middle, such that the resulting window
 still satisfies power complementarity [PRINCEN86].  The IMDCT and
 windowing are performed by mdct_backward (mdct.c).

4.3.7.1. Post-Filter

 The output of the inverse MDCT (after weighted overlap-add) is sent
 to the post-filter.  Although the post-filter is applied at the end,
 the post-filter parameters are encoded at the beginning, just after
 the silence flag.  The post-filter can be switched on or off using
 one bit (logp=1).  If the post-filter is enabled, then the octave is
 decoded as an integer value between 0 and 6 of uniform probability.
 Once the octave is known, the fine pitch within the octave is decoded
 using 4+octave raw bits.  The final pitch period is equal to
 (16<<octave)+fine_pitch-1 so it is bounded between 15 and 1022,
 inclusively.  Next, the gain is decoded as three raw bits and is
 equal to G=3*(int_gain+1)/32.  The set of post-filter taps is decoded
 last, using a pdf equal to {2, 1, 1}/4.  Tapset zero corresponds to
 the filter coefficients g0 = 0.3066406250, g1 = 0.2170410156, g2 =
 0.1296386719.  Tapset one corresponds to the filter coefficients g0 =
 0.4638671875, g1 = 0.2680664062, g2 = 0, and tapset two uses filter
 coefficients g0 = 0.7998046875, g1 = 0.1000976562, g2 = 0.

Valin, et al. Standards Track [Page 121] RFC 6716 Interactive Audio Codec September 2012

 The post-filter response is thus computed as:
           y(n) = x(n) + G*(g0*y(n-T) + g1*(y(n-T+1)+y(n-T+1))
                                      + g2*(y(n-T+2)+y(n-T+2)))
 During a transition between different gains, a smooth transition is
 calculated using the square of the MDCT window.  It is important that
 values of y(n) be interpolated one at a time such that the past value
 of y(n) used is interpolated.

4.3.7.2. De-emphasis

 After the post-filter, the signal is de-emphasized using the inverse
 of the pre-emphasis filter used in the encoder:
                          1            1
                         ---- = ---------------
                         A(z)                -1
                                1 - alpha_p*z
 where alpha_p=0.8500061035.

4.4. Packet Loss Concealment (PLC)

 Packet Loss Concealment (PLC) is an optional decoder-side feature
 that SHOULD be included when receiving from an unreliable channel.
 Because PLC is not part of the bitstream, there are many acceptable
 ways to implement PLC with different complexity/quality trade-offs.
 The PLC in the reference implementation depends on the mode of last
 packet received.  In CELT mode, the PLC finds a periodicity in the
 decoded signal and repeats the windowed waveform using the pitch
 offset.  The windowed waveform is overlapped in such a way as to
 preserve the time-domain aliasing cancellation with the previous
 frame and the next frame.  This is implemented in celt_decode_lost()
 (mdct.c).  In SILK mode, the PLC uses LPC extrapolation from the
 previous frame, implemented in silk_PLC() (PLC.c).

4.4.1. Clock Drift Compensation

 Clock drift refers to the gradual desynchronization of two endpoints
 whose sample clocks run at different frequencies while they are
 streaming live audio.  Differences in clock frequencies are generally
 attributable to manufacturing variation in the endpoints' clock
 hardware.  For long-lived streams, the time difference between sender
 and receiver can grow without bound.

Valin, et al. Standards Track [Page 122] RFC 6716 Interactive Audio Codec September 2012

 When the sender's clock runs slower than the receiver's, the effect
 is similar to packet loss: too few packets are received.  The
 receiver can distinguish between drift and loss if the transport
 provides packet timestamps.  A receiver for live streams SHOULD
 conceal the effects of drift, and it MAY do so by invoking the PLC.
 When the sender's clock runs faster than the receiver's, too many
 packets will be received.  The receiver MAY respond by skipping any
 packet (i.e., not submitting the packet for decoding).  This is
 likely to produce a less severe artifact than if the frame were
 dropped after decoding.
 A decoder MAY employ a more sophisticated drift compensation method.
 For example, the NetEQ component [GOOGLE-NETEQ] of the Google WebRTC
 codebase [GOOGLE-WEBRTC] compensates for drift by adding or removing
 one period when the signal is highly periodic.  The reference
 implementation of Opus allows a caller to learn whether the current
 frame's signal is highly periodic, and if so what the period is,
 using the OPUS_GET_PITCH() request.

4.5. Configuration Switching

 Switching between the Opus coding modes, audio bandwidths, and
 channel counts requires careful consideration to avoid audible
 glitches.  Switching between any two configurations of the CELT-only
 mode, any two configurations of the Hybrid mode, or from WB SILK to
 Hybrid mode does not require any special treatment in the decoder, as
 the MDCT overlap will smooth the transition.  Switching from Hybrid
 mode to WB SILK requires adding in the final contents of the CELT
 overlap buffer to the first SILK-only packet.  This can be done by
 decoding a 2.5 ms silence frame with the CELT decoder using the
 channel count of the SILK-only packet (and any choice of audio
 bandwidth), which will correctly handle the cases when the channel
 count changes as well.
 When changing the channel count for SILK-only or Hybrid packets, the
 encoder can avoid glitches by smoothly varying the stereo width of
 the input signal before or after the transition, and it SHOULD do so.
 However, other transitions between SILK-only packets or between NB or
 MB SILK and Hybrid packets may cause glitches, because neither the
 LSF coefficients nor the LTP, LPC, stereo unmixing, and resampler
 buffers are available at the new sample rate.  These switches SHOULD
 be delayed by the encoder until quiet periods or transients, where
 the inevitable glitches will be less audible.  Additionally, the
 bitstream MAY include redundant side information ("redundancy"), in
 the form of additional CELT frames embedded in each of the Opus
 frames around the transition.

Valin, et al. Standards Track [Page 123] RFC 6716 Interactive Audio Codec September 2012

 The other transitions that cannot be easily handled are those where
 the lower frequencies switch between the SILK LP-based model and the
 CELT MDCT model.  However, an encoder may not have an opportunity to
 delay such a switch to a convenient point.  For example, if the
 content switches from speech to music, and the encoder does not have
 enough latency in its analysis to detect this in advance, there may
 be no convenient silence period during which to make the transition
 for quite some time.  To avoid or reduce glitches during these
 problematic mode transitions, and between audio bandwidth changes in
 the SILK-only modes, transitions MAY include redundant side
 information ("redundancy"), in the form of an additional CELT frame
 embedded in the Opus frame.
 A transition between coding the lower frequencies with the LP model
 and the MDCT model or a transition that involves changing the SILK
 bandwidth is only normatively specified when it includes redundancy.
 For those without redundancy, it is RECOMMENDED that the decoder use
 a concealment technique (e.g., make use of a PLC algorithm) to "fill
 in" the gap or discontinuity caused by the mode transition.
 Therefore, PLC MUST NOT be applied during any normative transition,
 i.e., when
 o  A packet includes redundancy for this transition (as described
    below),
 o  The transition is between any WB SILK packet and any Hybrid
    packet, or vice versa,
 o  The transition is between any two Hybrid mode packets, or
 o  The transition is between any two CELT mode packets,
 unless there is actual packet loss.

4.5.1. Transition Side Information (Redundancy)

 Transitions with side information include an extra 5 ms "redundant"
 CELT frame within the Opus frame.  This frame is designed to fill in
 the gap or discontinuity in the different layers without requiring
 the decoder to conceal it.  For transitions from CELT-only to SILK-
 only or Hybrid, the redundant frame is inserted in the first Opus
 frame after the transition (i.e., the first SILK-only or Hybrid
 frame).  For transitions from SILK-only or Hybrid to CELT-only, the
 redundant frame is inserted in the last Opus frame before the
 transition (i.e., the last SILK-only or Hybrid frame).

Valin, et al. Standards Track [Page 124] RFC 6716 Interactive Audio Codec September 2012

4.5.1.1. Redundancy Flag

 The presence of redundancy is signaled in all SILK-only and Hybrid
 frames, not just those involved in a mode transition.  This allows
 the frames to be decoded correctly even if an adjacent frame is lost.
 For SILK-only frames, this signaling is implicit, based on the size
 of the Opus frame and the number of bits consumed decoding the SILK
 portion of it.  After decoding the SILK portion of the Opus frame,
 the decoder uses ec_tell() (see Section 4.1.6.1) to check if there
 are at least 17 bits remaining.  If so, then the frame contains
 redundancy.
 For Hybrid frames, this signaling is explicit.  After decoding the
 SILK portion of the Opus frame, the decoder uses ec_tell() (see
 Section 4.1.6.1) to ensure there are at least 37 bits remaining.  If
 so, it reads a symbol with the PDF in Table 64, and if the value is
 1, then the frame contains redundancy.  Otherwise (if there were
 fewer than 37 bits left or the value was 0), the frame does not
 contain redundancy.
                          +----------------+
                          | PDF            |
                          +----------------+
                          | {4095, 1}/4096 |
                          +----------------+
                     Table 64: Redundancy Flag PDF

4.5.1.2. Redundancy Position Flag

 Since the current frame is a SILK-only or a Hybrid frame, it must be
 at least 10 ms.  Therefore, it needs an additional flag to indicate
 whether the redundant 5 ms CELT frame should be mixed into the
 beginning of the current frame, or the end.  After determining that a
 frame contains redundancy, the decoder reads a 1 bit symbol with a
 uniform PDF (Table 65).
                             +----------+
                             | PDF      |
                             +----------+
                             | {1, 1}/2 |
                             +----------+
                   Table 65: Redundancy Position PDF

Valin, et al. Standards Track [Page 125] RFC 6716 Interactive Audio Codec September 2012

 If the value is zero, this is the first frame in the transition, and
 the redundancy belongs at the end.  If the value is one, this is the
 second frame in the transition, and the redundancy belongs at the
 beginning.  There is no way to specify that an Opus frame contains
 separate redundant CELT frames at both the beginning and the end.

4.5.1.3. Redundancy Size

 Unlike the CELT portion of a Hybrid frame, the redundant CELT frame
 does not use the same entropy coder state as the rest of the Opus
 frame, because this would break the CELT bit allocation mechanism in
 Hybrid frames.  Thus, a redundant CELT frame always starts and ends
 on a byte boundary, even in SILK-only frames, where this is not
 strictly necessary.
 For SILK-only frames, the number of bytes in the redundant CELT frame
 is simply the number of whole bytes remaining, which must be at least
 2, due to the space check in Section 4.5.1.1.  For Hybrid frames, the
 number of bytes is equal to 2, plus a decoded unsigned integer less
 than 256 (see Section 4.1.5).  This may be more than the number of
 whole bytes remaining in the Opus frame, in which case the frame is
 invalid.  However, a decoder is not required to ignore the entire
 frame, as this may be the result of a bit error that desynchronized
 the range coder.  There may still be useful data before the error,
 and a decoder MAY keep any audio decoded so far instead of invoking
 the PLC, but it is RECOMMENDED that the decoder stop decoding and
 discard the rest of the current Opus frame.
 It would have been possible to avoid these invalid states in the
 design of Opus by limiting the range of the explicit length decoded
 from Hybrid frames by the actual number of whole bytes remaining.
 However, this would require an encoder to determine the rate
 allocation for the MDCT layer up front, before it began encoding that
 layer.  By allowing some invalid sizes, the encoder is able to defer
 that decision until much later.  When encoding Hybrid frames that do
 not include redundancy, the encoder must still decide up front if it
 wishes to use the minimum 37 bits required to trigger encoding of the
 redundancy flag, but this is a much looser restriction.
 After determining the size of the redundant CELT frame, the decoder
 reduces the size of the buffer currently in use by the range coder by
 that amount.  The MDCT layer reads any raw bits from the end of this
 reduced buffer, and all calculations of the number of bits remaining
 in the buffer must be done using this new, reduced size, rather than
 the original size of the Opus frame.

Valin, et al. Standards Track [Page 126] RFC 6716 Interactive Audio Codec September 2012

4.5.1.4. Decoding the Redundancy

 The redundant frame is decoded like any other CELT-only frame, with
 the exception that it does not contain a TOC byte.  The frame size is
 fixed at 5 ms, the channel count is set to that of the current frame,
 and the audio bandwidth is also set to that of the current frame,
 with the exception that for MB SILK frames, it is set to WB.
 If the redundancy belongs at the beginning (in a CELT-only to SILK-
 only or Hybrid transition), the final reconstructed output uses the
 first 2.5 ms of audio output by the decoder for the redundant frame
 as is, discarding the corresponding output from the SILK-only or
 Hybrid portion of the frame.  The remaining 2.5 ms is cross-lapped
 with the decoded SILK/Hybrid signal using the CELT's power-
 complementary MDCT window to ensure a smooth transition.
 If the redundancy belongs at the end (in a SILK-only or Hybrid to
 CELT-only transition), only the second half (2.5 ms) of the audio
 output by the decoder for the redundant frame is used.  In that case,
 the second half of the redundant frame is cross-lapped with the end
 of the SILK/Hybrid signal, again using CELT's power-complementary
 MDCT window to ensure a smooth transition.

4.5.2. State Reset

 When a transition occurs, the state of the SILK or the CELT decoder
 (or both) may need to be reset before decoding a frame in the new
 mode.  This avoids reusing "out of date" memory, which may not have
 been updated in some time or may not be in a well-defined state due
 to, e.g., PLC.  The SILK state is reset before every SILK-only or
 Hybrid frame where the previous frame was CELT-only.  The CELT state
 is reset every time the operating mode changes and the new mode is
 either Hybrid or CELT-only, except when the transition uses
 redundancy as described above.  When switching from SILK-only or
 Hybrid to CELT-only with redundancy, the CELT state is reset before
 decoding the redundant CELT frame embedded in the SILK-only or Hybrid
 frame, but it is not reset before decoding the following CELT-only
 frame.  When switching from CELT-only mode to SILK-only or Hybrid
 mode with redundancy, the CELT decoder is not reset for decoding the
 redundant CELT frame.

Valin, et al. Standards Track [Page 127] RFC 6716 Interactive Audio Codec September 2012

4.5.3. Summary of Transitions

 Figure 18 illustrates all of the normative transitions involving a
 mode change, an audio bandwidth change, or both.  Each one uses an S,
 H, or C to represent an Opus frame in the corresponding mode.  In
 addition, an R indicates the presence of redundancy in the Opus frame
 with which it is cross-lapped.  Its location in the first or last
 5 ms is assumed to correspond to whether it is the frame before or
 after the transition.  Other uses of redundancy are non-normative.
 Finally, a c indicates the contents of the CELT overlap buffer after
 the previously decoded frame (i.e., as extracted by decoding a
 silence frame).

Valin, et al. Standards Track [Page 128] RFC 6716 Interactive Audio Codec September 2012

  SILK to SILK with Redundancy:             S -> S -> S
                                                      &
                                                     !R -> R
                                                           &
                                                          ;S -> S -> S
  NB or MB SILK to Hybrid with Redundancy:  S -> S -> S
                                                      &
                                                     !R ->;H -> H -> H
  WB SILK to Hybrid:                        S -> S -> S ->!H -> H -> H
  SILK to CELT with Redundancy:             S -> S -> S
                                                      &
                                                     !R -> C -> C -> C
  Hybrid to NB or MB SILK with Redundancy:  H -> H -> H
                                                      &
                                                     !R -> R
                                                           &
                                                          ;S -> S -> S
  Hybrid to WB SILK:                        H -> H -> H -> c
                                                        \  +
                                                         > S -> S -> S
  Hybrid to CELT with Redundancy:           H -> H -> H
                                                      &
                                                     !R -> C -> C -> C
  CELT to SILK with Redundancy:             C -> C -> C -> R
                                                           &
                                                          ;S -> S -> S
  CELT to Hybrid with Redundancy:           C -> C -> C -> R
                                                           &
                                                          |H -> H -> H
  Key:
  S   SILK-only frame                 ;   SILK decoder reset
  H   Hybrid frame                    |   CELT and SILK decoder resets
  C   CELT-only frame                 !   CELT decoder reset
  c   CELT overlap                    +   Direct mixing
  R   Redundant CELT frame            &   Windowed cross-lap
                   Figure 18: Normative Transitions

Valin, et al. Standards Track [Page 129] RFC 6716 Interactive Audio Codec September 2012

 The first two and the last two Opus frames in each example are
 illustrative, i.e., there is no requirement that a stream remain in
 the same configuration for three consecutive frames before or after a
 switch.
 The behavior of transitions without redundancy where PLC is allowed
 is non-normative.  An encoder might still wish to use these
 transitions if, for example, it doesn't want to add the extra bitrate
 required for redundancy or if it makes a decision to switch after it
 has already transmitted the frame that would have had to contain the
 redundancy.  Figure 19 illustrates the recommended cross-lapping and
 decoder resets for these transitions.
  SILK to SILK (audio bandwidth change):    S -> S -> S   ;S -> S -> S
  NB or MB SILK to Hybrid:                  S -> S -> S   |H -> H -> H
  SILK to CELT without Redundancy:          S -> S -> S -> P
                                                           &
                                                          !C -> C -> C
  Hybrid to NB or MB SILK:                  H -> H -> H -> c
                                                           +
                                                          ;S -> S -> S
  Hybrid to CELT without Redundancy:        H -> H -> H -> P
                                                           &
                                                          !C -> C -> C
  CELT to SILK without Redundancy:          C -> C -> C -> P
                                                           &
                                                          ;S -> S -> S
  CELT to Hybrid without Redundancy:        C -> C -> C -> P
                                                           &
                                                          |H -> H -> H
  Key:
  S   SILK-only frame                 ;   SILK decoder reset
  H   Hybrid frame                    |   CELT and SILK decoder resets
  C   CELT-only frame                 !   CELT decoder reset
  c   CELT overlap                    +   Direct mixing
  P   Packet Loss Concealment         &   Windowed cross-lap
           Figure 19: Recommended Non-Normative Transitions
 Encoders SHOULD NOT use other transitions, e.g., those that involve
 redundancy in ways not illustrated in Figure 18.

Valin, et al. Standards Track [Page 130] RFC 6716 Interactive Audio Codec September 2012

5. Opus Encoder

 Just like the decoder, the Opus encoder also normally consists of two
 main blocks: the SILK encoder and the CELT encoder.  However, unlike
 the case of the decoder, a valid (though potentially suboptimal) Opus
 encoder is not required to support all modes and may thus only
 include a SILK encoder module or a CELT encoder module.  The output
 bitstream of the Opus encoding contains bits from the SILK and CELT
 encoders, though these are not separable due to the use of a range
 coder.  A block diagram of the encoder is illustrated below.
                      +------------+    +---------+
                      |   Sample   |    |  SILK   |------+
                   +->|    Rate    |--->| Encoder |      V
    +-----------+  |  | Conversion |    |         | +---------+
    | Optional  |  |  +------------+    +---------+ |  Range  |
  ->| High-pass |--+                                | Encoder |---->
    |  Filter   |  |  +--------------+  +---------+ |         | Bit-
    +-----------+  |  |    Delay     |  |  CELT   | +---------+ stream
                   +->| Compensation |->| Encoder |      ^
                      |              |  |         |------+
                      +--------------+  +---------+
                        Figure 20: Opus Encoder
 For a normal encoder where both the SILK and the CELT modules are
 included, an optimal encoder should select which coding mode to use
 at run-time depending on the conditions.  In the reference
 implementation, the frame size is selected by the application, but
 the other configuration parameters (number of channels, bandwidth,
 mode) are automatically selected (unless explicitly overridden by the
 application) depending on the following:
 o  Requested bitrate
 o  Input sampling rate
 o  Type of signal (speech vs. music)
 o  Frame size in use
 The type of signal currently needs to be provided by the application
 (though it can be changed in real-time).  An Opus encoder
 implementation could also do automatic detection, but since Opus is
 an interactive codec, such an implementation would likely have to
 either delay the signal (for non-interactive applications) or delay
 the mode switching decisions (for interactive applications).

Valin, et al. Standards Track [Page 131] RFC 6716 Interactive Audio Codec September 2012

 When the encoder is configured for voice over IP applications, the
 input signal is filtered by a high-pass filter to remove the lowest
 part of the spectrum that contains little speech energy and may
 contain background noise.  This is a second order Auto Regressive
 Moving Average (i.e., with poles and zeros) filter with a cut-off
 frequency around 50 Hz.  In the future, a music detector may also be
 used to lower the cut-off frequency when the input signal is detected
 to be music rather than speech.

5.1. Range Encoder

 The range coder acts as the bit-packer for Opus.  It is used in three
 different ways: to encode
 o  Entropy-coded symbols with a fixed probability model using
    ec_encode() (entenc.c),
 o  Integers from 0 to (2**M - 1) using ec_enc_uint() or ec_enc_bits()
    (entenc.c),
 o  Integers from 0 to (ft - 1) (where ft is not a power of two) using
    ec_enc_uint() (entenc.c).
 The range encoder maintains an internal state vector composed of the
 four-tuple (val, rng, rem, ext) representing the low end of the
 current range, the size of the current range, a single buffered
 output byte, and a count of additional carry-propagating output
 bytes.  Both val and rng are 32-bit unsigned integer values, rem is a
 byte value or less than 255 or the special value -1, and ext is an
 unsigned integer with at least 11 bits.  This state vector is
 initialized at the start of each frame to the value
 (0, 2**31, -1, 0).  After encoding a sequence of symbols, the value
 of rng in the encoder should exactly match the value of rng in the
 decoder after decoding the same sequence of symbols.  This is a
 powerful tool for detecting errors in either an encoder or decoder
 implementation.  The value of val, on the other hand, represents
 different things in the encoder and decoder, and is not expected to
 match.
 The decoder has no analog for rem and ext.  These are used to perform
 carry propagation in the renormalization loop below.  Each iteration
 of this loop produces 9 bits of output, consisting of 8 data bits and
 a carry flag.  The encoder cannot determine the final value of the
 output bytes until it propagates these carry flags.  Therefore, the
 reference implementation buffers a single non-propagating output byte
 (i.e., one less than 255) in rem and keeps a count of additional

Valin, et al. Standards Track [Page 132] RFC 6716 Interactive Audio Codec September 2012

 propagating (i.e., 255) output bytes in ext.  An implementation may
 choose to use any mathematically equivalent scheme to perform carry
 propagation.

5.1.1. Encoding Symbols

 The main encoding function is ec_encode() (entenc.c), which encodes
 symbol k in the current context using the same three-tuple
 (fl[k], fh[k], ft) as the decoder to describe the range of the symbol
 (see Section 4.1).
 ec_encode() updates the state of the encoder as follows.  If fl[k] is
 greater than zero, then
                                     rng
                   val = val + rng - --- * (ft - fl)
                                     ft
                         rng
                   rng = --- * (fh - fl)
                         ft
 Otherwise, val is unchanged and
                                  rng
                      rng = rng - --- * (fh - fl)
                                  ft
 The divisions here are integer division.

5.1.1.1. Renormalization

 After this update, the range is normalized using a procedure very
 similar to that of Section 4.1.2.1, implemented by ec_enc_normalize()
 (entenc.c).  The following process is repeated until rng > 2**23.
 First, the top 9 bits of val, (val>>23), are sent to the carry
 buffer, described in Section 5.1.1.2.  Then, the encoder sets
                      val = (val<<8) & 0x7FFFFFFF
                      rng = rng<<8

5.1.1.2. Carry Propagation and Output Buffering

 The function ec_enc_carry_out() (entenc.c) implements carry
 propagation and output buffering.  It takes, as input, a 9-bit
 unsigned value, c, consisting of 8 data bits and an additional carry

Valin, et al. Standards Track [Page 133] RFC 6716 Interactive Audio Codec September 2012

 bit.  If c is equal to the value 255, then ext is simply incremented,
 and no other state updates are performed.  Otherwise, let b = (c>>8)
 be the carry bit.  Then,
 o  If the buffered byte rem contains a value other than -1, the
    encoder outputs the byte (rem + b).  Otherwise, if rem is -1, no
    byte is output.
 o  If ext is non-zero, then the encoder outputs ext bytes -- all with
    a value of 0 if b is set, or 255 if b is unset -- and sets ext to
    0.
 o  rem is set to the 8 data bits:
                             rem = c & 255

5.1.2. Alternate Encoding Methods

 The reference implementation uses three additional encoding methods
 that are exactly equivalent to the above, but make assumptions and
 simplifications that allow for a more efficient implementation.

5.1.2.1. ec_encode_bin()

 The first is ec_encode_bin() (entenc.c), defined using the parameter
 ftb instead of ft.  It is mathematically equivalent to calling
 ec_encode() with ft = (1<<ftb), but it avoids using division.

5.1.2.2. ec_enc_bit_logp()

 The next is ec_enc_bit_logp() (entenc.c), which encodes a single
 binary symbol.  The context is described by a single parameter, logp,
 which is the absolute value of the base-2 logarithm of the
 probability of a "1".  It is mathematically equivalent to calling
 ec_encode() with the 3-tuple (fl[k] = 0, fh[k] = (1<<logp) - 1,
 ft = (1<<logp)) if k is 0 and with (fl[k] = (1<<logp) - 1,
 fh[k] = ft = (1<<logp)) if k is 1.  The implementation requires no
 multiplications or divisions.

5.1.2.3. ec_enc_icdf()

 The last is ec_enc_icdf() (entenc.c), which encodes a single binary
 symbol with a table-based context of up to 8 bits.  This uses the
 same icdf table as ec_dec_icdf() from Section 4.1.3.3.  The function

Valin, et al. Standards Track [Page 134] RFC 6716 Interactive Audio Codec September 2012

 is mathematically equivalent to calling ec_encode() with
 fl[k] = (1<<ftb) - icdf[k-1] (or 0 if k == 0), fh[k] = (1<<ftb) -
  icdf[k], and ft = (1<<ftb).  This only saves a few arithmetic
 operations over ec_encode_bin(), but it allows the encoder to use the
 same icdf tables as the decoder.

5.1.3. Encoding Raw Bits

 The raw bits used by the CELT layer are packed at the end of the
 buffer using ec_enc_bits() (entenc.c).  Because the raw bits may
 continue into the last byte output by the range coder if there is
 room in the low-order bits, the encoder must be prepared to merge
 these values into a single byte.  The procedure in Section 5.1.5 does
 this in a way that ensures both the range coded data and the raw bits
 can be decoded successfully.

5.1.4. Encoding Uniformly Distributed Integers

 The function ec_enc_uint() (entenc.c) encodes one of ft equiprobable
 symbols in the range 0 to (ft - 1), inclusive, each with a frequency
 of 1, where ft may be as large as (2**32 - 1).  Like the decoder (see
 Section 4.1.5), it splits up the value into a range coded symbol
 representing up to 8 of the high bits, and, if necessary, raw bits
 representing the remainder of the value.
 ec_enc_uint() takes a two-tuple (t, ft), where t is the unsigned
 integer to be encoded, 0 <= t < ft, and ft is not necessarily a power
 of two.  Let ftb = ilog(ft - 1), i.e., the number of bits required to
 store (ft - 1) in two's complement notation.  If ftb is 8 or less,
 then t is encoded directly using ec_encode() with the three-tuple (t,
 t + 1, ft).
 If ftb is greater than 8, then the top 8 bits of t are encoded using
 the three-tuple (t>>(ftb - 8), (t>>(ftb - 8)) + 1,
 ((ft - 1)>>(ftb - 8)) + 1), and the remaining bits,
 (t & ((1<<(ftb - 8)) - 1), are encoded as raw bits with
 ec_enc_bits().

5.1.5. Finalizing the Stream

 After all symbols are encoded, the stream must be finalized by
 outputting a value inside the current range.  Let end be the unsigned
 integer in the interval [val, val + rng) with the largest number of
 trailing zero bits, b, such that (end + (1<<b) - 1) is also in the
 interval [val, val + rng).  This choice of end allows the maximum
 number of trailing bits to be set to arbitrary values while still
 ensuring the range coded part of the buffer can be decoded correctly.

Valin, et al. Standards Track [Page 135] RFC 6716 Interactive Audio Codec September 2012

 Then, while end is not zero, the top 9 bits of end, i.e., (end>>23),
 are passed to the carry buffer in accordance with the procedure in
 Section 5.1.1.2, and end is updated via
                      end = (end<<8) & 0x7FFFFFFF
 Finally, if the buffered output byte, rem, is neither zero nor the
 special value -1, or the carry count, ext, is greater than zero, then
 9 zero bits are sent to the carry buffer to flush it to the output
 buffer.  When outputting the final byte from the range coder, if it
 would overlap any raw bits already packed into the end of the output
 buffer, they should be ORed into the same byte.  The bit allocation
 routines in the CELT layer should ensure that this can be done
 without corrupting the range coder data so long as end is chosen as
 described above.  If there is any space between the end of the range
 coder data and the end of the raw bits, it is padded with zero bits.
 This entire process is implemented by ec_enc_done() (entenc.c).

5.1.6. Current Bit Usage

 The bit allocation routines in Opus need to be able to determine a
 conservative upper bound on the number of bits that have been used to
 encode the current frame thus far.  This drives allocation decisions
 and ensures that the range coder and raw bits will not overflow the
 output buffer.  This is computed in the reference implementation to
 whole-bit precision by the function ec_tell() (entcode.h) and to
 fractional 1/8th bit precision by the function ec_tell_frac()
 (entcode.c).  Like all operations in the range coder, it must be
 implemented in a bit-exact manner, and it must produce exactly the
 same value returned by the same functions in the decoder after
 decoding the same symbols.

5.2. SILK Encoder

 In many respects, the SILK encoder mirrors the SILK decoder described
 in Section 4.2.  Details such as the quantization and range coder
 tables can be found there, while this section describes the high-
 level design choices that were made.  The diagram below shows the
 basic modules of the SILK encoder.
             +----------+    +--------+    +---------+
             |  Sample  |    | Stereo |    |  SILK   |
      ------>|   Rate   |--->| Mixing |--->|  Core   |---------->
      Input  |Conversion|    |        |    | Encoder |  Bitstream
             +----------+    +--------+    +---------+
                        Figure 21: SILK Encoder

Valin, et al. Standards Track [Page 136] RFC 6716 Interactive Audio Codec September 2012

5.2.1. Sample Rate Conversion

 The input signal's sampling rate is adjusted by a sample rate
 conversion module so that it matches the SILK internal sampling rate.
 The input to the sample rate converter is delayed by a number of
 samples depending on the sample rate ratio, such that the overall
 delay is constant for all input and output sample rates.

5.2.2. Stereo Mixing

 The stereo mixer is only used for stereo input signals.  It converts
 a stereo left-right signal into an adaptive mid-side representation.
 The first step is to compute non-adaptive mid-side signals as half
 the sum and difference between left and right signals.  The side
 signal is then minimized in energy by subtracting a prediction of it
 based on the mid signal.  This prediction works well when the left
 and right signals exhibit linear dependency, for instance, for an
 amplitude-panned input signal.  Like in the decoder, the prediction
 coefficients are linearly interpolated during the first 8 ms of the
 frame.  The mid signal is always encoded, whereas the residual side
 signal is only encoded if it has sufficient energy compared to the
 mid signal's energy.  If it has not, the "mid_only_flag" is set
 without encoding the side signal.
 The predictor coefficients are coded regardless of whether the side
 signal is encoded.  For each frame, two predictor coefficients are
 computed, one that predicts between low-passed mid and side channels,
 and one that predicts between high-passed mid and side channels.  The
 low-pass filter is a simple three-tap filter and creates a delay of
 one sample.  The high-pass filtered signal is the difference between
 the mid signal delayed by one sample and the low-passed signal.
 Instead of explicitly computing the high-passed signal, it is
 computationally more efficient to transform the prediction
 coefficients before applying them to the filtered mid signal, as
 follows:
             pred(n) = LP(n) * w0 + HP(n) * w1
                     = LP(n) * w0 + (mid(n-1) - LP(n)) * w1
                     = LP(n) * (w0 - w1) + mid(n-1) * w1
 where w0 and w1 are the low-pass and high-pass prediction
 coefficients, mid(n-1) is the mid signal delayed by one sample, LP(n)
 and HP(n) are the low-passed and high-passed signals and pred(n) is
 the prediction signal that is subtracted from the side signal.

Valin, et al. Standards Track [Page 137] RFC 6716 Interactive Audio Codec September 2012

5.2.3. SILK Core Encoder

 What follows is a description of the core encoder and its components.
 For simplicity, the core encoder is referred to simply as the encoder
 in the remainder of this section.  An overview of the encoder is
 given in Figure 22.
                                                              +---+
                           +--------------------------------->|   |
      +---------+          |      +---------+                 |   |
      |Voice    |          |      |LTP      |12               |   |
  +-->|Activity |--+       +----->|Scaling  |-----------+---->|   |
  |   |Detection|3 |       |      |Control  |<--+       |     |   |
  |   +---------+  |       |      +---------+   |       |     |   |
  |                |       |      +---------+   |       |     |   |
  |                |       |      |Gains    |   |       |     |   |
  |                |       |  +-->|Processor|---|---+---|---->| R |
  |                |       |  |   |         |11 |   |   |     | a |
  |               \/       |  |   +---------+   |   |   |     | n |
  |          +---------+   |  |   +---------+   |   |   |     | g |
  |          |Pitch    |   |  |   |LSF      |   |   |   |     | e |
  |       +->|Analysis |---+  |   |Quantizer|---|---|---|---->|   |
  |       |  |         |4  |  |   |         |8  |   |   |     | E |-->
  |       |  +---------+   |  |   +---------+   |   |   |     | n | 2
  |       |                |  |    9/\  10|     |   |   |     | c |
  |       |                |  |     |    \/     |   |   |     | o |
  |       |  +---------+   |  |   +----------+  |   |   |     | d |
  |       |  |Noise    |   +--|-->|Prediction|--+---|---|---->| e |
  |       +->|Shaping  |---|--+   |Analysis  |7 |   |   |     | r |
  |       |  |Analysis |5  |  |   |          |  |   |   |     |   |
  |       |  +---------+   |  |   +----------+  |   |   |     |   |
  |       |                |  |        /\       |   |   |     |   |
  |       |     +----------|--|--------+        |   |   |     |   |
  |       |     |         \/  \/               \/  \/  \/     |   |
  |       |     |      +----------+          +------------+   |   |
  |       |     |      |          |          |Noise       |   |   |
 -+-------+-----+----->|Pre-filter|--------->|Shaping     |-->|   |
 1                     |          | 6        |Quantization|13 |   |
                       +----------+          +------------+   +---+
 1:  Input speech signal
 2:  Range encoded bitstream
 3:  Voice activity estimate
 4:  Pitch lags (per 5 ms) and voicing decision (per 20 ms)
 5:  Noise shaping quantization coefficients
   - Short-term synthesis and analysis
     noise shaping coefficients (per 5 ms)

Valin, et al. Standards Track [Page 138] RFC 6716 Interactive Audio Codec September 2012

  1. Long-term synthesis and analysis noise

shaping coefficients (per 5 ms and for voiced speech only)

  1. Noise shaping tilt (per 5 ms)
  2. Quantizer gain/step size (per 5 ms)

6: Input signal filtered with analysis noise shaping filters

 7:  Short- and Long-Term Prediction coefficients
     LTP (per 5 ms) and LPC (per 20 ms)
 8:  LSF quantization indices
 9:  LSF coefficients
 10: Quantized LSF coefficients
 11: Processed gains, and synthesis noise shape coefficients
 12: LTP state scaling coefficient.  Controlling error
     propagation / prediction gain trade-off
 13: Quantized signal
                     Figure 22: SILK Core Encoder

5.2.3.1. Voice Activity Detection

 The input signal is processed by a Voice Activity Detection (VAD)
 algorithm to produce a measure of voice activity, spectral tilt, and
 signal-to-noise estimates for each frame.  The VAD uses a sequence of
 half-band filterbanks to split the signal into four subbands:
 0...Fs/16, Fs/16...Fs/8, Fs/8...Fs/4, and Fs/4...Fs/2, where Fs is
 the sampling frequency (8, 12, 16, or 24 kHz).  The lowest subband,
 from 0 - Fs/16, is high-pass filtered with a first-order moving
 average (MA) filter (with transfer function H(z) = 1-z**(-1)) to
 reduce the energy at the lowest frequencies.  For each frame, the
 signal energy per subband is computed.  In each subband, a noise
 level estimator tracks the background noise level and a Signal-to-
 Noise Ratio (SNR) value is computed as the logarithm of the ratio of
 energy-to-noise level.  Using these intermediate variables, the
 following parameters are calculated for use in other SILK modules:
 o  Average SNR.  The average of the subband SNR values.
 o  Smoothed subband SNRs.  Temporally smoothed subband SNR values.
 o  Speech activity level.  Based on the average SNR and a weighted
    average of the subband energies.
 o  Spectral tilt.  A weighted average of the subband SNRs, with
    positive weights for the low subbands and negative weights for the
    high subbands.

Valin, et al. Standards Track [Page 139] RFC 6716 Interactive Audio Codec September 2012

5.2.3.2. Pitch Analysis

 The input signal is processed by the open loop pitch estimator shown
 in Figure 23.
                                  +--------+  +----------+
                                  |2 x Down|  |Time-     |
                               +->|sampling|->|Correlator|     |
                               |  |        |  |          |     |4
                               |  +--------+  +----------+    \/
                               |                    | 2    +-------+
                               |                    |  +-->|Speech |5
     +---------+    +--------+ |                   \/  |   |Type   |->
     |LPC      |    |Down    | |              +----------+ |       |
  +->|Analysis | +->|sample  |-+------------->|Time-     | +-------+
  |  |         | |  |to 8 kHz|                |Correlator|----------->
  |  +---------+ |  +--------+                |__________|          6
  |       |      |                                  |3
  |      \/      |                                 \/
  |  +---------+ |                            +----------+
  |  |Whitening| |                            |Time-     |
 -+->|Filter   |-+--------------------------->|Correlator|----------->
 1   |         |                              |          |          7
     +---------+                              +----------+
 1: Input signal
 2: Lag candidates from stage 1
 3: Lag candidates from stage 2
 4: Correlation threshold
 5: Voiced/unvoiced flag
 6: Pitch correlation
 7: Pitch lags
            Figure 23: Block Diagram of the Pitch Estimator
 The pitch analysis finds a binary voiced/unvoiced classification,
 and, for frames classified as voiced, four pitch lags per frame --
 one for each 5 ms subframe -- and a pitch correlation indicating the
 periodicity of the signal.  The input is first whitened using a
 Linear Prediction (LP) whitening filter, where the coefficients are
 computed through standard Linear Predictive Coding (LPC) analysis.
 The order of the whitening filter is 16 for best results, but is
 reduced to 12 for medium complexity and 8 for low complexity modes.
 The whitened signal is analyzed to find pitch lags for which the time
 correlation is high.  The analysis consists of three stages for
 reducing the complexity:

Valin, et al. Standards Track [Page 140] RFC 6716 Interactive Audio Codec September 2012

 o  In the first stage, the whitened signal is downsampled to 4 kHz
    (from 8 kHz), and the current frame is correlated to a signal
    delayed by a range of lags, starting from a shortest lag
    corresponding to 500 Hz, to a longest lag corresponding to 56 Hz.
 o  The second stage operates on an 8 kHz signal (downsampled from 12,
    16, or 24 kHz) and measures time correlations only near the lags
    corresponding to those that had sufficiently high correlations in
    the first stage.  The resulting correlations are adjusted for a
    small bias towards short lags to avoid ending up with a multiple
    of the true pitch lag.  The highest adjusted correlation is
    compared to a threshold depending on:
  • Whether the previous frame was classified as voiced.
  • The speech activity level.
  • The spectral tilt.
    If the threshold is exceeded, the current frame is classified as
    voiced and the lag with the highest adjusted correlation is stored
    for a final pitch analysis of the highest precision in the third
    stage.
 o  The last stage operates directly on the whitened input signal to
    compute time correlations for each of the four subframes
    independently in a narrow range around the lag with highest
    correlation from the second stage.

5.2.3.3. Noise Shaping Analysis

 The noise shaping analysis finds gains and filter coefficients used
 in the pre-filter and noise shaping quantizer.  These parameters are
 chosen such that they will fulfill several requirements:
 o  Balancing quantization noise and bitrate.  The quantization gains
    determine the step size between reconstruction levels of the
    excitation signal.  Therefore, increasing the quantization gain
    amplifies quantization noise, but also reduces the bitrate by
    lowering the entropy of the quantization indices.
 o  Spectral shaping of the quantization noise; the noise shaping
    quantizer is capable of reducing quantization noise in some parts
    of the spectrum at the cost of increased noise in other parts
    without substantially changing the bitrate.  By shaping the noise
    such that it follows the signal spectrum, it becomes less audible.
    In practice, best results are obtained by making the shape of the
    noise spectrum slightly flatter than the signal spectrum.

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 o  De-emphasizing spectral valleys; by using different coefficients
    in the analysis and synthesis part of the pre-filter and noise
    shaping quantizer, the levels of the spectral valleys can be
    decreased relative to the levels of the spectral peaks such as
    speech formants and harmonics.  This reduces the entropy of the
    signal, which is the difference between the coded signal and the
    quantization noise, thus lowering the bitrate.
 o  Matching the levels of the decoded speech formants to the levels
    of the original speech formants; an adjustment gain and a first
    order tilt coefficient are computed to compensate for the effect
    of the noise shaping quantization on the level and spectral tilt.
               / \   ___
                |   // \\
                |  //   \\     ____
                |_//     \\___//  \\         ____
                | /  ___  \   /    \\       //  \\
              P |/  /   \  \_/      \\_____//    \\
              o |  /     \     ____  \     /      \\
              w | /       \___/    \  \___/  ____  \\___ 1
              e |/                  \       /    \  \
              r |                    \_____/      \  \__ 2
                |                                  \
                |                                   \___ 3
                |
                +---------------------------------------->
                                 Frequency
             1: Input signal spectrum
             2: De-emphasized and level matched spectrum
             3: Quantization noise spectrum
    Figure 24: Noise Shaping and Spectral De-emphasis Illustration
 Figure 24 shows an example of an input signal spectrum (1).  After
 de-emphasis and level matching, the spectrum has deeper valleys (2).
 The quantization noise spectrum (3) more or less follows the input
 signal spectrum, while having slightly less pronounced peaks.  The
 entropy, which provides a lower bound on the bitrate for encoding the
 excitation signal, is proportional to the area between the de-
 emphasized spectrum (2) and the quantization noise spectrum (3).
 Without de-emphasis, the entropy is proportional to the area between
 input spectrum (1) and quantization noise (3) -- clearly higher.
 The transformation from input signal to de-emphasized signal can be
 described as a filtering operation with a filter

Valin, et al. Standards Track [Page 142] RFC 6716 Interactive Audio Codec September 2012

  1. 1 Wana(z)

H(z) = G * ( 1 - c_tilt * z ) * ——-

                                                 Wsyn(z)
 having an adjustment gain G, a first order tilt adjustment filter
 with tilt coefficient c_tilt, and where
                     16                           d
                     __            -k        -L  __            -k
      Wana(z) = (1 - \ a_ana(k) * z  )*(1 - z  * \ b_ana(k) * z  )
                     /_                          /_
                     k=1                         k=-d
 is the analysis part of the de-emphasis filter, consisting of the
 short-term shaping filter with coefficients a_ana(k) and the long-
 term shaping filter with coefficients b_ana(k) and pitch lag L.  The
 parameter d determines the number of long-term shaping filter taps.
 Similarly, but without the tilt adjustment, the synthesis part can be
 written as
                     16                           d
                     __            -k        -L  __            -k
      Wsyn(z) = (1 - \ a_syn(k) * z  )*(1 - z  * \ b_syn(k) * z  )
                     /_                          /_
                     k=1                         k=-d
 All noise shaping parameters are computed and applied per subframe of
 5 ms.  First, an LPC analysis is performed on a windowed signal block
 of 15 ms.  The signal block has a look-ahead of 5 ms relative to the
 current subframe, and the window is an asymmetric sine window.  The
 LPC analysis is done with the autocorrelation method, with an order
 of between 8, in lowest-complexity mode, and 16, for best quality.
 Optionally, the LPC analysis and noise shaping filters are warped by
 replacing the delay elements by first-order allpass filters.  This
 increases the frequency resolution at low frequencies and reduces it
 at high ones, which better matches the human auditory system and
 improves quality.  The warped analysis and filtering comes at a cost
 in complexity and is therefore only done in higher complexity modes.
 The quantization gain is found by taking the square root of the
 residual energy from the LPC analysis and multiplying it by a value
 inversely proportional to the coding quality control parameter and
 the pitch correlation.

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 Next, the two sets of short-term noise shaping coefficients a_ana(k)
 and a_syn(k) are obtained by applying different amounts of bandwidth
 expansion to the coefficients found in the LPC analysis.  This
 bandwidth expansion moves the roots of the LPC polynomial towards the
 origin, using the formulas
                                            k
                       a_ana(k) = a(k)*g_ana   and
                                            k
                       a_syn(k) = a(k)*g_syn
 where a(k) is the k'th LPC coefficient, and the bandwidth expansion
 factors g_ana and g_syn are calculated as
                       g_ana = 0.95 - 0.01*C  and
                       g_syn = 0.95 + 0.01*C
 where C is the coding quality control parameter between 0 and 1.
 Applying more bandwidth expansion to the analysis part than to the
 synthesis part gives the desired de-emphasis of spectral valleys in
 between formants.
 The long-term shaping is applied only during voiced frames.  It uses
 a three-tap filter, described by
                 b_ana = F_ana * [0.25, 0.5, 0.25]  and
                 b_syn = F_syn * [0.25, 0.5, 0.25].
 For unvoiced frames, these coefficients are set to 0.  The
 multiplication factors F_ana and F_syn are chosen between 0 and 1,
 depending on the coding quality control parameter, as well as the
 calculated pitch correlation and smoothed subband SNR of the lowest
 subband.  By having F_ana less than F_syn, the pitch harmonics are
 emphasized relative to the valleys in between the harmonics.
 The tilt coefficient c_tilt is for unvoiced frames chosen as
                             c_tilt = 0.25
 and as
                       c_tilt = 0.25 + 0.2625 * V
 for voiced frames, where V is the voice activity level between 0 and
 1.

Valin, et al. Standards Track [Page 144] RFC 6716 Interactive Audio Codec September 2012

 The adjustment gain G serves to correct any level mismatch between
 the original and decoded signals that might arise from the noise
 shaping and de-emphasis.  This gain is computed as the ratio of the
 prediction gain of the short-term analysis and synthesis filter
 coefficients.  The prediction gain of an LPC synthesis filter is the
 square root of the output energy when the filter is excited by a
 unit-energy impulse on the input.  An efficient way to compute the
 prediction gain is by first computing the reflection coefficients
 from the LPC coefficients through the step-down algorithm, and
 extracting the prediction gain from the reflection coefficients as
                                  K
                                 ___          2  -0.5
                    predGain = ( | | 1 - (r_k)  )
                                 k=1
 where r_k is the k'th reflection coefficient.
 Initial values for the quantization gains are computed as the square
 root of the residual energy of the LPC analysis, adjusted by the
 coding quality control parameter.  These quantization gains are later
 adjusted based on the results of the prediction analysis.

5.2.3.4. Prediction Analysis

 The prediction analysis is performed in one of two ways depending on
 how the pitch estimator classified the frame.  The processing for
 voiced and unvoiced speech is described in Section 5.2.3.4.1 and
 Section 5.2.3.4.2, respectively.  Inputs to this function include the
 pre-whitened signal from the pitch estimator (see Section 5.2.3.2).

5.2.3.4.1. Voiced Speech

 For a frame of voiced speech, the pitch pulses will remain dominant
 in the pre-whitened input signal.  Further whitening is desirable as
 it leads to higher quality at the same available bitrate.  To achieve
 this, a Long-Term Prediction (LTP) analysis is carried out to
 estimate the coefficients of a fifth-order LTP filter for each of
 four subframes.  The LTP coefficients are quantized using the method
 described in Section 5.2.3.6, and the quantized LTP coefficients are
 used to compute the LTP residual signal.  This LTP residual signal is
 the input to an LPC analysis where the LPC coefficients are estimated
 using Burg's method [BURG], such that the residual energy is
 minimized.  The estimated LPC coefficients are converted to a Line
 Spectral Frequency (LSF) vector and quantized as described in
 Section 5.2.3.5.  After quantization, the quantized LSF vector is
 converted back to LPC coefficients using the full procedure in
 Section 4.2.7.5.  By using quantized LTP coefficients and LPC

Valin, et al. Standards Track [Page 145] RFC 6716 Interactive Audio Codec September 2012

 coefficients derived from the quantized LSF coefficients, the encoder
 remains fully synchronized with the decoder.  The quantized LPC and
 LTP coefficients are also used to filter the input signal and measure
 residual energy for each of the four subframes.

5.2.3.4.2. Unvoiced Speech

 For a speech signal that has been classified as unvoiced, there is no
 need for LTP filtering, as it has already been determined that the
 pre-whitened input signal is not periodic enough within the allowed
 pitch period range for LTP analysis to be worth the cost in terms of
 complexity and bitrate.  The pre-whitened input signal is therefore
 discarded, and, instead, the input signal is used for LPC analysis
 using Burg's method.  The resulting LPC coefficients are converted to
 an LSF vector and quantized as described in the following section.
 They are then transformed back to obtain quantized LPC coefficients,
 which are then used to filter the input signal and measure residual
 energy for each of the four subframes.

5.2.3.4.2.1. Burg's Method

 The main purpose of linear prediction in SILK is to reduce the
 bitrate by minimizing the residual energy.  At least at high
 bitrates, perceptual aspects are handled independently by the noise
 shaping filter.  Burg's method is used because it provides higher
 prediction gain than the autocorrelation method and, unlike the
 covariance method, produces stable filters (assuming numerical errors
 don't spoil that).  SILK's implementation of Burg's method is also
 computationally faster than the autocovariance method.  The
 implementation of Burg's method differs from traditional
 implementations in two aspects.  The first difference is that it
 operates on autocorrelations, similar to the Schur algorithm [SCHUR],
 but with a simple update to the autocorrelations after finding each
 reflection coefficient to make the result identical to Burg's method.
 This brings down the complexity of Burg's method to near that of the
 autocorrelation method.  The second difference is that the signal in
 each subframe is scaled by the inverse of the residual quantization
 step size.  Subframes with a small quantization step size will, on
 average, spend more bits for a given amount of residual energy than
 subframes with a large step size.  Without scaling, Burg's method
 minimizes the total residual energy in all subframes, which doesn't
 necessarily minimize the total number of bits needed for coding the
 quantized residual.  The residual energy of the scaled subframes is a
 better measure for that number of bits.

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5.2.3.5. LSF Quantization

 Unlike many other speech codecs, SILK uses variable bitrate coding
 for the LSFs.  This improves the average rate-distortion (R-D) trade-
 off and reduces outliers.  The variable bitrate coding minimizes a
 linear combination of the weighted quantization errors and the
 bitrate.  The weights for the quantization errors are the Inverse
 Harmonic Mean Weighting (IHMW) function proposed by Laroia et al.
 (see [LAROIA-ICASSP]).  These weights are referred to here as Laroia
 weights.
 The LSF quantizer consists of two stages.  The first stage is an
 (unweighted) vector quantizer (VQ), with a codebook size of 32
 vectors.  The quantization errors for the codebook vector are sorted,
 and for the N best vectors a second stage quantizer is run.  By
 varying the number N, a trade-off is made between R-D performance and
 computational efficiency.  For each of the N codebook vectors, the
 Laroia weights corresponding to that vector (and not to the input
 vector) are calculated.  Then, the residual between the input LSF
 vector and the codebook vector is scaled by the square roots of these
 Laroia weights.  This scaling partially normalizes error sensitivity
 for the residual vector so that a uniform quantizer with fixed step
 sizes can be used in the second stage without too much performance
 loss.  Additionally, by scaling with Laroia weights determined from
 the first-stage codebook vector, the process can be reversed in the
 decoder.
 The second stage uses predictive delayed decision scalar
 quantization.  The quantization error is weighted by Laroia weights
 determined from the LSF input vector.  The predictor multiplies the
 previous quantized residual value by a prediction coefficient that
 depends on the vector index from the first stage VQ and on the
 location in the LSF vector.  The prediction is subtracted from the
 LSF residual value before quantizing the result and is added back
 afterwards.  This subtraction can be interpreted as shifting the
 quantization levels of the scalar quantizer, and as a result the
 quantization error of each value depends on the quantization decision
 of the previous value.  This dependency is exploited by the delayed
 decision mechanism to search for a quantization sequency with best
 R-D performance with a Viterbi-like algorithm [VITERBI].  The
 quantizer processes the residual LSF vector in reverse order (i.e.,
 it starts with the highest residual LSF value).  This is done because
 the prediction works slightly better in the reverse direction.

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 The quantization index of the first stage is entropy coded.  The
 quantization sequence from the second stage is also entropy coded,
 where for each element the probability table is chosen depending on
 the vector index from the first stage and the location of that
 element in the LSF vector.

5.2.3.5.1. LSF Stabilization

 If the input is stable, finding the best candidate usually results in
 a quantized vector that is also stable.  Because of the two-stage
 approach, however, it is possible that the best quantization
 candidate is unstable.  The encoder applies the same stabilization
 procedure applied by the decoder (see Section 4.2.7.5.4) to ensure
 the LSF parameters are within their valid range, increasingly sorted,
 and have minimum distances between each other and the border values.

5.2.3.6. LTP Quantization

 For voiced frames, the prediction analysis described in
 Section 5.2.3.4.1 resulted in four sets (one set per subframe) of
 five LTP coefficients, plus four weighting matrices.  The LTP
 coefficients for each subframe are quantized using entropy
 constrained vector quantization.  A total of three vector codebooks
 are available for quantization, with different rate-distortion trade-
 offs.  The three codebooks have 10, 20, and 40 vectors and average
 rates of about 3, 4, and 5 bits per vector, respectively.
 Consequently, the first codebook has larger average quantization
 distortion at a lower rate, whereas the last codebook has smaller
 average quantization distortion at a higher rate.  Given the
 weighting matrix W_ltp and LTP vector b, the weighted rate-distortion
 measure for a codebook vector cb_i with rate r_i is give by
             RD = u * (b - cb_i)' * W_ltp * (b - cb_i) + r_i
 where u is a fixed, heuristically determined parameter balancing the
 distortion and rate.  Which codebook gives the best performance for a
 given LTP vector depends on the weighting matrix for that LTP vector.
 For example, for a low valued W_ltp, it is advantageous to use the
 codebook with 10 vectors as it has a lower average rate.  For a large
 W_ltp, on the other hand, it is often better to use the codebook with
 40 vectors, as it is more likely to contain the best codebook vector.
 The weighting matrix W_ltp depends mostly on two aspects of the input
 signal.  The first is the periodicity of the signal; the more
 periodic, the larger W_ltp.  The second is the change in signal
 energy in the current subframe, relative to the signal one pitch lag
 earlier.  A decaying energy leads to a larger W_ltp than an
 increasing energy.  Both aspects fluctuate relatively slowly, which
 causes the W_ltp matrices for different subframes of one frame often

Valin, et al. Standards Track [Page 148] RFC 6716 Interactive Audio Codec September 2012

 to be similar.  Because of this, one of the three codebooks typically
 gives good performance for all subframes.  Therefore, the codebook
 search for the subframe LTP vectors is constrained to only allow
 codebook vectors to be chosen from the same codebook, resulting in a
 rate reduction.
 To find the best codebook, each of the three vector codebooks is used
 to quantize all subframe LTP vectors and produce a combined weighted
 rate-distortion measure for each vector codebook.  The vector
 codebook with the lowest combined rate-distortion over all subframes
 is chosen.  The quantized LTP vectors are used in the noise shaping
 quantizer, and the index of the codebook plus the four indices for
 the four subframe codebook vectors are passed on to the range
 encoder.

5.2.3.7. Pre-filter

 In the pre-filter, the input signal is filtered using the spectral
 valley de-emphasis filter coefficients from the noise shaping
 analysis (see Section 5.2.3.3).  By applying only the noise shaping
 analysis filter to the input signal, it provides the input to the
 noise shaping quantizer.

5.2.3.8. Noise Shaping Quantizer

 The noise shaping quantizer independently shapes the signal and
 coding noise spectra to obtain a perceptually higher quality at the
 same bitrate.
 The pre-filter output signal is multiplied with a compensation gain G
 computed in the noise shaping analysis.  Then, the output of a
 synthesis shaping filter is added, and the output of a prediction
 filter is subtracted to create a residual signal.  The residual
 signal is multiplied by the inverse quantized quantization gain from
 the noise shaping analysis and input to a scalar quantizer.  The
 quantization indices of the scalar quantizer represent a signal of
 pulses that is input to the pyramid range encoder.  The scalar
 quantizer also outputs a quantization signal, which is multiplied by
 the quantized quantization gain from the noise shaping analysis to
 create an excitation signal.  The output of the prediction filter is
 added to the excitation signal to form the quantized output signal
 y(n).  The quantized output signal y(n) is input to the synthesis
 shaping and prediction filters.

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 Optionally, the noise shaping quantizer operates in a delayed
 decision mode.  In this mode, it uses a Viterbi algorithm to keep
 track of multiple rounding choices in the quantizer and select the
 best one after a delay of 32 samples.  This improves the rate/
 distortion performance of the quantizer.

5.2.3.9. Constant Bitrate Mode

 SILK was designed to run in variable bitrate (VBR) mode.  However,
 the reference implementation also has a constant bitrate (CBR) mode
 for SILK.  In CBR mode, SILK will attempt to encode each packet with
 no more than the allowed number of bits.  The Opus wrapper code then
 pads the bitstream if any unused bits are left in SILK mode, or it
 encodes the high band with the remaining number of bits in Hybrid
 mode.  The number of payload bits is adjusted by changing the
 quantization gains and the rate/distortion trade-off in the noise
 shaping quantizer, in an iterative loop around the noise shaping
 quantizer and entropy coding.  Compared to the SILK VBR mode, the CBR
 mode has lower audio quality at a given average bitrate and has
 higher computational complexity.

5.3. CELT Encoder

 Most of the aspects of the CELT encoder can be directly derived from
 the description of the decoder.  For example, the filters and
 rotations in the encoder are simply the inverse of the operation
 performed by the decoder.  Similarly, the quantizers generally
 optimize for the mean square error (because noise shaping is part of
 the bitstream itself), so no special search is required.  For this
 reason, only the less straightforward aspects of the encoder are
 described here.

5.3.1. Pitch Pre-filter

 The pitch pre-filter is applied after the pre-emphasis.  It is
 applied in such a way as to be the inverse of the decoder's post-
 filter.  The main non-obvious aspect of the pre-filter is the
 selection of the pitch period.  The pitch search should be optimized
 for the following criteria:
 o  continuity: it is important that the pitch period does not change
    abruptly between frames; and
 o  avoidance of pitch multiples: when the period used is a multiple
    of the real period (lower frequency fundamental), the post-filter
    loses most of its ability to reduce noise

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5.3.2. Bands and Normalization

 The MDCT output is divided into bands that are designed to match the
 ear's critical bands for the smallest (2.5 ms) frame size.  The
 larger frame sizes use integer multiples of the 2.5 ms layout.  For
 each band, the encoder computes the energy that will later be
 encoded.  Each band is then normalized by the square root of the
 *unquantized* energy, such that each band now forms a unit vector X.
 The energy and the normalization are computed by
 compute_band_energies() and normalise_bands() (bands.c),
 respectively.

5.3.3. Energy Envelope Quantization

 Energy quantization (both coarse and fine) can be easily understood
 from the decoding process.  For all useful bitrates, the coarse
 quantizer always chooses the quantized log energy value that
 minimizes the error for each band.  Only at very low rate does the
 encoder allow larger errors to minimize the rate and avoid using more
 bits than are available.  When the available CPU requirements allow
 it, it is best to try encoding the coarse energy both with and
 without inter-frame prediction such that the best prediction mode can
 be selected.  The optimal mode depends on the coding rate, the
 available bitrate, and the current rate of packet loss.
 The fine energy quantizer always chooses the quantized log energy
 value that minimizes the error for each band because the rate of the
 fine quantization depends only on the bit allocation and not on the
 values that are coded.

5.3.4. Bit Allocation

 The encoder must use exactly the same bit allocation process as used
 by the decoder and described in Section 4.3.3.  The three mechanisms
 that can be used by the encoder to adjust the bitrate on a frame-by-
 frame basis are band boost, allocation trim, and band skipping.

5.3.4.1. Band Boost

 The reference encoder makes a decision to boost a band when the
 energy of that band is significantly higher than that of the
 neighboring bands.  Let E_j be the log-energy of band j, we define
    D_j = 2*E_j - E_j-1 - E_j+1
 The allocation of band j is boosted once if D_j > t1 and twice if D_j
 > t2.  For LM>=1, t1=2 and t2=4, while for LM<1, t1=3 and t2=5.

Valin, et al. Standards Track [Page 151] RFC 6716 Interactive Audio Codec September 2012

5.3.4.2. Allocation Trim

 The allocation trim is a value between 0 and 10 (inclusively) that
 controls the allocation balance between the low and high frequencies.
 The encoder starts with a safe "default" of 5 and deviates from that
 default in two different ways.  First, the trim can deviate by +/- 2
 depending on the spectral tilt of the input signal.  For signals with
 more low frequencies, the trim is increased by up to 2, while for
 signals with more high frequencies, the trim is decreased by up to 2.
 For stereo inputs, the trim value can be decreased by up to 4 when
 the inter-channel correlation at low frequency (first 8 bands) is
 high.

5.3.4.3. Band Skipping

 The encoder uses band skipping to ensure that the shape of the bands
 is only coded if there is at least 1/2 bit per sample available for
 the PVQ.  If not, then no bit is allocated and folding is used
 instead.  To ensure continuity in the allocation, some amount of
 hysteresis is added to the process, such that a band that received
 PVQ bits in the previous frame only needs 7/16 bit/sample to be coded
 for the current frame, while a band that did not receive PVQ bits in
 the previous frames needs at least 9/16 bit/sample to be coded.

5.3.5. Stereo Decisions

 Because CELT applies mid-side stereo coupling in the normalized
 domain, it does not suffer from important stereo image problems even
 when the two channels are completely uncorrelated.  For this reason,
 it is always safe to use stereo coupling on any audio frame.  That
 being said, there are some frames for which dual (independent) stereo
 is still more efficient.  This decision is made by comparing the
 estimated entropy with and without coupling over the first 13 bands,
 taking into account the fact that all bands with more than two MDCT
 bins require one extra degree of freedom when coded in mid-side.  Let
 L1_ms and L1_lr be the L1-norm of the mid-side vector and the L1-norm
 of the left-right vector, respectively.  The decision to use mid-side
 is made if and only if
                          L1_ms          L1_lr
                         --------    <   -----
                         bins + E        bins
 where bins is the number of MDCT bins in the first 13 bands and E is
 the number of extra degrees of freedom for mid-side coding.  For
 LM>1, E=13, otherwise E=5.

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 The reference encoder decides on the intensity stereo threshold based
 on the bitrate alone.  After taking into account the frame size by
 subtracting 80 bits per frame for coarse energy, the first band using
 intensity coding is as follows:
                   +------------------+------------+
                   | bitrate (kbit/s) | start band |
                   +------------------+------------+
                   |        <35       |      8     |
                   |                  |            |
                   |       35-50      |     12     |
                   |                  |            |
                   |       50-68      |     16     |
                   |                  |            |
                   |       84-84      |     18     |
                   |                  |            |
                   |      84-102      |     19     |
                   |                  |            |
                   |      102-130     |     20     |
                   |                  |            |
                   |       >130       |  disabled  |
                   +------------------+------------+
               Table 66: Thresholds for Intensity Stereo

5.3.6. Time-Frequency Decision

 The choice of time-frequency resolution used in Section 4.3.4.5 is
 based on R-D optimization.  The distortion is the L1-norm (sum of
 absolute values) of each band after each TF resolution under
 consideration.  The L1 norm is used because it represents the entropy
 for a Laplacian source.  The number of bits required to code a change
 in TF resolution between two bands is higher than the cost of having
 those two bands use the same resolution, which is what requires the
 R-D optimization.  The optimal decision is computed using the Viterbi
 algorithm.  See tf_analysis() in celt/celt.c.

5.3.7. Spreading Values Decision

 The choice of the spreading value in Table 59 has an impact on the
 nature of the coding noise introduced by CELT.  The larger the f_r
 value, the lower the impact of the rotation, and the more tonal the
 coding noise.  The more tonal the signal, the more tonal the noise
 should be, so the CELT encoder determines the optimal value for f_r
 by estimating how tonal the signal is.  The tonality estimate is
 based on discrete pdf (4-bin histogram) of each band.  Bands that

Valin, et al. Standards Track [Page 153] RFC 6716 Interactive Audio Codec September 2012

 have a large number of small values are considered more tonal and a
 decision is made by combining all bands with more than 8 samples.
 See spreading_decision() in celt/bands.c.

5.3.8. Spherical Vector Quantization

 CELT uses a Pyramid Vector Quantizer (PVQ) [PVQ] for quantizing the
 details of the spectrum in each band that have not been predicted by
 the pitch predictor.  The PVQ codebook consists of all sums of K
 signed pulses in a vector of N samples, where two pulses at the same
 position are required to have the same sign.  Thus, the codebook
 includes all integer codevectors y of N dimensions that satisfy
 sum(abs(y(j))) = K.
 In bands where there are sufficient bits allocated, PVQ is used to
 encode the unit vector that results from the normalization in
 Section 5.3.2 directly.  Given a PVQ codevector y, the unit vector X
 is obtained as X = y/||y||, where ||.|| denotes the L2 norm.

5.3.8.1. PVQ Search

 The search for the best codevector y is performed by alg_quant()
 (vq.c).  There are several possible approaches to the search, with a
 trade-off between quality and complexity.  The method used in the
 reference implementation computes an initial codeword y1 by
 projecting the normalized spectrum X onto the codebook pyramid of K-1
 pulses:
 y0 = truncate_towards_zero( (K-1) * X / sum(abs(X)))
 Depending on N, K and the input data, the initial codeword y0 may
 contain from 0 to K-1 non-zero values.  All the remaining pulses,
 with the exception of the last one, are found iteratively with a
 greedy search that minimizes the normalized correlation between y and
 X:
                                 T
                           J = -X * y / ||y||
 The search described above is considered to be a good trade-off
 between quality and computational cost.  However, there are other
 possible ways to search the PVQ codebook and the implementers MAY use
 any other search methods.  See alg_quant() in celt/vq.c.

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5.3.8.2. PVQ Encoding

 The vector to encode, X, is converted into an index i such that
 0 <= i < V(N,K) as follows.  Let i = 0 and k = 0.  Then, for
 j = (N - 1) down to 0, inclusive, do:
 1.  If k > 0, set i = i + (V(N-j-1,k-1) + V(N-j,k-1))/2.
 2.  Set k = k + abs(X[j]).
 3.  If X[j] < 0, set i = i + (V(N-j-1,k) + V(N-j,k))/2.
 The index i is then encoded using the procedure in Section 5.1.4 with
 ft = V(N,K).

6. Conformance

 It is our intention to allow the greatest possible choice of freedom
 in implementing the specification.  For this reason, outside of the
 exceptions noted in this section, conformance is defined through the
 reference implementation of the decoder provided in Appendix A.
 Although this document includes a prose description of the codec,
 should the description contradict the source code of the reference
 implementation, the latter shall take precedence.
 Compliance with this specification means that, in addition to
 following the normative keywords in this document, a decoder's output
 MUST also be within the thresholds specified by the opus_compare.c
 tool (included with the code) when compared to the reference
 implementation for each of the test vectors provided (see
 Appendix A.4) and for each output sampling rate and channel count
 supported.  In addition, a compliant decoder implementation MUST have
 the same final range decoder state as that of the reference decoder.
 It is therefore RECOMMENDED that the decoder implement the same
 functional behavior as the reference.  A decoder implementation is
 not required to support all output sampling rates or all output
 channel counts.

6.1. Testing

 Using the reference code provided in Appendix A, a test vector can be
 decoded with
    opus_demo -d <rate> <channels> testvectorX.bit testX.out
 where <rate> is the sampling rate and can be 8000, 12000, 16000,
 24000, or 48000, and <channels> is 1 for mono or 2 for stereo.

Valin, et al. Standards Track [Page 155] RFC 6716 Interactive Audio Codec September 2012

 If the range decoder state is incorrect for one of the frames, the
 decoder will exit with "Error: Range coder state mismatch between
 encoder and decoder".  If the decoder succeeds, then the output can
 be compared with the "reference" output with
    opus_compare -s -r <rate> testvectorX.dec testX.out
 for stereo or
    opus_compare -r <rate> testvectorX.dec testX.out
 for mono.
 In addition to indicating whether the test vector comparison passes,
 the opus_compare tool outputs an "Opus quality metric" that indicates
 how well the tested decoder matches the reference implementation.  A
 quality of 0 corresponds to the passing threshold, while a quality of
 100 is the highest possible value and means that the output of the
 tested decoder is identical to the reference implementation.  The
 passing threshold (quality 0) was calibrated in such a way that it
 corresponds to additive white noise with a 48 dB SNR (similar to what
 can be obtained on a cassette deck).  It is still possible for an
 implementation to sound very good with such a low quality measure
 (e.g., if the deviation is due to inaudible phase distortion), but
 unless this is verified by listening tests, it is RECOMMENDED that
 implementations achieve a quality above 90 for 48 kHz decoding.  For
 other sampling rates, it is normal for the quality metric to be lower
 (typically, as low as 50 even for a good implementation) because of
 harmless mismatch with the delay and phase of the internal sampling
 rate conversion.
 On POSIX environments, the run_vectors.sh script can be used to
 verify all test vectors.  This can be done with
    run_vectors.sh <exec path> <vector path> <rate>
 where <exec path> is the directory where the opus_demo and
 opus_compare executables are built and <vector path> is the directory
 containing the test vectors.

6.2. Opus Custom

 Opus Custom is an OPTIONAL part of the specification that is defined
 to handle special sample rates and frame rates that are not supported
 by the main Opus specification.  Use of Opus Custom is discouraged
 for all but very special applications for which a frame size
 different from 2.5, 5, 10, or 20 ms is needed (for either complexity
 or latency reasons).  Because Opus Custom is optional, streams

Valin, et al. Standards Track [Page 156] RFC 6716 Interactive Audio Codec September 2012

 encoded using Opus Custom cannot be expected to be decodable by all
 Opus implementations.  Also, because no in-band mechanism exists for
 specifying the sampling rate and frame size of Opus Custom streams,
 out-of-band signaling is required.  In Opus Custom operation, only
 the CELT layer is available, using the opus_custom_* function calls
 in opus_custom.h.

7. Security Considerations

 Like any other audio codec, Opus should not be used with insecure
 ciphers or cipher-modes that are vulnerable to known plaintext
 attacks.  In addition to the zeros used in Opus padding, digital
 silence frames generate predictable compressed results and the TOC
 byte may have an easily predictable value.
 Implementations of the Opus codec need to take appropriate security
 considerations into account, as outlined in [DOS].  It is extremely
 important for the decoder to be robust against malicious payloads.
 Malicious payloads must not cause the decoder to overrun its
 allocated memory or to take an excessive amount of resources to
 decode.  Although problems in encoders are typically rarer, the same
 applies to the encoder.  Malicious audio streams must not cause the
 encoder to misbehave because this would allow an attacker to attack
 transcoding gateways.
 The reference implementation contains no known buffer overflow or
 cases where a specially crafted packet or audio segment could cause a
 significant increase in CPU load.  However, on certain CPU
 architectures where denormalized floating-point operations are much
 slower than normal floating-point operations, it is possible for some
 audio content (e.g., silence or near silence) to cause an increase in
 CPU load.  Denormals can be introduced by reordering operations in
 the compiler and depend on the target architecture, so it is
 difficult to guarantee that an implementation avoids them.  For
 architectures on which denormals are problematic, adding very small
 floating-point offsets to the affected signals to prevent significant
 numbers of denormalized operations is RECOMMENDED.  Alternatively, it
 is often possible to configure the hardware to treat denormals as
 zero (DAZ).  No such issue exists for the fixed-point reference
 implementation.
 The reference implementation was validated in the following
 conditions:
 1.  Sending the decoder valid packets generated by the reference
     encoder and verifying that the decoder's final range coder state
     matches that of the encoder.

Valin, et al. Standards Track [Page 157] RFC 6716 Interactive Audio Codec September 2012

 2.  Sending the decoder packets generated by the reference encoder
     and then subjected to random corruption.
 3.  Sending the decoder random packets.
 4.  Sending the decoder packets generated by a version of the
     reference encoder modified to make random coding decisions
     (internal fuzzing), including mode switching, and verifying that
     the range coder final states match.
 In all of the conditions above, both the encoder and the decoder were
 run inside the Valgrind [VALGRIND] memory debugger, which tracks
 reads and writes to invalid memory regions as well as the use of
 uninitialized memory.  There were no errors reported on any of the
 tested conditions.

8. Acknowledgements

 Thanks to all other developers, including Henrik Astrom, Jon
 Bergenheim, Raymond Chen, Soren Skak Jensen, Gregory Maxwell,
 Christopher Montgomery, and Karsten Vandborg Sorensen.  We would also
 like to thank Igor Dyakonov, Hoang Thi Minh Nguyet, Christian Hoene,
 Gian-Carlo Pascutto, and Jan Skoglund for their help with testing of
 the Opus codec.  Thanks to Andrew D'Addesio, Elwyn Davies, Ralph
 Giles, John Ridges, Ben Schwartz, Kat Walsh, Mark Warner, Keith Yan,
 and many others on the Opus and CELT mailing lists for their bug
 reports and feedback.  At last, the authors would like to thank
 Robert Sparks, Cullen Jennings, and Jonathan Rosenberg for their
 support throughout the standardization process.

Valin, et al. Standards Track [Page 158] RFC 6716 Interactive Audio Codec September 2012

9. References

9.1. Normative References

 [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
            Requirement Levels", BCP 14, RFC 2119, March 1997.

9.2. Informative References

 [BURG]     Burg, J., "Maximum Entropy Spectral Analysis", Proceedings
            of the 37th Annual International SEG Meeting, Vol.
            6, 1975.
 [CELT]     Valin, JM., Terriberry, T., Maxwell, G., and C.
            Montgomery, "Constrained-Energy Lapped Transform (CELT)
            Codec", Work in Progress, July 2010.
 [CODING-THESIS]
            Pasco, R., "Source coding algorithms for fast data
            compression", Ph.D. thesis Dept. of Electrical
            Engineering, Stanford University, May 1976.
 [DOS]      Handley, M., Rescorla, E., and IAB, "Internet Denial-of-
            Service Considerations", RFC 4732, December 2006.
 [FFT]      Wikipedia, "Fast Fourier Transform",
            <http://en.wikipedia.org/w/
            index.php?title=Fast_Fourier_transform&oldid=508004516>.
 [GOOGLE-NETEQ]
            "Google NetEQ code", <http://code.google.com/p/webrtc/
            source/browse/trunk/src/modules/audio_coding/NetEQ/main/
            source/?r=583>.
 [GOOGLE-WEBRTC]
            "Google WebRTC code", <http://code.google.com/p/webrtc/>.
 [HADAMARD] Wikipedia, "Hadamard Transform", <http://en.wikipedia.org/
            w/index.php?title=Hadamard_transform&oldid=508252957>.
 [KABAL86]  Kabal, P. and R. Ramachandran, "The Computation of Line
            Spectral Frequencies Using Chebyshev Polynomials", IEEE
            Trans. Acoustics, Speech, Signal Processing, Vol. 34, no.
            6, pp. 1419-1426, December 1986.

Valin, et al. Standards Track [Page 159] RFC 6716 Interactive Audio Codec September 2012

 [LAROIA-ICASSP]
            Laroia, R., Phamdo, N., and N. Farvardin, "Robust and
            Efficient Quantization of Speech LSP Parameters Using
            Structured Vector Quantization", ICASSP-1991, Proc. IEEE
            Int. Conf. Acoust., Speech, Signal Processing, pp. 641-
            644, October 1991.
 [LPC]      Wikipedia, "Linear Prediction", <http://en.wikipedia.org/
            w/index.php?title=Linear_prediction&oldid=497201278>.
 [MARTIN79] Martin, G., "Range encoding: An algorithm for removing
            redundancy from a digitised message", Proc. Institution of
            Electronic and Radio Engineers International Conference on
            Video and Data Recording, 1979.
 [MATROSKA-WEBSITE]
            "Matroska website", <http://matroska.org/>.
 [MDCT]     Wikipedia, "Modified Discrete Cosine Transform", <http://
            en.wikipedia.org/w/
            index.php?title=Modified_discrete_cosine_
            transform&oldid=490295438>.
 [OPUS-GIT] "Opus Git Repository", <https://git.xiph.org/opus.git>.
 [OPUS-WEBSITE]
            "Opus website", <http://opus-codec.org/>.
 [PRINCEN86]
            Princen, J. and A. Bradley, "Analysis/Synthesis Filter
            Bank Design Based on Time Domain Aliasing Cancellation",
            IEEE Trans. Acoustics, Speech, and Siginal Processing,
            ASSP-34 (5), pp. 1153-1161, October, 1986.
 [PVQ]      Fischer, T., "A Pyramid Vector Quantizer", IEEE Trans. on
            Information Theory, Vol. 32, pp. 568-583, July 1986.
 [RANGE-CODING]
            Wikipedia, "Range Coding", <http://en.wikipedia.org/w/
            index.php?title=Range_encoding&oldid=509582757>.
 [REQUIREMENTS]
            Valin, JM. and K. Vos, "Requirements for an Internet Audio
            Codec", RFC 6366, August 2011.

Valin, et al. Standards Track [Page 160] RFC 6716 Interactive Audio Codec September 2012

 [RFC3533]  Pfeiffer, S., "The Ogg Encapsulation Format Version 0",
            RFC 3533, May 2003.
 [RFC3550]  Schulzrinne, H., Casner, S., Frederick, R., and V.
            Jacobson, "RTP: A Transport Protocol for Real-Time
            Applications", STD 64, RFC 3550, July 2003.
 [SCHUR]    Le Roux, J. and C. Gueguen, "A fixed point computation of
            partial correlation coefficients", ICASSP-1977, Proc. IEEE
            Int. Conf. Acoustics, Speech, and Signal Processing, pp.
            257-259, June 1977.
 [SILK]     Vos, K., Jensen, S., and K. Sorensen, "SILK Speech Codec",
            Work in Progress, September 2010.
 [SPECTRAL-PAIRS]
            Wikipedia, "Line Spectral Pairs", <http://
            en.wikipedia.org/w/
            index.php?title=Line_spectral_pairs&oldid=365426016>.
 [SRTP-VBR] Perkins, C. and JM. Valin, "Guidelines for the Use of
            Variable Bit Rate Audio with Secure RTP", RFC 6562,
            March 2012.
 [VALGRIND] "Valgrind website", <http://valgrind.org/>.
 [VALIN2010]
            Valin, JM., Terriberry, T., Montgomery, C., and G.
            Maxwell, "A High-Quality Speech and Audio Codec With Less
            Than 10 ms Delay", IEEE Trans. on Audio, Speech, and
            Language Processing, Vol. 18, No. 1, pp. 58-67 2010.
 [VECTORS-PROC]
            "Opus Testvectors (proceedings)", <http://www.ietf.org/
            proceedings/83/slides/slides-83-codec-0.gz>.
 [VECTORS-WEBSITE]
            "Opus Testvectors (website)",
            <http://opus-codec.org/testvectors/>.
 [VITERBI]  Wikipedia, "Viterbi Algorithm", <http://en.wikipedia.org/
            w/index.php?title=Viterbi_algorithm&oldid=508835871>.
 [VORBIS-WEBSITE]
            "Vorbis website", <http://xiph.org/vorbis/>.

Valin, et al. Standards Track [Page 161] RFC 6716 Interactive Audio Codec September 2012

 [WHITENING]
            Wikipedia, "White Noise", <http://en.wikipedia.org/w/
            index.php?title=White_noise&oldid=497791998>.
 [Z-TRANSFORM]
            Wikipedia, "Z-transform", <http://en.wikipedia.org/w/
            index.php?title=Z-transform&oldid=508392884>.
 [ZWICKER61]
            Zwicker, E., "Subdivision of the Audible Frequency Range
            into Critical Bands", The Journal of the Acoustical
            Society of America, Vol. 33, No 2 pp. 248, February 1961.

Valin, et al. Standards Track [Page 162] RFC 6716 Interactive Audio Codec September 2012

Appendix A. Reference Implementation

 This appendix contains the complete source code for the reference
 implementation of the Opus codec written in C.  By default, this
 implementation relies on floating-point arithmetic, but it can be
 compiled to use only fixed-point arithmetic by defining the
 FIXED_POINT macro.  The normative behavior is defined as the output
 using the floating-point configuration.  Information on building and
 using the reference implementation is available in the README file.
 The implementation can be compiled with either a C89 or a C99
 compiler.  It is reasonably optimized for most platforms such that
 only architecture-specific optimizations are likely to be useful.
 The Fast Fourier Transform (FFT) [FFT] used is a slightly modified
 version of the KISS-FFT library, but it is easy to substitute any
 other FFT library.
 While the reference implementation does not rely on any _undefined
 behavior_ as defined by C89 or C99, it relies on common
 _implementation-defined behavior_ for two's complement architectures:
 o  Right shifts of negative values are consistent with two's
    complement arithmetic, so that a>>b is equivalent to
    floor(a/(2**b)),
 o  For conversion to a signed integer of N bits, the value is reduced
    modulo 2**N to be within range of the type,
 o  The result of integer division of a negative value is truncated
    towards zero, and
 o  The compiler provides a 64-bit integer type (a C99 requirement
    which is supported by most C89 compilers).
 In its current form, the reference implementation also requires the
 following architectural characteristics to obtain acceptable
 performance:
 o  Two's complement arithmetic,
 o  At least a 16 bit by 16 bit integer multiplier (32-bit result),
    and
 o  At least a 32-bit adder/accumulator.

Valin, et al. Standards Track [Page 163] RFC 6716 Interactive Audio Codec September 2012

A.1. Extracting the Source

 The complete source code can be extracted from this document, by
 running the following command line:
 o  cat rfc6716.txt | grep '^\ \ \ ###' | sed -e 's/...###//' | base64
    --decode > opus-rfc6716.tar.gz
 o  tar xzvf opus-rfc6716.tar.gz
 o  cd opus-rfc6716
 o  make
 On systems where the provided Makefile does not work, the following
 command line may be used to compile the source code:
 o  cc -O2 -g -o opus_demo src/opus_demo.c `cat *.mk | grep -v fixed |
    sed -e 's/.*=//' -e 's/\\\\//'` -DOPUS_BUILD -Iinclude -Icelt
    -Isilk -Isilk/float -DUSE_ALLOCA -Drestrict= -lm
 On systems where the base64 utility is not present, the following
 commands can be used instead:
 o  cat rfc6716.txt | grep '^\ \ \ ###' | sed -e 's/...###//' >
    opus.b64
 o  openssl base64 -d -in opus.b64 > opus-rfc6716.tar.gz
 The SHA1 hash of the opus-rfc6716.tar.gz file is
 86a927223e73d2476646a1b933fcd3fffb6ecc8c.

A.2. Up-to-Date Implementation

 As of the time of publication of this memo, an up-to-date
 implementation conforming to this RFC is available in a Git
 repository [OPUS-GIT].  Releases and other resources are available at
 [OPUS-WEBSITE].  However, although that implementation is expected to
 remain conformant with the RFC, it is the code in this document that
 shall remain normative.

A.3. Base64-Encoded Source Code

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Valin, et al. Standards Track [Page 164] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 165] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 166] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 167] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 168] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 169] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 170] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 171] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 172] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 173] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 174] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 176] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 177] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 178] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 179] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 182] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 183] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 184] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 185] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 186] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 187] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 188] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 189] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 190] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 191] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 192] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 193] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 194] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 195] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 199] RFC 6716 Interactive Audio Codec September 2012

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 ###iz6Wth4zcNyc7oEdM5wtmgDHFmspuMpd/AM621RNMtMkEoXXW/D67d7hKcs3xn9p

Valin, et al. Standards Track [Page 201] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 202] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 205] RFC 6716 Interactive Audio Codec September 2012

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 ###2rGskGpPqxbFgvWTvGxQldM66RiODGyQJFBMAVu49L2QjhVTdHA368KS9sMs7OLB

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Valin, et al. Standards Track [Page 211] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 212] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 213] RFC 6716 Interactive Audio Codec September 2012

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 ###b7owKhY32Fiy8nHXDRFW9MOZvf8iNVUxXLrkuY6xFAhKiIVqXIAGxYynlmWgMj4a

Valin, et al. Standards Track [Page 214] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 215] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 216] RFC 6716 Interactive Audio Codec September 2012

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 ###OIgj50fffopj9U/HYyUQ/eAazQSBHyrPYKmNH4AGKhisg378oP5o0A8lc4B698OX

Valin, et al. Standards Track [Page 217] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 218] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 219] RFC 6716 Interactive Audio Codec September 2012

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 ###aqUC7FupNdrjAmyHANL+etweYJcOzjNgjgqFzmiEQSfDMkoSZXlmzyuASAx0h74j

Valin, et al. Standards Track [Page 220] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 221] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 222] RFC 6716 Interactive Audio Codec September 2012

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 ###6CtobwMJXERzmjFhtQsv32w1DY8NPSTpZ5+ohaLezM19YLJab5/F7f8xeOaspmvo

Valin, et al. Standards Track [Page 223] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 224] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 225] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 226] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 227] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 228] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 229] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 230] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 231] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 232] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 233] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 234] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 235] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 237] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 238] RFC 6716 Interactive Audio Codec September 2012

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 ###N/T8L+vraxuF/1fh/1X4fxX+X4X/V+H/Vfh/Ff5fPy/+K1KDJHBF+aKqSPqchCkq

Valin, et al. Standards Track [Page 239] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 240] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 241] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 242] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 243] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 244] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 245] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 246] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 247] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 248] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 249] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 250] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 251] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 253] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 254] RFC 6716 Interactive Audio Codec September 2012

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 ###ZYTZlMUbftyUamtbrCRlHqFPoq80VRGbVWke1qW8ZA64vFvLGKbBGmn9mUJH7VeX

Valin, et al. Standards Track [Page 255] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 256] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 257] RFC 6716 Interactive Audio Codec September 2012

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 ###l9ZzA2/i9o44XtjUtBJ+jQGnrLdkFgr3R8PtH/3WbNj1db5adRPfLRejHqK97BiG

Valin, et al. Standards Track [Page 258] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 259] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 261] RFC 6716 Interactive Audio Codec September 2012

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 ###NBjvdLS5ZYtMybFB1fbWzo42jbG9oEMExGiGejIIBRhrKtngwTf0JudRt/+J9L88

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Valin, et al. Standards Track [Page 263] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 265] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 266] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 267] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 269] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 270] RFC 6716 Interactive Audio Codec September 2012

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 ###OoeubhBhyCtOXGgWEKnlz3vlXaitIpafCbpR8jUriTJfFbcQLUrJeL0snPnvUTjz

Valin, et al. Standards Track [Page 271] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 272] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 273] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 274] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 277] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 278] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 279] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 280] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 282] RFC 6716 Interactive Audio Codec September 2012

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 ###XxxAPqaCCGGcYMoKHDPRstyyUZJ4Mh1heSgqfq2VsssAjqJfR8n+oJORrulpiSoo

Valin, et al. Standards Track [Page 284] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 285] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 286] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 287] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 288] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 289] RFC 6716 Interactive Audio Codec September 2012

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 ###N8VtkzxkjFEHVBfzGUqBu6H2XbvV/9uYmEaf99wRvJLONv72fjD6xAcQPodHxz81

Valin, et al. Standards Track [Page 290] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 291] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 293] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 294] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 295] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 297] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 298] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 299] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 300] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 301] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 304] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 305] RFC 6716 Interactive Audio Codec September 2012

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 ###b/eBeI4viYi8SvN1A0kpCdajLgmqHFLa9LCmBCJNnrsmyaRigTNrhaj6pNMunNvN

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Valin, et al. Standards Track [Page 307] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 308] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 309] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 310] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 313] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 314] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 315] RFC 6716 Interactive Audio Codec September 2012

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 ###mhtrpRUPzFfloMsr11q9s2l8XnXA2rosNWi0knJFp3GtxSmBESZ6Gh+Q/edP40bL

Valin, et al. Standards Track [Page 316] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 317] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 318] RFC 6716 Interactive Audio Codec September 2012

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 ###M63+uytpO7PwDmQAXtURTFbJyEiMXVfr2shf0PoguohvBnO2Lg3sBW2z52jOlg37

Valin, et al. Standards Track [Page 319] RFC 6716 Interactive Audio Codec September 2012

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Valin, et al. Standards Track [Page 320] RFC 6716 Interactive Audio Codec September 2012

A.4. Test Vectors

 Because of size constraints, the Opus test vectors are not
 distributed in this document.  They are available in the proceedings
 of the 83rd IETF meeting (Paris) [VECTORS-PROC] and from the Opus
 codec website at [VECTORS-WEBSITE].  These test vectors were created
 specifically to exercise all aspects of the decoder.  Therefore, the
 audio quality of the decoded output is significantly lower than what
 Opus can achieve in normal operation.
 The SHA1 hash of the files in the test vector package are
 e49b2862ceec7324790ed8019eb9744596d5be01  testvector01.bit
 b809795ae1bcd606049d76de4ad24236257135e0  testvector02.bit
 e0c4ecaeab44d35a2f5b6575cd996848e5ee2acc  testvector03.bit
 a0f870cbe14ebb71fa9066ef3ee96e59c9a75187  testvector04.bit
 9b3d92b48b965dfe9edf7b8a85edd4309f8cf7c8  testvector05.bit
 28e66769ab17e17f72875283c14b19690cbc4e57  testvector06.bit
 bacf467be3215fc7ec288f29e2477de1192947a6  testvector07.bit
 ddbe08b688bbf934071f3893cd0030ce48dba12f  testvector08.bit
 3932d9d61944dab1201645b8eeaad595d5705ecb  testvector09.bit
 521eb2a1e0cc9c31b8b740673307c2d3b10c1900  testvector10.bit
 6bc8f3146fcb96450c901b16c3d464ccdf4d5d96  testvector11.bit
 338c3f1b4b97226bc60bc41038becbc6de06b28f  testvector12.bit
 a20a2122d42de644f94445e20185358559623a1f  testvector01.dec
 48ac1ff1995250a756e1e17bd32acefa8cd2b820  testvector02.dec
 d15567e919db2d0e818727092c0af8dd9df23c95  testvector03.dec
 1249dd28f5bd1e39a66fd6d99449dca7a8316342  testvector04.dec
 93eee37e5d26a456d2c24483060132ff7eae2143  testvector05.dec
 a294fc17e3157768c46c5ec0f2116de0d2c37ee2  testvector06.dec
 2bf550e2f072e0941438db3f338fe99444385848  testvector07.dec
 2695c1f2d1f9748ea0bf07249c70fd7b87f61680  testvector08.dec
 12862add5d53a9d2a7079340a542a2f039b992bb  testvector09.dec
 a081252bb2b1a902fdc500530891f47e2a373d84  testvector10.dec
 dfd0f844f2a42df506934fac2100a3c03beec711  testvector11.dec
 8c16b2a1fb60e3550ba165068f9d7341357fdb63  testvector12.dec

Appendix B. Self-Delimiting Framing

 To use the internal framing described in Section 3, the decoder must
 know the total length of the Opus packet, in bytes.  This section
 describes a simple variation of that framing that can be used when
 the total length of the packet is not known.  Nothing in the encoding
 of the packet itself allows a decoder to distinguish between the
 regular, undelimited framing and the self-delimiting framing
 described in this appendix.  Which one is used and where must be

Valin, et al. Standards Track [Page 321] RFC 6716 Interactive Audio Codec September 2012

 established by context at the transport layer.  It is RECOMMENDED
 that a transport layer choose exactly one framing scheme, rather than
 allowing an encoder to signal which one it wants to use.
 For example, although a regular Opus stream does not support more
 than two channels, a multi-channel Opus stream may be formed from
 several one- and two-channel streams.  To pack an Opus packet from
 each of these streams together in a single packet at the transport
 layer, one could use the self-delimiting framing for all but the last
 stream, and then the regular, undelimited framing for the last one.
 Reverting to the undelimited framing for the last stream saves
 overhead (because the total size of the transport-layer packet will
 still be known), and ensures that a "multi-channel" stream that only
 has a single Opus stream uses the same framing as a regular Opus
 stream does.  This avoids the need for signaling to distinguish these
 two cases.
 The self-delimiting framing is identical to the regular, undelimited
 framing from Section 3, except that each Opus packet contains one
 extra length field, encoded using the same one- or two-byte scheme
 from Section 3.2.1.  This extra length immediately precedes the
 compressed data of the first Opus frame in the packet, and is
 interpreted in the various modes as follows:
 o  Code 0 packets: It is the length of the single Opus frame (see
    Figure 25).
 o  Code 1 packets: It is the length used for both of the Opus frames
    (see Figure 26).
 o  Code 2 packets: It is the length of the second Opus frame (see
    Figure 27).
 o  CBR Code 3 packets: It is the length used for all of the Opus
    frames (see Figure 28).
 o  VBR Code 3 packets: It is the length of the last Opus frame (see
    Figure 29).

Valin, et al. Standards Track [Page 322] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|0|0| N1 (1-2 bytes):                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
   |               Compressed frame 1 (N1 bytes)...                :
   :                                                               |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
               Figure 25: A Self-Delimited Code 0 Packet
    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|0|1| N1 (1-2 bytes):                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
   |               Compressed frame 1 (N1 bytes)...                |
   :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                               |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
   |               Compressed frame 2 (N1 bytes)...                |
   :                                               +-+-+-+-+-+-+-+-+
   |                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
               Figure 26: A Self-Delimited Code 1 Packet
    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|0| N1 (1-2 bytes): N2 (1-2 bytes :               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+               :
   |               Compressed frame 1 (N1 bytes)...                |
   :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                               |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
   |               Compressed frame 2 (N2 bytes)...                :
   :                                                               |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
               Figure 27: A Self-Delimited Code 2 Packet

Valin, et al. Standards Track [Page 323] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|1|0|p|     M     | Pad len (Opt) : N1 (1-2 bytes):
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 1 (N1 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 2 (N1 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :                              ...                              :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame M (N1 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :                  Opus Padding (Optional)...                   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
             Figure 28: A Self-Delimited CBR Code 3 Packet

Valin, et al. Standards Track [Page 324] RFC 6716 Interactive Audio Codec September 2012

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | config  |s|1|1|1|p|     M     | Padding length (Optional)     :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   : N1 (1-2 bytes):     ...       :     N[M-1]    |     N[M]      :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 1 (N1 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :               Compressed frame 2 (N2 bytes)...                :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :                              ...                              :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   :              Compressed frame M (N[M] bytes)...               :
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   :                  Opus Padding (Optional)...                   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
             Figure 29: A Self-Delimited VBR Code 3 Packet

Valin, et al. Standards Track [Page 325] RFC 6716 Interactive Audio Codec September 2012

Authors' Addresses

 Jean-Marc Valin
 Mozilla Corporation
 650 Castro Street
 Mountain View, CA  94041
 USA
 Phone: +1 650 903-0800
 EMail: jmvalin@jmvalin.ca
 Koen Vos
 Skype Technologies S.A.
 Soder Malarstrand 43
 Stockholm,   11825
 SE
 Phone: +46 73 085 7619
 EMail: koenvos74@gmail.com
 Timothy B. Terriberry
 Mozilla Corporation
 650 Castro Street
 Mountain View, CA  94041
 USA
 Phone: +1 650 903-0800
 EMail: tterriberry@mozilla.com

Valin, et al. Standards Track [Page 326]

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