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Versions: (draft-ietf-codec-harmony) 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 RFC 6716

Network Working Group                                          JM. Valin
Internet-Draft                                       Mozilla Corporation
Intended status: Standards Track                                  K. Vos
Expires: May 3, 2012                             Skype Technologies S.A.
                                                           T. Terriberry
                                                     Mozilla Corporation
                                                        October 31, 2011


                   Definition of the Opus Audio Codec
                        draft-ietf-codec-opus-10

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 bit-rate narrowband speech at 6 kb/s to very high quality stereo
   music at 510 kb/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 Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at http://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on May 3, 2012.

Copyright Notice

   Copyright (c) 2011 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



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   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.


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 bit-rate  . . . . . . . . . . . . .  12
       2.1.9.  Discontinuous transmission (DTX)  . . . . . . . . . .  12
   3.  Internal Framing  . . . . . . . . . . . . . . . . . . . . . .  13
     3.1.  The TOC Byte  . . . . . . . . . . . . . . . . . . . . . .  13
     3.2.  Frame Packing . . . . . . . . . . . . . . . . . . . . . .  15
       3.2.1.  Frame Length Coding . . . . . . . . . . . . . . . . .  15
       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 . . . . . . . . . . . . . . . . . . .  16
       3.2.4.  Code 2: Two Frames in the Packet, with Different
               Compressed Sizes  . . . . . . . . . . . . . . . . . .  16
       3.2.5.  Code 3: An Arbitrary Number of Frames in the Packet .  17
     3.3.  Examples  . . . . . . . . . . . . . . . . . . . . . . . .  20
     3.4.  Extending Opus  . . . . . . . . . . . . . . . . . . . . .  21
   4.  Opus Decoder  . . . . . . . . . . . . . . . . . . . . . . . .  22
     4.1.  Range Decoder . . . . . . . . . . . . . . . . . . . . . .  22
       4.1.1.  Decoding Symbols  . . . . . . . . . . . . . . . . . .  24
       4.1.2.  Alternate Decoding Methods  . . . . . . . . . . . . .  25
       4.1.3.  Decoding Raw Bits . . . . . . . . . . . . . . . . . .  27
       4.1.4.  Decoding Uniformly Distributed Integers . . . . . . .  27
       4.1.5.  Current Bit Usage . . . . . . . . . . . . . . . . . .  28
     4.2.  SILK Decoder  . . . . . . . . . . . . . . . . . . . . . .  30
       4.2.1.  SILK Decoder Modules  . . . . . . . . . . . . . . . .  30
       4.2.2.  LP Layer Organization . . . . . . . . . . . . . . . .  32
       4.2.3.  Header Bits . . . . . . . . . . . . . . . . . . . . .  34
       4.2.4.  Per-Frame LBRR Flags  . . . . . . . . . . . . . . . .  35
       4.2.5.  LBRR Frames . . . . . . . . . . . . . . . . . . . . .  35



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       4.2.6.  Regular SILK Frames . . . . . . . . . . . . . . . . .  36
       4.2.7.  SILK Frame Contents . . . . . . . . . . . . . . . . .  36
         4.2.7.1.  Stereo Prediction Weights . . . . . . . . . . . .  39
         4.2.7.2.  Mid-only Flag . . . . . . . . . . . . . . . . . .  41
         4.2.7.3.  Frame Type  . . . . . . . . . . . . . . . . . . .  42
         4.2.7.4.  Subframe Gains  . . . . . . . . . . . . . . . . .  43
         4.2.7.5.  Normalized Line Spectral Frequency (LSF) and
                   Linear Predictive Coding (LPC) Coefficients . . .  45
         4.2.7.6.  Long-Term Prediction (LTP) Parameters . . . . . .  72
         4.2.7.7.  Linear Congruential Generator (LCG) Seed  . . . .  84
         4.2.7.8.  Excitation  . . . . . . . . . . . . . . . . . . .  84
         4.2.7.9.  SILK Frame Reconstruction . . . . . . . . . . . .  96
       4.2.8.  Stereo Unmixing . . . . . . . . . . . . . . . . . . .  99
       4.2.9.  Resampling  . . . . . . . . . . . . . . . . . . . . . 100
     4.3.  CELT Decoder  . . . . . . . . . . . . . . . . . . . . . . 101
       4.3.1.  Transient Decoding  . . . . . . . . . . . . . . . . . 103
       4.3.2.  Energy Envelope Decoding  . . . . . . . . . . . . . . 104
       4.3.3.  Bit Allocation  . . . . . . . . . . . . . . . . . . . 105
       4.3.4.  Shape Decoding  . . . . . . . . . . . . . . . . . . . 110
       4.3.5.  Anti-Collapse Processing  . . . . . . . . . . . . . . 114
       4.3.6.  Denormalization . . . . . . . . . . . . . . . . . . . 114
       4.3.7.  Inverse MDCT  . . . . . . . . . . . . . . . . . . . . 114
     4.4.  Packet Loss Concealment (PLC) . . . . . . . . . . . . . . 116
       4.4.1.  Clock Drift Compensation  . . . . . . . . . . . . . . 116
     4.5.  Configuration Switching . . . . . . . . . . . . . . . . . 117
       4.5.1.  Transition Side Information (Redundancy)  . . . . . . 118
       4.5.2.  State Reset . . . . . . . . . . . . . . . . . . . . . 121
       4.5.3.  Summary of Transitions  . . . . . . . . . . . . . . . 121
   5.  Opus Encoder  . . . . . . . . . . . . . . . . . . . . . . . . 124
     5.1.  Range Coder . . . . . . . . . . . . . . . . . . . . . . . 125
       5.1.1.  Encoding Symbols  . . . . . . . . . . . . . . . . . . 125
       5.1.2.  Encoding Raw Bits . . . . . . . . . . . . . . . . . . 126
       5.1.3.  Encoding Uniformly Distributed Integers . . . . . . . 127
       5.1.4.  Finalizing the Stream . . . . . . . . . . . . . . . . 127
       5.1.5.  Current Bit Usage . . . . . . . . . . . . . . . . . . 127
     5.2.  SILK Encoder  . . . . . . . . . . . . . . . . . . . . . . 128
       5.2.1.  Sample Rate Conversion  . . . . . . . . . . . . . . . 128
       5.2.2.  Stereo Mixing . . . . . . . . . . . . . . . . . . . . 128
       5.2.3.  SILK Core Encoder . . . . . . . . . . . . . . . . . . 129
     5.3.  CELT Encoder  . . . . . . . . . . . . . . . . . . . . . . 142
       5.3.1.  Pitch Prefilter . . . . . . . . . . . . . . . . . . . 142
       5.3.2.  Bands and Normalization . . . . . . . . . . . . . . . 143
       5.3.3.  Energy Envelope Quantization  . . . . . . . . . . . . 143
       5.3.4.  Bit allocation  . . . . . . . . . . . . . . . . . . . 143
       5.3.5.  Stereo decisions  . . . . . . . . . . . . . . . . . . 144
       5.3.6.  Time-Frequency Decision . . . . . . . . . . . . . . . 145
       5.3.7.  Spreading Values Decision . . . . . . . . . . . . . . 146
       5.3.8.  Spherical Vector Quantization . . . . . . . . . . . . 146



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   6.  Conformance . . . . . . . . . . . . . . . . . . . . . . . . . 148
     6.1.  Testing . . . . . . . . . . . . . . . . . . . . . . . . . 148
     6.2.  Opus Custom . . . . . . . . . . . . . . . . . . . . . . . 149
   7.  Security Considerations . . . . . . . . . . . . . . . . . . . 150
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . . 152
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . 153
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . . 154
     10.1. Normative References  . . . . . . . . . . . . . . . . . . 154
     10.2. Informative References  . . . . . . . . . . . . . . . . . 154
   Appendix A.  Reference Implementation . . . . . . . . . . . . . . 157
     A.1.  Extracting the source . . . . . . . . . . . . . . . . . . 157
     A.2.  Development Versions  . . . . . . . . . . . . . . . . . . 158
     A.3.  Base64-encoded source code  . . . . . . . . . . . . . . . 158
     A.4.  Test vectors  . . . . . . . . . . . . . . . . . . . . . . 316
   Appendix B.  Self-Delimiting Framing  . . . . . . . . . . . . . . 318
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 322



































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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 linear prediction (LP)-based layer and a Modified Discrete Cosine
   Transform (MDCT)-based layer.  The main idea behind using two layers
   is that in speech, linear prediction techniques (such as 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, by combining
   them, achieve better quality than 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.  The decoder contains significant amounts of
   integer and fixed-point arithmetic which must be performed exactly,
   including all rounding considerations, so any useful specification
   must make extensive use of 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 English 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
   multiplication in the text.  In general, the text is focused on the
   "what" and "why" while the symbolic representation most clearly
   provides the "how".




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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].

   Even when using floating-point, various operations in the codec
   require bit-exact fixed-point behavior.  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.  E.g., 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" is used to
   indicate 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, the lower bound is the one that is
   enforced.

1.1.4.  sign(x)

   The sign of x, i.e.,

                                   ( -1,  x < 0 ,
                         sign(x) = <  0,  x == 0 ,
                                   (  1,  x > 0 .








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1.1.5.  log2(f)

   The base-two logarithm of f.

1.1.6.  ilog(n)

   The minimum number of bits required to store a positive integer n in
   two's complement notation, 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

   o  ilog(7) = 3























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2.  Opus Codec Overview

   The Opus codec scales from 6 kb/s narrowband mono speech to 510 kb/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 degredations 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.

   The LP layer is based on the SILK [1] codec [SILK].  It supports NB,



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   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 1.2 ms) may be required for sampling rate
   conversion.  Like Vorbis 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 CELT [2] 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, so for music only the MDCT layer should be used.

   A "Hybrid" mode allows the use of both layers simultaneously with a
   frame size of 10 or 20 ms and a SWB or FB audio bandwidth.  Each
   frame is split into a low frequency signal and a high frequency
   signal, with a cutoff of 8 kHz.  The LP layer then codes the low
   frequency signal, followed by the MDCT layer coding the high
   frequency signal.  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 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.

   After conversion to the common, desired output sample rate, the
   decoder simply adds the output from the two layers together.  To



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   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 which 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
   bit-stream is signalled in-band such that a decoder can decode any
   Opus stream without any out-of-band signalling.  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 kb/s to 510 kb/s.  All other
   parameters being equal, higher bit-rate 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 kb/s for narrowband speech

   o  16-20 kb/s for wideband speech

   o  28-40 kb/s for fullband speech

   o  48-64 kb/s for fullband mono music

   o  64-128 kb/s for fullband stereo music








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2.1.2.  Number of channels (mono/stereo)

   Opus can transmit either mono or stereo audio within one stream.
   When decoding a mono stream in stereo, the left and right channels
   will be identical and when decoding a stereo channel in mono, the
   mono output will be the average of the encoded left and right
   channels.  In some cases it is desirable to encode a stereo input
   stream in mono (e.g. because the bit-rate is insufficient for good
   quality stereo).  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 encoded
   at any bandwidth.  For example, any Opus decoder operating at 8 kHz
   can decode a fullband Opus stream and any Opus decoder operating at
   48 kHz can decode a narrowband stream.  Similarly, the reference
   encoder can take a 48 kHz input signal and encode it in narrowband.
   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.  Because of the
   overhead from IP/UDP/RTP headers, sending fewer packets per second
   reduces the bitrate, but increases latency and sensitivity to packet
   losses as losing one packet constitutes a loss of a bigger chunk of
   audio signal.  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 tend to be 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 filter order of the pitch analysis whitening filter the short-
      term noise shaping filter;



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   o  The number of states in delayed decision quantization of the
      residual signal;

   o  The use of certain bit-stream features such as variable time-
      frequency resolution and 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 bit-rate

   Opus is more efficient when operating with variable bitrate (VBR),
   which is the default.  However, in some (rare) applications, constant
   bit-rate (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

   o  When security is important _and_ the input audio not a normal
      conversation but is highly constrained (e.g. yes/no, recorded
      prompts) [SRTP-VBR]

   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
   and AAC call CBR (i.e. not true CBR due to the bit reservoir).

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.




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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 higher layer (such as UDP or RTP or Ogg or Matroska) 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.

3.1.  The TOC Byte

   An Opus packet begins with a single-byte table-of-contents (TOC)
   header that signals which of the various modes and configurations a
   given packet uses.  It is composed of a frame count code, "c", a
   stereo flag, "s", and a configuration number, "config", arranged as
   illustrated in Figure 1.  A description of each of these fields
   follows.

                              0
                              0 1 2 3 4 5 6 7
                             +-+-+-+-+-+-+-+-+
                             | c |s| config  |
                             +-+-+-+-+-+-+-+-+

                          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 layer and MDCT layer can be combined in
   three possible operating modes:

   1.  An LP-only mode for use in low bitrate connections with an audio
       bandwidth of WB or less,

   2.  A Hybrid (LP+MDCT) mode for SWB or FB speech at medium bitrates,
       and





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   3.  An MDCT-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.

   +-----------------------+-----------+-----------+-------------------+
   | 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", is used to signal 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





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   o  2: 2 frames in the packet, with different compressed sizes

   o  3: an arbitrary number of frames in the packet

   This draft refers to a packet as a code 0 packet, code 1 packet,
   etc., based on the value of "c".

   A well-formed Opus packet MUST contain at least one byte with the TOC
   information, though the frame(s) within a packet MAY be zero bytes
   long.

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
      (len[1]*4)+len[0]

   The maximum representable length is 255*4+255=1275 bytes.  For 20 ms
   frames, this represents a bitrate of 510 kb/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, to allow for repacketization by gateways, conference
   bridges, or other software.







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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.

      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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |0|0|s| config  |                                               |
     +-+-+-+-+-+-+-+-+                                               |
     |                    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.

      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|s| config  |                                               |
     +-+-+-+-+-+-+-+-+                                               :
     |             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 the



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   figure below), 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.
   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.  This makes, for example, a 2-byte code 2 packet with a
   second byte in the range 1...250 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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |0|1|s| config  | N1 (1-2 bytes):                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
     |               Compressed frame 1 (N1 bytes)...                |
     :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                               |                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
     |                     Compressed frame 2...                     :
     :                                                               |
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                         Figure 4: A Code 2 Packet

3.2.5.  Code 3: An Arbitrary Number of Frames in the Packet

   Code 3 packets may encode an arbitrary 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.  The TOC byte is
   followed by a byte encoding the number of frames in the packet in
   bits 0 to 5 (marked "M" in the figure below), with bit 6 indicating
   whether or not Opus padding is inserted (marked "p" in the figure
   below), and bit 7 indicating VBR (marked "v" in the figure below).  M
   MUST NOT be zero, and the audio duration contained within a packet
   MUST NOT exceed 120 ms.  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.








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                              0
                              0 1 2 3 4 5 6 7
                             +-+-+-+-+-+-+-+-+
                             |     M     |p|v|
                             +-+-+-+-+-+-+-+-+

                      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
   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.  By using the value 255
   multiple times, it is possible to create a packet of any specific,
   desired size.  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.  Let P be the total amount of padding, including both
   the trailing padding bytes themselves and the header bytes used to
   indicate how many trailing bytes there are.  Then P MUST be no more
   than N-2.

   In the CBR case, the compressed length of each frame in bytes is
   equal to the number of remaining bytes in the packet after
   subtracting the (optional) padding, (N-2-P), divided by M. This
   number MUST be an integer multiple of M. The compressed data for all
   M frames then follows, each of size (N-2-P)/M bytes, as illustrated
   in Figure 6.





















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      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|1|s| config  |     M     |p|0|  Padding length (Optional)    :
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                                                               |
     :            Compressed frame 1 ((N-2-P)/M bytes)...            :
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                                                               |
     :            Compressed frame 2 ((N-2-P)/M bytes)...            :
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                                                               |
     :                              ...                              :
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                                                               |
     :            Compressed frame M ((N-2-P)/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 the figure below),
   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.  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
   length of the first M-1 frames themselves, plus the length of the
   padding MUST be no larger than N, the total size of the packet.












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      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|1|s| config  |     M     |p|1| 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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |0|0|0|    1    |               compressed data...              :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |1|0|0|   29    |               compressed data...              :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Two FB mono 20 ms Hybrid frames of different compressed size:




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    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|1|0|   15    |     2     |0|1|      N1       |               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+               |
   |                       compressed data...                      :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |1|1|1|   31    |     4     |0|0|      compressed data...       :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

3.4.  Extending Opus

   A receiver MUST NOT process packets which violate any of the rules
   above as normal Opus packets.  They are reserved for future
   applications, such as in-band headers (containing metadata, etc.).
   These constraints are summarized here for reference:

   o  Packets are at least one byte.

   o  No implicit frame length is larger than 1275 bytes.

   o  Code 1 packets have an odd total length, N, so that (N-1)/2 is an
      integer.

   o  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.

   o  Code 3 packets contain at least one frame, but no more than 120 ms
      of audio total.

   o  The length of a CBR code 3 packet, N, is at least two bytes, the
      size of the padding, P (including both the padding length bytes in
      the header and the trailing padding bytes) is no more than N-2,
      and the frame count, M, satisfies the constraint that (N-2-P) is
      an integer multiple of M.

   o  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.




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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], 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 instead of with bits,
   so it is faster when using larger bases (i.e., an octet).  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 8.  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.










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                    0               1               2               3
      7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     | Range coder data (packed MSB to LSB) ->                       :
     +                                                               +
     :                                                               :
     +     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     :     | <- Boundary occurs at an arbitrary bit position         :
     +-+-+-+                                                         +
     :                          <- Raw bits data (packed LSB to MSB) |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

    Figure 8: 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), 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 these 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

                  k-1                                      n-1
                  __                                       __
          fl[k] = \  f[i],  fh[k] = fl[k] + f[k],  ft[k] = \  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),
   representing the difference between the high end of the current range
   and the actual coded value, minus one, and the size of the current
   range, respectively.  Both val and rng are 32-bit unsigned integer
   values.  The decoder initializes rng to 128 and initializes val to
   127 minus the top 7 bits of the first input octet.  It saves the
   remaining bit for use in the renormalization procedure described in
   Section 4.1.1.1, which the decoder invokes immediately after
   initialization to read additional bits and establish the invariant
   that rng > 2**23.




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4.1.1.  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

                    fs = ft - min(val/(rng/ft)+1, ft) .

   The divisions here are exact 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

                     val = val - (rng/ft)*(ft-fh[k]) .

   If fl[k] is greater than zero, then the decoder updates rng using

                       rng = (rng/ft)*(fh[k]-fl[k]) .

   Otherwise, it updates rng using

                      rng = rng - (rng/ft)*(ft-fh[k]).

   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 4.1.2 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.







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4.1.1.1.  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 octet of the
   payload and combines it with the left-over bit buffered from the
   previous octet to form the 8-bit value sym.  It takes the left-over
   bit as the high bit (bit 7) of sym, and the top 7 bits of the octet
   it just read as the other 7 bits of sym.  The remaining bit in the
   octet just read is buffered for use in the next iteration.  If no
   more input octets remain, it uses zero bits instead.  Then, it sets

                val = ((val<<8) + (255-sym)) & 0x7FFFFFFF .

   It is normal and expected that the range decoder will read several
   bytes into the raw bits data (if any) at the end of the packet by the
   time the frame is completely decoded, as illustrated in Figure 9.
   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
   decoder will decode the intended values regardless of the data
   contained in the raw bits.  Section 5.1.4 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
      7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     :     | <----------- Overlap region ------------> |             :
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
           ^                                           ^
           |   End of data buffered by the range coder |
     ...-----------------------------------------------+
           |
           | End of data consumed by raw bits
           +-------------------------------------------------------...

    Figure 9: Illustrative example of raw bits overlapping range coder
                                   data

4.1.2.  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.




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4.1.2.1.  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 avoids one of the divisions.

4.1.2.2.  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
   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.

4.1.2.3.  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 draft
   lists the latter, not the former, when describing the context in



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   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 adding the constant offset of the
   first value with a non-zero probability to its return value.

4.1.3.  Decoding Raw Bits

   The raw bits used by the CELT layer are packed at the end of the
   packet, 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 4.1.1.1, 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.

4.1.4.  Decoding Uniformly Distributed Integers

   The ec_dec_uint() (entdec.c) function 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, this is split up into a range coded
   symbol representing up to 8 of the high bits of the value, and, if
   necessary, raw bits representing the remaining bits.  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



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   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 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 and/or report to the application that a problem has
   occurred.

4.1.5.  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 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 was 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



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   ensure that decoding the symbol will not bust the budget.

   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 33 just after the initial range
   renormalization process completes (or equivalently, it can be
   initialized to 9 before 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.

4.1.5.1.  ec_tell()

   The whole number of bits buffered in rng may be estimated via l =
   ilog(rng). ec_tell() then becomes a simple matter of removing these
   bits from the total.  It returns (nbits_total - l).

   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.

4.1.5.2.  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, l must be at least 24.  Let


                          r_Q15 = rng >> (l-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 l.  First, update


                       r_Q15 = (r_Q15*r_Q15) >> 15 .

   Then add the 16th bit of r_Q15 to l via


                         l = 2*l + (r_Q15 >> 16) .

   Finally, if this bit was a 1, reduce r_Q15 by a factor of two via




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                            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 l to 1/8th bit
   precision. ec_tell_frac() then returns (nbits_total*8 - l).

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 10.
































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        +---------+    +------------+
     -->| Range   |--->| Decode     |---------------------------+
      1 | Decoder | 2  | Parameters |----------+       5        |
        +---------+    +------------+     4    |                |
                            3 |                |                |
                             \/               \/               \/
                       +------------+   +------------+   +------------+
                       | Generate   |-->| LTP        |-->| LPC        |
                       | Excitation |   | Synthesis  |   | Synthesis  |
                       +------------+   +------------+   +------------+
                                               ^                |
                                               |                |
                           +-------------------+----------------+
                           |                                      6
                           |   +------------+   +-------------+
                           +-->| Stereo     |-->| Sample Rate |-->
                             8 | Unmixing   | 7 | Conversion  | 8
                               +------------+   +-------------+

     1: Range encoded bitstream
     2: Coded parameters
     3: Pulses, LSBs, and signs
     4: Pitch lags, LTP coefficients
     5: LPC coefficients and gains
     6: Decoded signal (mono or mid-side stereo)
     7: Unmixed signal (mono or left-right stereo)
     8: Resampled signal


                          Decoder block diagram.

                                 Figure 10

   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.







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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 draft 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
   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 11 and 12 illustrate the ordering of the various SILK
   frames for a 60 ms Opus frame, for both mono and stereo,
   respectively.









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         +-----------------------+---------------+---------------+
         |       Symbol(s)       |     PDF(s)    |   Condition   |
         +-----------------------+---------------+---------------+
         |       VAD flags       |    {1, 1}/2   |               |
         |                       |               |               |
         |       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 |               |
         +-----------------------+---------------+---------------+

             Organization of the SILK layer of an Opus frame.

                                  Table 3


                    +---------------------------------+
                    |            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 11: A 60 ms Mono Frame











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                 +---------------------------------------+
                 |             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 12: 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.



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   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



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   20 ms Opus frame, potentially leaving a hole that needs to be
   concealed from even a single packet loss.  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.

   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),

   o  An LSF interpolation weight (Section 4.2.7.5.5),





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   o  Long-term prediction 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.












































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   +---------------------------+-------------------+-------------------+
   |         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     | Section 4.2.7.5.5 |
   |    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     | Section 4.2.7.8.5 |
   +---------------------------+-------------------+-------------------+

             Order of the symbols in an individual SILK frame.

                                  Table 5




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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).

   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() (decode_stereo_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



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   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 exact integer division.  The range of these
   indices is 0 to 14, inclusive.  Let w[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.

































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                         +-------+--------------+
                         | 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




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   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 Section 4.2.3
         and Section 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()
   (decode_stereo_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.

                             +---------------+
                             | 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
   the second PDF in Table 9.  Table 10 translates between the value of



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   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 the pitch gains 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

   o  This is the first subframe in the current SILK frame, and





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   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

   For subframes which 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 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                                                            |
   +-------------------------------------------------------------------+




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             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 = min(max(2*gain_index - 16,
                          previous_log_gain + gain_index - 4), 63) .

   The value here is not clamped at 0, and may reach values as low as
   -16 over the course of consecutive subframes within a single Opus
   frame.

   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.  If i < 16, then

               (1<<i) + (((-174*f*(128-f)>>16)+f)>>7)*(1<<i)

   yields the approximate exponential.  Otherwise, silk_log2lin uses

              (1<<i) + ((-174*f*(128-f)>>16)+f)*((1<<i)>>7) .

   The final Q16 gain values lies between 4096 and 1686110208, inclusive
   (representing scale factors of 0.0625 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.
   Once decoded, the normalized LSFs form an increasing list of Q15
   values between 0 and 1.  These represent the interleaved zeros on the
   unit circle between 0 and pi (hence "normalized") in the standard
   decomposition 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)
   (Section 4.2.7.5.1 and Section 4.2.7.5.2).  NB and MB frames use an



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   order-10 predictor, while WB frames use an order-16 predictor, and
   thus have different sets 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.  Stage 1 Normalized LSF 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.  This indexes an element in a coarse codebook,
   selects the PDFs for the second stage of the VQ, and selects the
   prediction weights used to remove intra-frame redundancy from the
   second stage.  The actual codebook elements are listed in Table 23
   and Table 24, but they are not needed until the last stages of
   reconstructing the LSF coefficients.

   +-----------+----------+--------------------------------------------+
   | 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 Index Stage-1 Decoding







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4.2.7.5.2.  Stage 2 Normalized LSF 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.

            +----------+--------------------------------------+
            | 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 Index Stage-2 Decoding




















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           +----------+---------------------------------------+
           | 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 Index Stage-2 Decoding

                       +----+---------------------+
                       | 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 |



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                       | 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 |
                       |    |                     |
                       | 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 Index Stage 2
                                 Decoding






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          +----+------------------------------------------------+
          | 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 |
          |    |                                                |
          | j0 | 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 |



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          | 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 Index Stage 2
                                 Decoding

   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



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   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.

                  +-------------+-----+-----+-----+-----+
                  | 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



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   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) ,

   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 |
                        |    |                   |



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                        | 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 |
                        |    |                   |
                        | 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 | A 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 |



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           | 5  | C  D  C  C  C  C  C  C  C  C  C  C  C  C  C |
           |    |                                             |
           | 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 |
           |    |                                             |



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           | 29 | C  C  C  C  C  C  C  C  C  C  C  C  C  C  D |
           |    |                                             |
           | 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
   exact 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 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.





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              +----+----------------------------------------+
              | 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 |



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              | 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: Codebook Vectors for NB/MB Normalized LSF Stage 1 Decoding

    +----+------------------------------------------------------------+
    | 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 |
    |    |                                                            |



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    | 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: Codebook Vectors for WB Normalized LSF Stage 1 Decoding

   Given the stage-1 codebook entry cb1_Q8[], the stage-2 residual



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   res_Q10[], and their corresponding weights, w_Q9[], the reconstructed
   normalized LSF coefficients are

         NLSF_Q15[k] = (cb1_Q8[k]<<7) + (res_Q10[k]<<14)/w_Q9[k] ,

   where the division is exact 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.  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_Q[k]-NLSF_Q[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.


























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                     +-------------+-----------+-----+
                     | 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 which
   attempt to minimize the amount of distortion introduced.  After 20
   such adjustments, it falls back to a more direct method which
   guarantees the constraints are enforced but may require large
   adjustments.

   Let NDeltaMin_Q15[k] be the minimum required spacing for the current
   audio bandwidth from Table 25.  First, the procedure finds the index



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   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[i]>>1) + \  NDeltaMin[k]
                                             /_
                                             k=0
                                                    d_LPC
                                                     __
        max_center_Q15 = 32768 - (NDeltaMin[i]>>1) - \  NDeltaMin[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 has 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]) .

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



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   (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).  For the first frame after
   a decoder reset (see Section 4.5), when no prior LSF coefficients are
   available, 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

   with

                                    -d_LPC-1      -1
                     P(z) = A(z) + z         * A(z  )

                                    -d_LPC-1      -1
                     Q(z) = A(z) - z         * A(z  ) .

   The even normalized LSF coefficients correspond to a pair of



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   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.
























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                     +-------------+-----------+----+
                     | 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_Q13[i]*256
                            + (cos_Q13[i+1]-cos_Q13[i])*f + 8) >> 4 ,

   where ordering[k] is the k'th entry of the column of Table 27



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   corresponding to the current audio bandwidth and cos_Q13[i] is the
   i'th entry of Table 28.

                  +-----+-------+-------+-------+-------+
                  |   i |    +0 |    +1 |    +2 |    +3 |
                  +-----+-------+-------+-------+-------+
                  |   0 |  8192 |  8190 |  8182 |  8170 |
                  |     |       |       |       |       |
                  |   4 |  8152 |  8130 |  8104 |  8072 |
                  |     |       |       |       |       |
                  |   8 |  8034 |  7994 |  7946 |  7896 |
                  |     |       |       |       |       |
                  |  12 |  7840 |  7778 |  7714 |  7644 |
                  |     |       |       |       |       |
                  |  16 |  7568 |  7490 |  7406 |  7318 |
                  |     |       |       |       |       |
                  |  20 |  7226 |  7128 |  7026 |  6922 |
                  |     |       |       |       |       |
                  |  24 |  6812 |  6698 |  6580 |  6458 |
                  |     |       |       |       |       |
                  |  28 |  6332 |  6204 |  6070 |  5934 |
                  |     |       |       |       |       |
                  |  32 |  5792 |  5648 |  5502 |  5352 |
                  |     |       |       |       |       |
                  |  36 |  5198 |  5040 |  4880 |  4718 |
                  |     |       |       |       |       |
                  |  40 |  4552 |  4382 |  4212 |  4038 |
                  |     |       |       |       |       |
                  |  44 |  3862 |  3684 |  3502 |  3320 |
                  |     |       |       |       |       |
                  |  48 |  3136 |  2948 |  2760 |  2570 |
                  |     |       |       |       |       |
                  |  52 |  2378 |  2186 |  1990 |  1794 |
                  |     |       |       |       |       |
                  |  56 |  1598 |  1400 |  1202 |  1002 |
                  |     |       |       |       |       |
                  |  60 |   802 |   602 |   402 |   202 |
                  |     |       |       |       |       |
                  |  64 |     0 |  -202 |  -402 |  -602 |
                  |     |       |       |       |       |
                  |  68 |  -802 | -1002 | -1202 | -1400 |
                  |     |       |       |       |       |
                  |  72 | -1598 | -1794 | -1990 | -2186 |
                  |     |       |       |       |       |
                  |  76 | -2378 | -2570 | -2760 | -2948 |
                  |     |       |       |       |       |
                  |  80 | -3136 | -3320 | -3502 | -3684 |
                  |     |       |       |       |       |



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                  |  84 | -3862 | -4038 | -4212 | -4382 |
                  |     |       |       |       |       |
                  |  88 | -4552 | -4718 | -4880 | -5040 |
                  |     |       |       |       |       |
                  |  92 | -5198 | -5352 | -5502 | -5648 |
                  |     |       |       |       |       |
                  |  96 | -5792 | -5934 | -6070 | -6204 |
                  |     |       |       |       |       |
                  | 100 | -6332 | -6458 | -6580 | -6698 |
                  |     |       |       |       |       |
                  | 104 | -6812 | -6922 | -7026 | -7128 |
                  |     |       |       |       |       |
                  | 108 | -7226 | -7318 | -7406 | -7490 |
                  |     |       |       |       |       |
                  | 112 | -7568 | -7644 | -7714 | -7778 |
                  |     |       |       |       |       |
                  | 116 | -7840 | -7896 | -7946 | -7994 |
                  |     |       |       |       |       |
                  | 120 | -8034 | -8072 | -8104 | -8130 |
                  |     |       |       |       |       |
                  | 124 | -8152 | -8170 | -8182 | -8190 |
                  |     |       |       |       |       |
                  | 128 | -8192 |       |       |       |
                  +-----+-------+-------+-------+-------+

               Table 28: Q13 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) .

   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



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   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

   where the division here is exact 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



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   large.

   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 18
   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()
   (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




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                     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

                  a32_Q24[d_LPC-1][n] = a32_Q12[n] << 12 .

   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.  Otherwise, row k-1 of a32_Q24 is computed from row
   k as












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         rc_Q31[k] = -a32_Q24[k][k] << 7 ,

        div_Q30[k] = (1<<30) - (rc_Q31[k]*rc_Q31[k] >> 32) ,

             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-1.  Here, rc_Q30[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,
   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 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, 1 <= i <= 18, if the filter passes these stability



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   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 - i*(i+9) .

   If, after the 18th round, the filter still fails these stability
   checks, then a_Q12[k] is set to 0 for all k.

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

   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 reads the high part using the 32-
   entry codebook in Table 29 and 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



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   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 = lag_prev + (delta_lag_index - 9)

   where lag_prev 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.







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   +-------------------------------------------------------------------+
   | 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














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                       +-------+------------------+
                       | 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










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                       +-------+------------------+
                       | 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 |
                       |       |                  |



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                       | 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 |
                       |       |                  |



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                       | 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, and thus 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.









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                           +------------------+
                           | 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























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                      +-------+---------------------+
                      | 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






























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                      +-------+---------------------+
                      | 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 |



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                      | 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 |
                      |       |                     |



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                      | 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).







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                           +-------------------+
                           | 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 Quantization (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



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   offset described below).  SILK also handles large codebooks by coding
   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
   particular SILK frame may depend on the exact distribution of the



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   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.

















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   +----------+--------------------------------------------------------+
   | 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
   pulses within a shell block.  All of the pulses may be placed in a



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   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.



































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   +------------+------------------------------------------------------+
   | 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









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   +------------+------------------------------------------------------+
   | 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









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   +------------+------------------------------------------------------+
   | 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









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   +------------+------------------------------------------------------+
   | 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



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   pulses were allocated, before proceeding to the next one.  They 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 quantizaton 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 |
   |             |                       |             |               |



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   | 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 |



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   | 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).




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   +-------------+--------------------------+--------------------------+
   | Signal Type | Quantization Offset Type |      Quantization Offset |
   |             |                          |                    (Q25) |
   +-------------+--------------------------+--------------------------+
   | Inactive    | Low                      |                      100 |
   |             |                          |                          |
   | Inactive    | High                     |                      240 |
   |             |                          |                          |
   | Unvoiced    | Low                      |                      100 |
   |             |                          |                          |
   | Unvoiced    | High                     |                      240 |
   |             |                          |                          |
   | Voiced      | Low                      |                       32 |
   |             |                          |                          |
   | Voiced      | High                     |                      100 |
   +-------------+--------------------------+--------------------------+

                 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_Q25 be the quantization offset from Table 53.
   Then the following procedure produces the final reconstructed
   excitation value, e_Q25[i]:

       e_Q25[i] = (e_raw[i] << 10) - sign(e_raw[i])*80 + offset_Q25;
           seed = (196314165*seed + 907633515) & 0xFFFFFFFF;
       e_Q25[i] = (seed & 0x80000000) ? -(e_Q25[i] + 1) : e_Q25[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 80 term does not get added. offset does not get
   added.  The final e_Q25[i] value may require more than 16 bits per
   sample, but will not require more than 25, 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
   steps, this section describes them in terms of a floating-point



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   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

   Voiced SILK frames (see Section 4.2.7.3) 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 LPCs
   may have changed, it obtains this residual by "rewhitening" the
   corresponding output signal using the LPCs from the current subframe.
   Let e_Q25[i] be the excitation, and out[i] be the fully reconstructed
   output signal from previous subframes (see Section 4.2.7.9.2), or
   zeros in the first subframe for this channel after either

   o  An uncoded regular SILK frame in the side channel, or

   o  A decoder reset (see Section 4.5).

   Let LTP_scale_Q14 be the LTP scaling parameter from Section 4.2.7.6.3
   for the first two subframes in any SILK frame, as well as the last
   two subframes in a 20 ms SILK frame where w_Q2 == 4.  Otherwise let
   LTP_scale_Q14 be 16384 (corresponding to 1.0).  Then, for i such that
   (j - pitch_lags[s] - d_LPC - 2) <= i < j, where pitch_lags[s] is the
   pitch lag for the current subframe from Section 4.2.7.6.1, out[i] is
   rewhitened into an LPC residual, res[i], via





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               4.0*LTP_scale_Q14
   res[i] = ------------------------ * clamp(-1.0,
            max(gain_Q16[s], 131076)

                                      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 of
   18 ms * 16 kHz samples of pitch lag, plus 2 samples for the width of
   the LTP filter, plus 16 samples for d_LPC.

   Let b_Q7[k] 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_Q25[i]    __                                  b_Q7[k]
    res[i] = ---------- + \  res[i - pitch_lags[s] + 2 - k] * ------- .
             33554432.0   /_                                   128.0
                          k=0

   For unvoiced frames, the LPC residual for j <= i < (j + n) is simply
   a normalized copy of the excitation signal, i.e.,

                                      e_Q25[i]
                            res[i] = ----------
                                     33554432.0

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 previous subframe, or zeros
   in the first subframe for this channel after either

   o  An uncoded regular SILK frame in the side channel, or

   o  A decoder reset (see Section 4.5).

   Then for i such that j <= i < (j + n), the result of LPC synthesis
   for the current subframe is





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                                       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.

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



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   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) .

   These formulas require twp 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 whenver 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 which requires more delay than allowed
   for here (e.g., to improve quality), but then it most delay the
   output of the MDCT layer by this extra amount.  Keeping as much delay
   as possible on the encoder side allows an encoder which 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 which
   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



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   Hybrid mode.

   Table 54 gives the maximum resampler delay in samples at 48 kHz for
   each SILK audio bandwidth.  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.  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.  Some resampling filters (including those used by the
   reference implementation) may add a delay that is not itself an exact
   integer at either rate.  However, such deviations are unlikely to be
   perceptible.  The delays listed here are the ones that should be
   targeted by the encoder.

             +-----------------+----------------------------+
             | Audio Bandwidth | Delay in Samples at 48 kHz |
             +-----------------+----------------------------+
             | NB              | 18                         |
             |                 |                            |
             | MB              | 32                         |
             |                 |                            |
             | WB              | 24                         |
             +-----------------+----------------------------+

                Table 54: SILK Resampler Delay Allocations

4.3.  CELT Decoder

   An overview of the decoder is given in Figure 13.






















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                       +---------+
                       | Coarse  |
                    +->| decoder |----+
                    |  +---------+    |
                    |                 |
                    |  +---------+    v
                    |  |  Fine   |  +---+
                    +->| decoder |->| + |
                    |  +---------+  +---+
                    |       ^         |
        +---------+ |       |         |
        |  Range  | | +----------+    v
        | Decoder |-+ |   Bit    | +-----+
        +---------+ | |Allocation| | 2^x |
                    | +----------+ +-----+
                    |       |         |
                    |       v         v               +--------+
                    |  +---------+  +---+  +-------+  | pitch  |
                    +->|   PVQ   |->| * |->| IMDCT |->| post-  |--->
                    |  | decoder |  +---+  +-------+  | filter |
                    |  +---------+                    +--------+
                    |                                      ^
                    +--------------------------------------+

                 Figure 13: Structure of the CELT decoder

   The decoder is based on the following symbols and sets of symbols:

   +---------------+-----------------------------------+---------------+
   |   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           |               |



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   |   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  |  {2, 2, 5, 10, 22, 46, 22, 10, 5, |               |
   |               |             2, 2}/128             |               |
   |               |                                   |               |
   |      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           |               |
   +---------------+-----------------------------------+---------------+

        Order of the symbols in the CELT section of the bitstream.

                                 Table 55

   The decoder extracts information from the range-coded bitstream in
   the order described in the figure above.  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.

4.3.1.  Transient Decoding

   The "transient" flag encoded in the bitstream has a probability of
   1/8.  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.  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



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   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
   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 of the prediction filter is:

                                            -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 do not suffer desynchronization.  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.  The coarse energy
   quantization is performed by unquant_coarse_energy() and
   unquant_coarse_energy_impl() (quant_bands.c).  The encoding of the
   Laplace-distributed values is implemented in ec_laplace_decode()
   (laplace.c).



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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).

4.3.3.  Bit Allocation

   Many codecs transmit significant amounts of side information for the
   purpose of controlling bit allocation within a frame.  Often this
   side information controls bit usage indirectly and must be carefully
   selected to achieve the desired rate constraints.

   The band-energy normalized structure of Opus MDCT mode ensures that a
   constant bit allocation for the shape content of a band will result
   in a roughly constant tone to noise ratio, which provides for fairly
   consistent perceptual performance.  The effectiveness of this
   approach is the result of two factors: that the band energy, which is
   understood to be perceptually important on its own, is always
   preserved regardless of the shape precision, and because the constant
   tone-to-noise ratio implies a constant intra-band noise to masking
   ratio.  Intra-band masking is the strongest of the perceptual masking
   effects.  This structure means that the ideal allocation is more
   consistent from frame to frame than it is for other codecs without an
   equivalent structure.

   Because the bit allocation is used to drive 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 which produces identical results.



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   Because all of the information required to decode a frame must be
   derived from that frame alone in order to retain robustness to packet
   loss, the overhead of explicitly signaling the allocation would be
   considerable, especially for low-latency (small frame size)
   applications, even though the allocation is relatively static.

   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
   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' in jointly coded
   bands, and a flag to deactivate joint coding.  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 which increase the effectiveness of these parameters.



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   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.

   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 which can not be put to any use.  The maximums specified by
   the codec reflect the average maximum.  In the reference the maximums
   are provided in partially computed form, in order to fit in less
   memory as a static table (see cache_caps50[] in
   static_modes_float.h).  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 (e.g. 0 for 120, 1 for
   240, 3 for 480), then set i to nbBands*(2*LM+stereo).  Then 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 truncating 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 which



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   are 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,
   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).  Since the 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): set 'width' to the number of MDCT bins in this band for
   all channels.  Take the larger of width and 64, then the minimum of
   that value and the width times eight and set 'quanta' to the result.
   This represents a boost step size of six bits subject to limits of
   1/bit/sample and 1/8th 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 a with dynalloc_loop_logp
   as the cost of a one, 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 while loop finishes boost
   contains the 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 2352 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 1269 of celt.c.

   The allocation trim is a 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



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   the trim, first set the trim value to 5, then iff 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 inverse CDF {127, 126, 124, 119, 109, 87, 41, 19, 9, 4, 2,
   0}.

   For 10 ms and 20 ms frames using short blocks and that have at least
   LM+2 bits left prior to the allocation process, then one anti-
   collapse bit is reserved in the allocation process so it can be
   decoded later.  Following the 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 is 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) iff 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).  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 twenty-four 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 bit per channel or 48 128th bits per MDCT bin,



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   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.
   Then 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 a vector
   quantization scheme called a "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 which 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.





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4.3.4.2.  PVQ Decoding

   Decoding of PVQ vectors is implemented in decode_pulses() (cwrs.c).
   The uique 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 is normalised such that its L2-norm equals one.

4.3.4.3.  Spreading

   The normalised vector decoded in Section 4.3.4.2 is then rotated for
   the purpose of avoiding tonal artefacts.  The rotation gain is equal
   to

                           g_r = N / (N + f_r*K)

   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 bit-stream.

                 +--------------+------------------------+
                 | Spread value | f_r                    |
                 +--------------+------------------------+
                 | 0            | infinite (no rotation) |
                 |              |                        |
                 | 1            | 15                     |
                 |              |                        |
                 | 2            | 10                     |
                 |              |                        |
                 | 3            | 5                      |
                 +--------------+------------------------+

                        Table 56: Spreading values



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   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 the
   following 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))

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
   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 frame size dependent limit.  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 tradeoff 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}/15 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.




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                       +-----------------+---+----+
                       | Frame size (ms) | 0 |  1 |
                       +-----------------+---+----+
                       |       2.5       | 0 | -1 |
                       |                 |   |    |
                       |        5        | 0 | -1 |
                       |                 |   |    |
                       |        10       | 0 | -2 |
                       |                 |   |    |
                       |        20       | 0 | -2 |
                       +-----------------+---+----+

          TF adjustments for non-transient frames and tf_select=0

                                 Table 57

                       +-----------------+---+----+
                       | Frame size (ms) | 0 |  1 |
                       +-----------------+---+----+
                       |       2.5       | 0 | -1 |
                       |                 |   |    |
                       |        5        | 0 | -2 |
                       |                 |   |    |
                       |        10       | 0 | -3 |
                       |                 |   |    |
                       |        20       | 0 | -3 |
                       +-----------------+---+----+

          TF adjustments for non-transient frames and tf_select=1

                                 Table 58

                       +-----------------+---+----+
                       | Frame size (ms) | 0 |  1 |
                       +-----------------+---+----+
                       |       2.5       | 0 | -1 |
                       |                 |   |    |
                       |        5        | 1 |  0 |
                       |                 |   |    |
                       |        10       | 2 |  0 |
                       |                 |   |    |
                       |        20       | 3 |  0 |
                       +-----------------+---+----+

            TF adjustments for transient frames and tf_select=0

                                 Table 59




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                       +-----------------+---+----+
                       | Frame size (ms) | 0 |  1 |
                       +-----------------+---+----+
                       |       2.5       | 0 | -1 |
                       |                 |   |    |
                       |        5        | 1 | -1 |
                       |                 |   |    |
                       |        10       | 1 | -1 |
                       |                 |   |    |
                       |        20       | 1 | -1 |
                       +-----------------+---+----+

            TF adjustments for transient frames and tf_select=1

                                 Table 60

   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.  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

   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
   energy corresponding to the min 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 like 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



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   samples, while scaling by 1/2.  A "low-overlap" window is used to
   reduce 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.  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.

   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.







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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.

   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 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.



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   A decoder MAY employ a more sophisticated drift compensation method.
   For example, the NetEQ component [3] of the WebRTC.org codebase [4]
   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 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 bit-
   stream MAY include redundant side information ("redundancy"), in the
   form of additional CELT frames embedded in each of the Opus frames
   around the transition.

   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 reduces glitches during these
   problematic mode transitions, and also 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.



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   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).

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 for SILK-only frames, this signaling is implicit, based on the
   size of the 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 4.1.5.1) 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 4.1.5.1) to ensure there are at least 37 bits remaining.  If



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   so, it reads a symbol with the PDF in Table 61, 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 61: 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 62).

                               +----------+
                               | PDF      |
                               +----------+
                               | {1, 1}/2 |
                               +----------+

                     Table 62: Redundancy Position PDF

   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



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   number of bytes is equal to 2, plus a decoded unsigned integer less
   than 256 (see Section 4.1.4).  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 which 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 CELT layer read 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.

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
   is 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



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   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.

4.5.3.  Summary of Transitions

   Figure 14 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 frames in the corresponding modes.  In
   addition, an R indicates the presence of redundancy in the Opus frame
   it is cross-lapped with.  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).


















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    SILK to SILK with Redundancy:             S -> S -> S   ;S -> S -> S
                                                        &    &
                                                       !R -> R

    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   ;S -> S -> S
                                                        &    &
                                                       !R -> R

    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 14: Normative Transitions

   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.




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   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 15 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
    P   Packet Loss Concealment         &   Windowed cross-lap

             Figure 15: Recommended Non-Normative Transitions

   Encoders SHOULD NOT use other transitions, e.g., those that involve
   redundancy in ways not illustrated in Figure 14.









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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
   bit-stream 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|--+
     +-----------+  |  |conversion|    |       |  |
     | Optional  |  |  +----------+    +-------+  |   +-------+
   ->| high-pass |--+                             +-->| Range |
     +  filter   +  |  +------------+  +-------+      |encoder|---->
     +-----------+  |  |   Delay    |  | CELT  |  +-->|       | bit-
                    +->|compensation|->|encoder|--+   +-------+ stream
                       |            |  |       |
                       +------------+  +-------+


   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) depend on the following:

   o  Requested bit-rate

   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 application) or delay
   the mode switching decisions (for interactive applications).




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   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 (ARMA) 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 Coder

   The range coder also 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 N-1 (where N 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 (low,rng,rem,ext) representing the low end of the current
   range, the size of the current range, a single buffered output octet,
   and a count of additional carry-propagating output octets.  Both rng
   and low are 32-bit unsigned integer values, rem is an octet value or
   the special value -1, and ext is an integer with at least 16 bits.
   This state vector is initialized at the start of each each frame to
   the value (0,2**31,-1,0).  The reference implementation re-uses the
   'val' field of the entropy coder structure to hold low, in order to
   allow the same structure to be used for encoding and decoding, but we
   maintain the distinction here for clarity.

5.1.1.  Encoding Symbols

   The main encoding function is ec_encode() (entenc.c), which takes as
   an argument a three-tuple (fl,fh,ft) describing the range of the
   symbol to be encoded in the current context, with 0 <= fl < fh <= ft
   <= 65535.  The values of this tuple are derived from the probability
   model for the symbol.  Let f(i) be the frequency of the i'th symbol
   in the current context.  Then the three-tuple corresponding to the
   k'th symbol is given by fl=sum(f(i),i<k), fh=fl+f(i), and
   ft=sum(f(i)).

   ec_encode() updates the state of the encoder as follows.  If fl is
   greater than zero, then low = low + rng - (rng/ft)*(ft-fl) and rng =



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   (rng/ft)*(fh-fl).  Otherwise, low is unchanged and rng = rng - (rng/
   ft)*(fh-fl).  The divisions here are exact integer division.  After
   this update, the range is normalized.

   To normalize the range, the following process is repeated until rng >
   2**23.  First, the top 9 bits of low, (low>>23), are placed into a
   carry buffer.  Then, low is set to (low << 8 & 0x7FFFFFFF) and rng is
   set to (rng<<8).  This process is carried out by ec_enc_normalize()
   (entenc.c).

   The 9 bits produced in each iteration of the normalization loop
   consist of 8 data bits and a carry flag.  The final value of the
   output bits is not determined until carry propagation is accounted
   for.  Therefore the reference implementation buffers a single (non-
   propagating) output octet and keeps a count of additional propagating
   (0xFF) output octets.  An implementation may choose to use any
   mathematically equivalent scheme to perform carry propagation.

   The function ec_enc_carry_out() (entenc.c) performs this buffering.
   It takes a 9-bit input value, c, from the normalization: 8 bits of
   output and a carry bit.  If c is 0xFF, then ext is incremented and no
   octets are output.  Otherwise, if rem is not the special value -1,
   then the octet (rem+(c>>8)) is output.  Then ext octets are output
   with the value 0 if the carry bit is set, or 0xFF if it is not, and
   rem is set to the lower 8 bits of c.  After this, ext is set to zero.

   In the reference implementation, a special version of ec_encode()
   called ec_encode_bin() (entenc.c) is defined to take a two-tuple
   (fl,ftb), where 0 <= fl < 2**ftb and ftb < 16.  It is mathematically
   equivalent to calling ec_encode() with the three-tuple (fl,fl+
   1,1<<ftb), but avoids using division.

5.1.2.  Encoding Raw Bits

   The CELT layer also allows directly encoding a series of raw bits,
   outside of the range coder, implemented in ec_enc_bits() (entenc.c).
   The raw bits are packed at the end of the packet, starting by storing
   the least significant bit of the value to be packed in the least
   significant bit of the last byte, filling up to the most significant
   bit in the last byte, and then continuing in the least significant
   bit of the penultimate byte, and so on.  This packing may continue
   into the last byte output by the range coder, though the format
   should render it impossible to overwrite any set bit produced by the
   range coder when the procedure in Section 5.1.4 is followed to
   finalize the stream.






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5.1.3.  Encoding Uniformly Distributed Integers

   The function ec_enc_uint() is based on ec_encode() and encodes one of
   N equiprobable symbols, each with a frequency of 1, where N may be as
   large as 2**32-1.  Because ec_encode() is limited to a total
   frequency of 2**16-1, this is done by encoding a series of symbols in
   smaller contexts.

   ec_enc_uint() (entenc.c) takes a two-tuple (fl,ft), where ft is not
   necessarily a power of two.  Let ftb be the location of the highest 1
   bit in the two's-complement representation of (ft-1), or -1 if no
   bits are set.  If ftb>8, then the top 8 bits of fl are encoded using
   ec_encode() with the three-tuple (fl>>ftb-8,(fl>>ftb-8)+
   1,(ft-1>>ftb-8)+1), and the remaining bits are encoded as raw bits.
   Otherwise, fl is encoded with ec_encode() directly using the three-
   tuple (fl,fl+1,ft).

5.1.4.  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 integer
   in the interval [low,low+rng) with the largest number of trailing
   zero bits, b, such that end+(1<<b)-1 is also in the interval
   [low,low+rng).  Then while end is not zero, the top 9 bits of end,
   e.g., (end>>23), are sent to the carry buffer, and end is replaced by
   (end<<8&0x7FFFFFFF).  Finally, if the value in carry buffer, 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.
   After the carry buffer is finished outputting octets, the rest of the
   output buffer (if any) is padded with zero bits, until it reaches the
   raw bits.  Finally, rem is set to the special value -1.  This process
   is implemented by ec_enc_done() (entenc.c).

5.1.5.  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 must produce exactly the same
   value returned by the same functions in the decoder after decoding
   the same symbols.





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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
                +----------+    +--------+    +---------+


   Silk Encoder.

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 convertor 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



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   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.

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 16.


                                                                +---+
                             +--------------------------------->|   |
        +---------+          |      +---------+                 |   |
        |Voice    |          |      |LTP      |12               |   |
    +-->|Activity |--+       +----->|Scaling  |-----------+---->|   |
    |   |Detector |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  |  |   |          |  |   |   |     |   |



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    |       |  +---------+   |  |   +----------+  |   |   |     |   |
    |       |                |  |        /\       |   |   |     |   |
    |       |     +----------|--|--------+        |   |   |     |   |
    |       |     |         \/  \/               \/  \/  \/     |   |
    |       |     |       +---------+          +------------+   |   |
    |       |     |       |         |          |Noise       |   |   |
   -+-------+-----+------>|Prefilter|--------->|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)
     - Long term synthesis and analysis noise
       shaping coefficients (per 5 ms and for voiced speech only)
     - Noise shaping tilt (per 5 ms)
     - 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


                            Silk Core Encoder.

                                 Figure 16

5.2.3.1.  Voice Activity Detection

   The input signal is processed by a Voice Activity Detector (VAD) 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



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   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.

5.2.3.2.  Pitch Analysis

   The input signal is processed by the open loop pitch estimator shown
   in Figure 17.






























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                                    +--------+  +----------+
                                    |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


                   Block diagram of the pitch estimator.

                                 Figure 17

   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 Prediction 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:

   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



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      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 prefilter 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.

   o  De-emphasizing spectral valleys; by using different coefficients
      in the analysis and synthesis part of the prefilter and noise



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      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


           Noise shaping and spectral de-emphasis illustration.

                                 Figure 18

   Figure 18 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.




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   The transformation from input signal to de-emphasized signal can be
   described as a filtering operation with a filter


                                            -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



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   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.

   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
   three filter taps, 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



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   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.

   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



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   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 LPCs are estimated using
   Burg's method, such that the residual energy is minimized.  The
   estimated LPCs 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 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.  Burgs method

   The main purpose of LPC coding 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



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   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, 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.

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 tradeoff 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 refered 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 tradeoff 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.  And 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



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   location in the LSF vector.  The prediction is subtracted from the
   LSF residual value before quantizing the result, and 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 .  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.

   The quantization index of the first stage is entropy coded.  The
   quantization sequence from the second stage is also entropy coded,
   where for each elemnt the probability table is chosen depending on
   the vector index from the first 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.  Therefore we apply an LSF stabilization
   method which ensures that the LSF parameters are within their valid
   range, increasingly sorted, and have minimum distances between each
   other and the border values that have been predetermined as the 0.01
   percentile distance values from a large training set.

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,




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   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
   to be similar.  Because of this, one of the three codebooks typically
   gives good performance for all subframes, and 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.  Prefilter

   In the prefilter 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 prefilter 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



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   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.

   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
   encodes the high band with the remaining number of bits in Hybrid
   mode.  If SILK is unable to encode the packet with less than the
   allowed number of bits, the Opus encoder temporarily codes the signal
   in CELT mode instead.  The number of payload bits is adjusted by
   changing the quantization gains and the rate/distortion tradeoff in
   the noise shaping quantizer, in an iterateve 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
   also 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 bit-stream 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 Prefilter

   The pitch prefilter 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



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   main non-obvious aspect of the prefilter is the selection of the
   pitch period.  The pitch search should be optimised 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

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 avaialble 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 bit-rate, 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 bit-rate on a frame-by-
   frame basis are band boost, allocation trim, and band skipping.




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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.

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



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   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 extra
   is the number of extra degrees of freedom for mid-side coding.  For
   LM>1, E=13, otherwise E=5.

   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 (kb/s) | start band |
                      +----------------+------------+
                      |       <35      |      8     |
                      |                |            |
                      |      35-50     |     12     |
                      |                |            |
                      |      50-68     |     16     |
                      |                |            |
                      |      84-84     |     18     |
                      |                |            |
                      |     84-102     |     19     |
                      |                |            |
                      |     102-130    |     20     |
                      |                |            |
                      |      >130      |  disabled  |
                      +----------------+------------+

                      Thresholds for intensity stereo

                                 Table 63

5.3.6.  Time-Frequency Decision

   The choice of time-frequency resolution used in Section 4.3.4.5 is
   based on rate-distortion (RD) 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 RD optimization.  The optimal
   decision is computed using the Viterbi algorithm.  See tf_analysis()



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   in celt/celt.c.

5.3.7.  Spreading Values Decision

   The choice of the spreading value in Table 56 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
   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 Quantization (PVQ) [PVQ] codebook 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 the 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



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                             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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6.  Conformance

   It is the intention to allow the greatest possible choice of freedom
   in implementing the specification.  For this reason, outside of a few
   exceptions noted in this section, conformance is defined through the
   reference implementation of the decoder provided in Appendix A.
   Although this document includes an English 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 a decoder's output MUST
   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).  Either the floating-point implementation or the
   fixed-point implementation can be used as a reference and being
   within the threshold for one of the two is sufficient.  In addition,
   a compilant decoder implementation MUST have the same final range
   decoder state as that of the reference decoder.

6.1.  Testing

   Using the reference code provided in Appendix A, a mono test vector
   can be decoded with

      opus_demo -d 48000 1 test_mono.bit test_mono.out

   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 test_mono.float test_mono.out

   or

      opus_compare test_mono.fixed test_mono.out

   For a stereo test vector, the command line for decoding is

      opus_demo -d 48000 2 test_stereo.bin test_stereo.out

   and the output can be compared with the reference output with

      opus_compare -s test_stereo.float test_stereo.out

   or




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      opus_compare -s test_stereo.fixed test_stereo.out

   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>

   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

   To complement the Opus specification, the "Opus Custom" codec 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).  Such applications will not be
   compatible with the "main" Opus codec.  In Opus Custom operation,
   only the CELT layer is available, which is available using the celt_*
   function calls in celt.h.





























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7.  Security Considerations

   Implementations of the Opus codec need to take appropriate security
   considerations into account, as outlined in [DOS] and [SECGUIDE].  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 a certain 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.

   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 [5] memory debugger, which tracks reads and
   writes to invalid memory regions as well as the use of uninitialized



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   memory.  There were no errors reported on any of the tested
   conditions.

















































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8.  IANA Considerations

   This document has no actions for IANA.
















































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9.  Acknowledgements

   Thanks to all other developers, including Raymond Chen, Soeren Skak
   Jensen, Gregory Maxwell, Christopher Montgomery, and Karsten Vandborg
   Soerensen.  We would also like to thank Igor Dyakonov, Jan Skoglund,
   and Christian Hoene for their help with subjective testing of the
   Opus codec.  Thanks to Ralph Giles, John Ridges, Ben Schwartz, Keith
   Yan, Christian Hoene, Kat Walsh, and many others on the Opus and CELT
   mailing lists for their bug reports and feedback.










































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10.  References

10.1.  Normative References

   [rfc2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", RFC 2119.

10.2.  Informative References

   [requirements]
              Valin, J., Vos, K., and IETF, "Requirements for an
              Internet Audio Codec", RFC 6366, August 2011.

   [SILK]     Vos, K., Jensen, S., and K. Soerensen, "SILK Speech
              Codec", draft-vos-silk-01 (work in progress), March 2010.

   [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.

   [CELT]     Valin, J-M., Terriberry, T.B., Maxwell, G., and C.
              Montgomery, "Constrained-Energy Lapped Transform (CELT)
              Codec", draft-valin-celt-codec-02 (work in progress),
              July 2010.

   [SRTP-VBR]
              Perkins, C. and J. Valin, "Guidelines for the use of
              Variable Bit Rate Audio with Secure RTP",
              draft-ietf-avtcore-srtp-vbr-audio-03 (work in progress),
              July 2011.

   [DOS]      Handley, M., Rescorla, E., and IAB, "Internet Denial-of-
              Service Considerations", RFC 4732, December 2006.

   [SECGUIDE]
              Rescorla, E. and B. Korver, "Guidelines for Writing RFC
              Text on Security Considerations", BCP 72, RFC 3552,
              July 2003.

   [range-coding]
              Nigel, G. and N. Martin, "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.



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   [coding-thesis]
              Pasco, R., "Source coding algorithms for fast data
              compression", Ph.D. thesis Dept. of Electrical
              Engineering, Stanford University, May 1976.

   [PVQ]      Fischer, T., "A Pyramid Vector Quantizer", IEEE Trans. on
              Information Theory, Vol. 32 pp. 568-583, July 1986.












































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URIs

   [1]  <http://developer.skype.com/silk>

   [2]  <http://www.celt-codec.org/>

   [3]  <http://code.google.com/p/webrtc/source/browse/trunk/src/
        modules/audio_coding/NetEQ/main/source/?r=583>

   [4]  <http://code.google.com/p/webrtc/>

   [5]  <http://valgrind.org/>

   [6]  <git://git.opus-codec.org/opus.git>





































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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.  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 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 values is truncated
      towards zero

   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)

   o  at least a 32-bit adder/accumulator

A.1.  Extracting the source

   The complete source code can be extracted from this draft, by running
   the following command line:





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   o  cat draft-ietf-codec-opus.txt | grep '^\ \ \ ###' | sed -e
      's/\s\s\s###//' | base64 -d > opus_source.tar.gz

   o  tar xzvf opus_source.tar.gz

   o  cd opus_source

   o  make

   On systems where the base64 utility is not present, the following
   commands can be used instead:

   o  cat draft-ietf-codec-opus.txt | grep '^\ \ \ ###' | sed -e
      's/\s\s\s###//' > opus.b64

   o  openssl base64 -d -in opus.b64 > opus_source.tar.gz

A.2.  Development Versions

   The current development version of the source code is available in a
   Git repository [6].  Development snapshots are provided at
   <http://opus-codec.org/>.

A.3.  Base64-encoded source code

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   ###h+tPv/4fhSE5iwAwIAA=

A.4.  Test vectors

   Because of size constraints, the Opus test vectors are not
   distributed in this draft.  They are available from the Opus codec
   website at <http://opus-codec.org/testvectors/> and will also be made
   available in IETF meeting proceedings.  These test vectors were
   created specifically to exercise all aspects of the decoder and
   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

        1c93c979fcdd3b690e7f026c7d3c0dd7ff18ce26  test1_mono.bit
        d081f04726a9b55139169e9102c0e8aefd3bc598  test1_mono.fixed
        52ef3919cb33f423ab5ad3d6eaec73c78d59ae47  test1_mono.float
        581b0a5dbc1cb624c79e4d881813793d819a43f0  test2_mono.bit
        46d4ddc49c0ce80861dcbbcc3264383ebe851bd9  test2_mono.fixed
        fc8d3609f7fe22463641b52acf71bda7e97ebc99  test2_mono.float
        512965134678ec8a2883796467cd27c9d2e6b2ac  test3_mono.bit
        d6401be4d5dc006bb6433c4aa1c4c018ddd4d25c  test3_mono.fixed
        d10310d657fde1dd23c1a50c4fb3fad8d8ce8d5f  test3_mono.float
        5d3819e5ac37ecfbd6a7ab7142b083279e1815ff  test4_mono.bit
        44881c834f03f810ffb2397de3ec850323f49513  test4_mono.fixed
        6538684f07dc435aa6877f5cf705936afce3aca9  test4_mono.float
        58515e06eee6bfb0981b0d09882e6903b2de3a26  test5_mono.bit
        5ae5eb782f911ff7bd1faf2369fd09e88122b356  test5_mono.fixed
        120217917cad910d6ea5d6855192210ac88881dd  test5_mono.float
        3a8e9c2136daee94f517c0e1bcb79ffee9b094e0  test1_stereo.bit
        0016f27e2792ac5651cf9a47abacd0ffc3e3aa6b  test1_stereo.fixed
        b63ed7377bd39a1ebd76e965ff77a32adad837bd  test1_stereo.float
        521eb2a1e0cc9c31b8b740673307c2d3b10c1900  test2_stereo.bit
        3dba673f3ff244fb3930cd712ebf14ab4d51808b  test2_stereo.fixed
        8aa4a5c7c2fbd4add2e4d4b76bb0c15c8e3ea8a8  test2_stereo.float
        5b50aa6d1c093c77c15e61d6fc466a5ff1f7c423  test3_stereo.bit
        165c6b92599ab1319acb8e5637b8123856c102b9  test3_stereo.fixed
        e6613f0af12f6faa16f4760b0b1a59a5cb5bfbfd  test3_stereo.float
        6bc8f3146fcb96450c901b16c3d464ccdf4d5d96  test4_stereo.bit
        01c6f02bc5d10a5a653a89b82f6c5f7807397074  test4_stereo.fixed



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        20ffcbf8b0eeaf4ff17ed29d1120b2d23ce50334  test4_stereo.float


















































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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 which 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
   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 which 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 19).

   o  Code 1 packets: It is the length used for both of the Opus frames
      (see Figure 20).

   o  Code 2 packets: It is the length of the second Opus frame (see
      Figure 21).

   o  CBR Code 3 packets: It is the length used for all of the Opus
      frames (see Figure 22).

   o  VBR Code 3 packets: It is the length of the last Opus frame (see
      Figure 23).




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      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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |0|0|s| config  | N1 (1-2 bytes):                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
     |               Compressed frame 1 (N1 bytes)...                :
     :                                                               |
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                 Figure 19: 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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |1|0|s| config  | N1 (1-2 bytes):                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
     |               Compressed frame 1 (N1 bytes)...                |
     :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                               |                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
     |               Compressed frame 2 (N1 bytes)...                |
     :                                               +-+-+-+-+-+-+-+-+
     |                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                 Figure 20: 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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |0|1|s| config  | N1 (1-2 bytes): N2 (1-2 bytes :               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+               :
     |               Compressed frame 1 (N1 bytes)...                |
     :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                               |                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
     |               Compressed frame 2 (N2 bytes)...                :
     :                                                               |
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                 Figure 21: A Self-Delimited Code 2 Packet






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      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|1|s| config  |     M     |p|0| 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 22: A Self-Delimited CBR Code 3 Packet


























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      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|1|s| config  |     M     |p|1| 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 23: A Self-Delimited VBR Code 3 Packet
























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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: koen.vos@skype.net


   Timothy B. Terriberry
   Mozilla Corporation
   650 Castro Street
   Mountain View, CA  94041
   USA

   Phone: +1 650 903-0800
   Email: tterriberry@mozilla.com





















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