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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                                              Octasic Inc.
Intended status: Standards Track                                  K. Vos
Expires: January 8, 2012                         Skype Technologies S.A.
                                                           T. Terriberry
                                                     Mozilla Corporation
                                                            July 7, 2011


                   Definition of the Opus Audio Codec
                        draft-ietf-codec-opus-07

Abstract

   This document defines the Opus codec, designed for interactive speech
   and audio transmission over the Internet.

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 January 8, 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
   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.



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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Notation and Conventions  . . . . . . . . . . . . . . . .   5
       1.1.1.  min(x,y)  . . . . . . . . . . . . . . . . . . . . . .   5
       1.1.2.  max(x,y)  . . . . . . . . . . . . . . . . . . . . . .   5
       1.1.3.  clamp(lo,x,hi)  . . . . . . . . . . . . . . . . . . .   5
       1.1.4.  sign(x) . . . . . . . . . . . . . . . . . . . . . . .   5
       1.1.5.  log2(f) . . . . . . . . . . . . . . . . . . . . . . .   6
       1.1.6.  ilog(n) . . . . . . . . . . . . . . . . . . . . . . .   6
   2.  Opus Codec Overview . . . . . . . . . . . . . . . . . . . . .   7
   3.  Codec Modes . . . . . . . . . . . . . . . . . . . . . . . . .   9
     3.1.  Examples  . . . . . . . . . . . . . . . . . . . . . . . .  15
   4.  Opus Decoder  . . . . . . . . . . . . . . . . . . . . . . . .  17
     4.1.  Range Decoder . . . . . . . . . . . . . . . . . . . . . .  17
       4.1.1.  Decoding Symbols  . . . . . . . . . . . . . . . . . .  18
       4.1.2.  Alternate Decoding Methods  . . . . . . . . . . . . .  20
       4.1.3.  Decoding Raw Bits . . . . . . . . . . . . . . . . . .  21
       4.1.4.  Decoding Uniformly Distributed Integers . . . . . . .  21
       4.1.5.  Current Bit Usage . . . . . . . . . . . . . . . . . .  22
     4.2.  SILK Decoder  . . . . . . . . . . . . . . . . . . . . . .  24
       4.2.1.  Decoder Modules . . . . . . . . . . . . . . . . . . .  25
       4.2.2.  Header Bits . . . . . . . . . . . . . . . . . . . . .  27
       4.2.3.  LBRR Flags  . . . . . . . . . . . . . . . . . . . . .  27
       4.2.4.  SILK Frame Contents . . . . . . . . . . . . . . . . .  28
       4.2.5.  LBRR Frames . . . . . . . . . . . . . . . . . . . . .  61
     4.3.  CELT Decoder  . . . . . . . . . . . . . . . . . . . . . .  61
       4.3.1.  Transient Decoding  . . . . . . . . . . . . . . . . .  63
       4.3.2.  Energy Envelope Decoding  . . . . . . . . . . . . . .  63
       4.3.3.  Bit allocation  . . . . . . . . . . . . . . . . . . .  64
       4.3.4.  Shape Decoder . . . . . . . . . . . . . . . . . . . .  69
       4.3.5.  Anti-collapse processing  . . . . . . . . . . . . . .  70
       4.3.6.  Denormalization . . . . . . . . . . . . . . . . . . .  70
       4.3.7.  Inverse MDCT  . . . . . . . . . . . . . . . . . . . .  71
       4.3.8.  Packet Loss Concealment (PLC) . . . . . . . . . . . .  72
     4.4.  Mode Switching  . . . . . . . . . . . . . . . . . . . . .  72
       4.4.1.  Switching Side Information  . . . . . . . . . . . . .  72
   5.  Codec Encoder . . . . . . . . . . . . . . . . . . . . . . . .  74
     5.1.  Range Coder . . . . . . . . . . . . . . . . . . . . . . .  74
       5.1.1.  Encoding Symbols  . . . . . . . . . . . . . . . . . .  74
       5.1.2.  Encoding Raw Bits . . . . . . . . . . . . . . . . . .  75
       5.1.3.  Encoding Uniformly Distributed Integers . . . . . . .  76
       5.1.4.  Finalizing the Stream . . . . . . . . . . . . . . . .  76
       5.1.5.  Current Bit Usage . . . . . . . . . . . . . . . . . .  76
     5.2.  SILK Encoder  . . . . . . . . . . . . . . . . . . . . . .  77
       5.2.1.  Voice Activity Detection  . . . . . . . . . . . . . .  78
       5.2.2.  High-Pass Filter  . . . . . . . . . . . . . . . . . .  79
       5.2.3.  Pitch Analysis  . . . . . . . . . . . . . . . . . . .  79



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       5.2.4.  Noise Shaping Analysis  . . . . . . . . . . . . . . .  81
       5.2.5.  Prefilter . . . . . . . . . . . . . . . . . . . . . .  85
       5.2.6.  Prediction Analysis . . . . . . . . . . . . . . . . .  85
       5.2.7.  LSF Quantization  . . . . . . . . . . . . . . . . . .  86
       5.2.8.  LTP Quantization  . . . . . . . . . . . . . . . . . .  89
       5.2.9.  Noise Shaping Quantizer . . . . . . . . . . . . . . .  90
       5.2.10. Range Encoder . . . . . . . . . . . . . . . . . . . .  91
     5.3.  CELT Encoder  . . . . . . . . . . . . . . . . . . . . . .  91
       5.3.1.  Pre-filter  . . . . . . . . . . . . . . . . . . . . .  91
       5.3.2.  Forward MDCT  . . . . . . . . . . . . . . . . . . . .  91
       5.3.3.  Bands and Normalization . . . . . . . . . . . . . . .  91
       5.3.4.  Energy Envelope Quantization  . . . . . . . . . . . .  92
       5.3.5.  Spherical Vector Quantization . . . . . . . . . . . .  93
       5.3.6.  Stereo support  . . . . . . . . . . . . . . . . . . .  94
       5.3.7.  Synthesis . . . . . . . . . . . . . . . . . . . . . .  94
       5.3.8.  Variable Bitrate (VBR)  . . . . . . . . . . . . . . .  95
   6.  Conformance . . . . . . . . . . . . . . . . . . . . . . . . .  96
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  97
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  98
   9.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .  99
   10. Informative References  . . . . . . . . . . . . . . . . . . . 100
   Appendix A.  Reference Implementation . . . . . . . . . . . . . . 102
     A.1.  Extracting the source . . . . . . . . . . . . . . . . . . 102
     A.2.  Development Versions  . . . . . . . . . . . . . . . . . . 102
     A.3.  Base64-encoded source code  . . . . . . . . . . . . . . . 102
   Appendix B.  opus_compare.c . . . . . . . . . . . . . . . . . . . 248
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 254
























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1.  Introduction

   The Opus codec is a real-time interactive audio codec 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.  In general, 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.

   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 syntax like "2**n" is used to
   indicate 2 raised to the power n.  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.  Internally, the codec always operates at a
   48 kHz sampling rate, though it allows input and output of various
   bandwidths, defined as follows:

   +----------------------+-----------------+--------------------------+
   | Abbreviation         | Audio Bandwidth |            Sampling 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 |
   +----------------------+-----------------+--------------------------+

   These 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.

   The LP layer is based on the SILK [1] codec [SILK].  It supports NB,
   MB, or WB audio and frame sizes from 10 ms to 60 ms, and requires an
   additional 5.2 ms look-ahead for noise shaping estimation (5 ms) and
   internal resampling (0.2 ms).  Like Vorbis and many other modern
   codecs, SILK is inherently designed for variable-bitrate (VBR)
   coding, though an encoder can with sufficient effort produce
   constant-bitrate (CBR) or near-CBR streams.

   The MDCT layer is based on the CELT [2] codec [CELT].  It supports
   sampling 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



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   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.

   At the decoder, the two decoder outputs are simply added together.
   To compensate for the different look-aheads 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
   lookahead 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.























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3.  Codec Modes

   As described, the two layers can be combined in three possible
   operating modes:

   1.  A 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

   3.  An MDCT-only mode for very low delay speech transmission as well
       as music transmission.

   A single packet may contain multiple audio frames, however they must
   share a common set of parameters, including the operating mode, audio
   bandwidth, frame size, and channel count.  A single-byte table-of-
   contents (TOC) header 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.  Table 1 lists the parameters for each configuration.

















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   +-----------------------+-----------+-----------+-------------------+
   | Configuration         | Mode      | Bandwidth | Frame Size(s)     |
   | Number(s)             |           |           |                   |
   +-----------------------+-----------+-----------+-------------------+
   | 0...3                 | LP-only   | NB        | 10, 20, 40, 60 ms |
   |                       |           |           |                   |
   | 4...7                 | LP-only   | MB        | 10, 20, 40, 60 ms |
   |                       |           |           |                   |
   | 8...11                | LP-only   | WB        | 10, 20, 40, 60 ms |
   |                       |           |           |                   |
   | 12...13               | Hybrid    | SWB       | 10, 20 ms         |
   |                       |           |           |                   |
   | 14...15               | Hybrid    | FB        | 10, 20 ms         |
   |                       |           |           |                   |
   | 16...19               | MDCT-only | NB        | 2.5, 5, 10, 20 ms |
   |                       |           |           |                   |
   | 20...23               | MDCT-only | WB        | 2.5, 5, 10, 20 ms |
   |                       |           |           |                   |
   | 24...27               | MDCT-only | SWB       | 2.5, 5, 10, 20 ms |
   |                       |           |           |                   |
   | 28...31               | MDCT-only | FB        | 2.5, 5, 10, 20 ms |
   +-----------------------+-----------+-----------+-------------------+

                Table 1: TOC Byte Configuration Parameters

   One additional bit, labeled "s", is used to signal mono vs. stereo,
   with 0 indicating mono and 1 indicating stereo.  The remaining two
   bits, labeled "c", code the number of frames per packet (codes 0 to
   3) as follows:

   o  0: 1 frame in the packet

   o  1: 2 frames in the packet, each with equal compressed size

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

   o  3: an arbitrary number of frames in the packet

   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.  It must also obey various additional rules indicated by
   "MUST", "MUST NOT", etc., in this section.  A receiver MUST NOT
   process packets which violate these rules as normal Opus packets.
   They are reserved for future applications, such as in-band headers
   (containing metadata, etc.) or multichannel support.

   When a packet contains multiple VBR frames, the compressed length of
   one or more of these frames is indicated with a one or two byte



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   sequence, with the meaning of the first byte as follows:

   o  0: No frame (DTX or lost packet)

   o  1...251: Size of the frame in bytes

   o  252...255: A second byte is needed.  The total size is
      (size[1]*4)+size[0]

   The maximum representable size 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 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, it MUST NOT exceed 1275 bytes, to allow for repacketization
   by gateways, conference bridges, or other software.

   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

   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.







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

                         Figure 3: A Code 1 Packet

   For code 2 packets, the TOC byte is followed by a one or two byte
   sequence indicating the the length of the first frame (marked N1 in
   the 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.  The
   length of the first frame, N1, MUST be no larger than the size of the
   payload remaining after decoding that length for all code 2 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
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |0|1|s| config  | N1 (1-2 bytes):                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               :
     |               Compressed frame 1 (N1 bytes)...                |
     :                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |                               |                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               |
     |                     Compressed frame 2...                     :
     :                                                               |
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                         Figure 4: A Code 2 Packet

   For code 3 packets, 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 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



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   illustrates the layout of the frame count byte.

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

                      Figure 5: The frame count byte

   When 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.  The additional padding
   bytes appear at the end of the packet, and SHOULD be set to zero by
   the encoder, however the decoder MUST accept any value for the
   padding bytes.  By using code 255 multiple times, it is possible to
   create a packet of any specific, desired size.  Let P be the total
   amount of padding, including both the trailing padding bytes
   themselves and the header bytes used to indicate how many there are.
   Then P MUST be no more than N-2 for CBR packets, or N-M-1 for VBR
   packets.

   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)...            :
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     :                     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 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...                     :
     |                                                               |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     :                     Padding (Optional)...                     |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                       Figure 7: A VBR Code 3 Packet

3.1.  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...       :
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+




































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4.  Opus Decoder

   The Opus decoder consists of two main blocks: the SILK decoder and
   the CELT decoder.  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 as illustrated in the block diagram
   below.  At any given time, one or both of the SILK and CELT decoders
   may be active.


                          +-------+    +----------+
                          | SILK  |    |  sample  |
                       +->|encoder|--->|   rate   |----+
   bit-    +-------+   |  |       |    |conversion|    v
   stream  | Range |---+  +-------+    +----------+  /---\  audio
   ------->|decoder|                                 | + |------>
           |       |---+  +-------+    +----------+  \---/
           +-------+   |  | CELT  |    | Delay    |    ^
                       +->|decoder|----| compens- |----+
                          |       |    | ation    |
                          +-------+    +----------+


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.  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.

   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.
   Opus only uses static contexts.  They are not adapted to the
   statistics of the data as it is coded.

   The parameters needed to encode or decode a symbol in a given context



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   are represented by a three-tuple (fl,fh,ft), with 0 <= fl < fh <= ft
   <= 65535.  The values of this tuple are derived from the probability
   model for the symbol, represented by traditional _frequency counts_
   (although, since Opus uses static contexts, these are not updated as
   symbols are decoded).  Let f[i] be the frequency of the _i_th symbol
   in a context with _n_ symbols total.  Then the three-tuple
   corresponding to the _k_th symbol is given by

                     k-1                             n-1
                     __                              __
                fl = \  f[i],  fh = fl + f[k],  ft = \  f[i]
                     /_                              /_
                     i=0                             i=0

   The range decoder extracts the symbols and integers encoded using the
   range encoder in Section 5.1.  The range decoder maintains an
   internal state vector composed of the two-tuple (val,rng),
   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 then
   immediately normalizes the range using the procedure described in
   Section 4.1.1.1.

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,fh,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 one whose three-tuple (fl,fh,ft)
   satisfies fl <= fs < fh.  It uses this tuple to update val according
   to val = val - (rng/ft)*(ft-fh).  If fl is greater than zero, then
   the decoder updates rng using rng = (rng/ft)*(fh-fl).  Otherwise, it
   updates rng using rng = rng - (rng/ft)*(ft-fh).  After these updates,
   implemented by ec_dec_update() (entdec.c), it normalizes the range
   using the procedure in the next section, and returns the index of the
   identified symbol.

   With this formulation, all the truncation error from using finite
   precision arithmetic accumulates in symbol 0.  This makes the cost of



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   coding a 0 slightly smaller, on average, than the negative log of its
   estimated probability 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.

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 8 bits of
   input into sym, using the remaining bit from the previous input octet
   as the high bit of sym, and the top 7 bits of the next octet as the
   remaining bits of sym.  If no more input octets remain, it uses zero
   bits instead.  Then, it sets val to (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 8.
   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 8: Illustrative example of raw bits overlapping range coder
                                   data








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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.

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 fl = 0, fh =
   (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 = (1<<logp)-1, fh = 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 for the kth symbol
   in the context, which is equal to (1<<ftb)-fl for the (k+1)st symbol.
   fl for the 0th symbol is assumed to be 0, and the table is terminated
   by a value of 0 (where fh == 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 = (1<<ftb)-icdf[k-1] (or 0 if k == 0), fh =
   (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



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   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
   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 to be
   packed in the least significant bit of the last byte, filling up to
   the most significant bit in the last byte, and 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



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   0 to ft-1, inclusive.

   ec_dec_uint() takes a single, positive parameter, ft, which is not
   necessarily a power of two, and returns an integer, t, whose value
   lies between 0 and ft-1, inclusive.  Let ftb = ilog(ft-1), i.e., the
   number of bits required to store ft-1 in two's complement notation.
   If ftb is 8 or less, then t is decoded with t = ec_decode(ft), and
   the range coder state is updated using the three-tuple (t,t+1,ft).

   If ftb is greater than 8, then the top 8 bits of t are decoded using
   t = ec_decode((ft-1>>ftb-8)+1), the decoder state is updated using
   the three-tuple (t,t+1,(ft-1>>ftb-8)+1), and the remaining bits are
   decoded as raw bits, setting 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 symbol that requires a whole
   number of bits (i.e., ft/(fh-fl) is a power of two, including values
   of ft larger than 2**8 with ec_dec_uint()), and there are at least p



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   1/8th bits available, decoding the symbol will never advance the
   decoder past the end of the frame, i.e., will never _bust_ the
   budget.  Frames contain a whole number of bits, and the return value
   of ec_tell_frac() will only advance by more than p 1/8th bits in this
   case if there was a fractional number of bits remaining, and by no
   more than the fractional part.  However, when p is not a whole number
   of bits, an extra 1/8th bit is required to ensure decoding the symbol
   will not bust.

   The reference implementation keeps track of the total number of whole
   bits that have been processed by the decoder so far in a 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 = rng>>(l-16), so that
   32768 <= r < 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 = r*r>>15.  Then add the 16th bit
   of r to l via l = 2*l + (r>>16).  Finally, if this bit was a 1,
   reduce r by a factor of two via r = r>>1, so that it once again lies
   in the range 32768 <= r < 65536.

   This procedure is repeated three times to extend l to 1/8th bit
   precision. ec_tell_frac() then returns (nbits_total*8 - l).





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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 in NB, MB, and WB modes internally.  When used in a
   hybrid frame in SWB or FB mode, the LP layer itself still only runs
   in WB mode.

   Internally, the LP layer of a single Opus frame is composed of either
   a single 10 ms SILK frame or between one and three 20 ms SILK frames.
   Each SILK frame is in turn composed of either two or four 5 ms
   subframes.  Optional Low Bit-Rate Redundancy (LBRR) frames, which are
   reduced-bitrate encodings of previous SILK frames, may appear 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, and from here on this draft will use "SILK frame" to refer
   to either one and "regular SILK frame" if it needs to draw a
   distinction between the two.

   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.
















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        +----------------------+-----------------+---------------+
        |       Symbol(s)      |       PDF       |   Condition   |
        +----------------------+-----------------+---------------+
        |       VAD flags      |     {1, 1}/2    |               |
        |                      |                 |               |
        |       LBRR flag      |     {1, 1}/2    |               |
        |                      |                 |               |
        | Per-frame LBRR flags |  Section 4.2.3  | Section 4.2.3 |
        |                      |                 |               |
        |      Frame Type      | Section 4.2.4.1 |               |
        |                      |                 |               |
        |      Gain index      | Section 4.2.4.2 |               |
        +----------------------+-----------------+---------------+

        Order of the symbols in the SILK section of the bit-stream.

                                  Table 2

4.2.1.  Decoder Modules

   An overview of the decoder is given in Figure 9.



      +---------+    +------------+
   -->| Range   |--->| Decode     |---------------------------+
    1 | Decoder | 2  | Parameters |----------+       5        |
      +---------+    +------------+     4    |                |
                          3 |                |                |
                           \/               \/               \/
                     +------------+   +------------+   +------------+
                     | Generate   |-->| LTP        |-->| LPC        |-->
                     | Excitation |   | Synthesis  |   | Synthesis  | 6
                     +------------+   +------------+   +------------+

   1: Range encoded bitstream
   2: Coded parameters
   3: Pulses and gains
   4: Pitch lags and LTP coefficients
   5: LPC coefficients
   6: Decoded signal


                          Decoder block diagram.

                                 Figure 9





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4.2.1.1.  Range Decoder

   The range decoder decodes the encoded parameters from the received
   bitstream.  Output from this function includes the pulses and gains
   for the excitation signal generation, as well as LTP and LSF codebook
   indices, which are needed for decoding LTP and LPC coefficients
   needed for LTP and LPC synthesis filtering the excitation signal,
   respectively.

4.2.1.2.  Decode Parameters

   Pulses and gains are decoded from the parameters that were decoded by
   the range decoder.

   When a voiced frame is decoded and LTP codebook selection and indices
   are received, LTP coefficients are decoded using the selected
   codebook by choosing the vector that corresponds to the given
   codebook index in that codebook.  This is done for each of the four
   subframes.  The LPC coefficients are decoded from the LSF codebook by
   first adding the chosen LSF vector and the decoded LSF residual
   signal.  The resulting LSF vector is stabilized using the same method
   that was used in the encoder, see Section 5.2.7.5.  The LSF
   coefficients are then converted to LPC coefficients, and passed on to
   the LPC synthesis filter.

4.2.1.3.  Generate Excitation

   The pulses signal is multiplied with the quantization gain to create
   the excitation signal.

4.2.1.4.  LTP Synthesis

   For voiced speech, the excitation signal e(n) is input to an LTP
   synthesis filter that will recreate the long term correlation that
   was removed in the LTP analysis filter and generate an LPC excitation
   signal e_LPC(n), according to


                                   d
                                  __
                e_LPC(n) = e(n) + \  e_LPC(n - L - i) * b_i,
                                  /_
                                 i=-d


   using the pitch lag L, and the decoded LTP coefficients b_i.  The
   number of LTP coefficients is 5, and thus d = 2.  For unvoiced
   speech, the output signal is simply a copy of the excitation signal,



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   i.e., e_LPC(n) = e(n).

4.2.1.5.  LPC Synthesis

   In a similar manner, the short-term correlation that was removed in
   the LPC analysis filter is recreated in the LPC synthesis filter.
   The LPC excitation signal e_LPC(n) is filtered using the LTP
   coefficients a_i, according to


                                     d_LPC
                                      __
                    y(n) = e_LPC(n) + \  y(n - i) * a_i,
                                      /_
                                      i=1


   where d_LPC is the LPC synthesis filter order, and y(n) is the
   decoded output signal.

4.2.2.  Header Bits

   The LP layer begins with two to eight header bits, decoded in
   silk_Decode() (silk_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
   flags correspond to the mid channel, and a second set of flags is
   included for the side channel.

   Because these are the first symbols decoded by the range coder, they
   can be extracted directly from the upper 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.

4.2.3.  LBRR Flags

   For Opus frames longer than 20 ms, a set of per-frame LBRR flags is
   decoded for each channel that has its LBRR flag set.  For 40 ms Opus
   frames the 2-frame LBRR flag PDF from Table 3 is used, and for 60 ms
   Opus frames the 3-frame LBRR flag PDF is used.  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.








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           +------------+-------------------------------------+
           | Frame Size | PDF                                 |
           +------------+-------------------------------------+
           | 40 ms      | {0, 53, 53, 150}/256                |
           |            |                                     |
           | 60 ms      | {0, 41, 20, 29, 41, 15, 28, 82}/256 |
           +------------+-------------------------------------+

                          Table 3: LBRR Flag PDFs

   LBRR frames do not include their own separate VAD flags.  An LBRR
   frame is only meant to be transmitted for active speech, thus all
   LBRR frames are treated as active.

4.2.4.  SILK Frame Contents

   Each SILK frame includes a set of side information that encodes the
   frame type, quantization type and gains, short-term prediction filter
   coefficients, LSF interpolation weight, long-term prediction filter
   lags and gains, and a pseudorandom number generator (PRNG) seed.
   This is followed by the quantized excitation signal.

4.2.4.1.  Frame Type

   Each SILK frame begins with 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 the value either 0 or 1 and is decoded using the
   first PDF in Table 4.  If the frame is an LBRR frame or a regular
   SILK frame whose VAD flag was set (an _active_ frame), then the
   symbol ranges from 2 to 5, inclusive, and is decoded using the second
   PDF in Table 4.  Table 5 translates between the value of the frame
   type symbol and the corresponding signal type and quantization offset
   type.

                +----------+-----------------------------+
                | VAD Flag | PDF                         |
                +----------+-----------------------------+
                | Inactive | {26, 230, 0, 0, 0, 0}/256   |
                |          |                             |
                | Active   | {0, 0, 24, 74, 148, 10}/256 |
                +----------+-----------------------------+

                         Table 4: Frame Type PDFs






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          +------------+-------------+--------------------------+
          | Frame Type | Signal Type | Quantization Offset Type |
          +------------+-------------+--------------------------+
          | 0          | Inactive    |                        0 |
          |            |             |                          |
          | 1          | Inactive    |                        1 |
          |            |             |                          |
          | 2          | Unvoiced    |                        0 |
          |            |             |                          |
          | 3          | Unvoiced    |                        1 |
          |            |             |                          |
          | 4          | Voiced      |                        0 |
          |            |             |                          |
          | 5          | Voiced      |                        1 |
          +------------+-------------+--------------------------+

     Table 5: Signal Type and Quantization Offset Type from Frame Type

4.2.4.2.  Sub-Frame 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.

   For the first LBRR frame, an LBRR frame where the previous LBRR frame
   was not coded, or the first regular SILK frame in an Opus frame, the
   first subframe uses an independent coding method.  The 3 most
   significant bits of the quantization gain are decoded using a PDF
   selected from Table 6 based on the decoded signal type.

           +-------------+------------------------------------+
           | 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 6: PDFs for Independent Quantization Gain MSb Coding

   The 3 least significant bits are decoded using a uniform PDF:




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                 +--------------------------------------+
                 | PDF                                  |
                 +--------------------------------------+
                 | {32, 32, 32, 32, 32, 32, 32, 32}/256 |
                 +--------------------------------------+

         Table 7: PDF for Independent Quantization Gain LSb Coding

   For all other subframes (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.  The PDF in
   Table 8 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                                                            |
   +-------------------------------------------------------------------+

              Table 8: 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)

   silk_gains_dequant() (silk_gain_quant.c) dequantizes the 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() (silk_log2lin.c) computes an
   approximation of of 2**(inLog_Q7/128.0), where inLog_Q7 is its Q7
   input.  Let i = inLog_Q7>>7 be the integer part of inLogQ7 and f =
   inLog_Q7&127 be the fractional part.  Then, 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) .







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4.2.4.3.  Normalized Line Spectral Frequencies

   Normalized Line Spectral Frequencies (LSFs) follow the quantization
   gains in the bitstream, and represent the Linear Prediction
   Coefficients (LPCs) for the current SILK frame.  Once decoded, they
   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 4.2.4.3.3).  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).  NB and MB frames use an order-10 predictor, while WB frames
   use an order-16 predictor, and thus have different sets of tables.
   The first VQ stage uses a 32-element codebook, coded with one of the
   PDFs in Table 9, 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 18 and Table 19, 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          |
   +-----------+----------+--------------------------------------------+




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          Table 9: PDFs for Normalized LSF Index Stage-1 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 10, and the
   8 (i...p) for WB frames given in Table 11.  Which PDF is used for
   which coefficient is driven by the index, I1, decoded in the first
   stage.  Table 12 lists the letter of the corresponding PDF for each
   normalized LSF coefficient for NB and MB, and Table 13 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 10: 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 11: 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 12: 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 13: 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 12 or Table 13, and subtracts 4 from the
   result to given 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 14, 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 14: PDF for Normalized LSF Index Extension Decoding

   The decoded indices from both stages are translated back into
   normalized LSF coefficients in silk_NLSF_decode()
   (silk_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 15



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   contains lists of prediction weights for each coefficient.  There are
   two lists for NB and MB, and another two lists for WB, giving two
   possible prediction weights for each coefficient.

                  +-------------+-----+-----+-----+-----+
                  | 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 15: Prediction Weights for Normalized LSF Decoding

   The prediction is undone using the procedure implemented in
   silk_NLSF_residual_dequant() (silk_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 16 gives the
   selections for each coefficient for NB and MB, and Table 17 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.



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   Then, the stage-2 residual for each coefficient is computed via

        res_Q10[k] = (k+1 < d_LPC ? (res_Q10[k+1]*pred_Q8[k])>>8 : 0)
                     + ((((I2[k]<<10) + sign(I2[k])*102)*qstep)>>16) ,

   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 16: 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 17: Prediction Weight Selection for WB Normalized LSF Decoding

   The spectral distortion introduced by the quantization of each LSF
   coefficient varies, so the stage-2 residual is weighted accordingly,
   using the low-complexity weighting 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 18 or Table 19.  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) to avoid computing them when decoding.  The reference
   implementation computes them on the fly in
   silk_NLSF_VQ_weights_laroia() (silk_NLSF_VQ_weights_laroia.c) and its
   caller, to reduce the amount of ROM required.

              +----+----------------------------------------+
              | I1 | Codebook                               |
              +----+----------------------------------------+
              |    |  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 |
              |    |                                        |



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              | 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 |
              |    |                                        |
              | 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 |
              |    |                                        |



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              | 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 18: Codebook Vectors for NB/MB Normalized LSF Stage 1 Decoding

    +----+------------------------------------------------------------+
    | I1 | Codebook                                                   |
    +----+------------------------------------------------------------+
    |    |  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 |
    |    |                                                            |
    | 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 |
    |    |                                                            |



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    | 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 19: Codebook Vectors for WB Normalized LSF Stage 1 Decoding

   Given the stage-1 codebook entry cb1_Q8[], the stage-2 residual
   res_Q10[], and their corresponding weights, w_Q9[], the reconstructed
   normalized LSF coefficients are

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

   where the division is exact integer division.  However, nothing thus
   far in the reconstruction process, nor in the quantization process in
   the encoder, guarantees that the coefficients are monotonically



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   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.

4.2.4.3.1.  Normalized LSF Stabilization

   The normalized LSF stabilization procedure is implemented in
   silk_NLSF_stabilize() (silk_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 20 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 20: 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 20.  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 executed 20 times, or
   until it stops because the coefficients satisfy all the constraints.

   After the 20th repetition of the above, 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]) .

4.2.4.3.2.  Normalized LSF Interpolation

   For 20 ms SILK frames, the first half of the frame (i.e., the first
   two sub-frames) may use normalized LSF coefficients that are
   interpolated between the decoded LSFs for the previous frame and the
   current frame.  A Q2 interpolation factor follows the LSF coefficient



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   indices in the bitstream, which is decoded using the PDF in Table 21.
   This happens in silk_decode_indices() (silk_decode_indices.c).  For
   the first frame after a decoder reset, 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 21: PDF for Normalized LSF Interpolation Index

   Let n2_Q15[k] be the normalized LSF coefficients decoded by the
   procedure in Section 4.2.4.3, 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()
   (silk_decode_parameters.c).

4.2.4.3.3.  Converting Normalized LSF Coefficients to LPCs

   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
   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.



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   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()
   (silk_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 true the inverse of the
   cosine function, when deriving the normalized LSF coefficients.

   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 approximated
   cosine, c_Q17[k], is

    c_Q17[k] = (cos_Q13[i]*256 + (cos_Q13[i+1]-cos_Q13[i])*f + 8) >> 4 ,

   where cos_Q13[i] is the corresponding entry of Table 22.

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



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                  |  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 |
                  |     |       |       |       |       |
                  |  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 |
                  |     |       |       |       |       |



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                  | 120 | -8034 | -8072 | -8104 | -8130 |
                  |     |       |       |       |       |
                  | 124 | -8152 | -8170 | -8182 | -8190 |
                  |     |       |       |       |       |
                  | 128 | -8192 |       |       |       |
                  +-----+-------+-------+-------+-------+

               Table 22: Q13 Cosine Table for LSF Conversion

   Given the list of cosine values, silk_NLSF2A_find_poly()
   (silk_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
   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.



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4.2.4.3.4.  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 the largest, breaking ties by using the lower
   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
   large.

   silk_bwexpander_32() (silk_bwexpander_32.c) peforms 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:



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       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 of the coefficients still overflow a 16-bit integer.
   This saturation is not performed if maxabs_Q12 drops to 32767 or less
   prior to the 10th round.

4.2.4.3.5.  Limiting the Prediction Gain of the LPC Filter

   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 LPC_inverse_pred_gain_QA()
   (silk_LPC_inv_pred_gain.c) to compute the reflection coefficients
   associated with the filter.  The filter is stable if and only if the
   magnitude of these coefficients is sufficiently less than one.  The
   reflection coefficients, rc[k], can be computed using a simple
   Levinson recurrence, initialized with the LPC coefficients a[d_LPC-
   1][n] = a[n], and then updated via

                     rc[k] = -a[k][k] ,

                             a[k][n] - a[k][k-n-1]*rc[k]
                 a[k-1][n] = --------------------------- .
                                              2
                                     1 - rc[k]

   However, LPC_inverse_pred_gain_QA() approximates this using fixed-
   point arithmetic to guarantee reproducible results across platforms
   and implementations.  It is important to run on the real Q12
   coefficients that will be used during reconstruction, because small
   changes in the coefficients can make a stable filter unstable, but
   increasing the precision back to Q16 allows more accurate computation
   of the reflection coefficients.  Thus, let

            a32_Q16[d_LPC-1][n] = ((a32_Q17[n] + 16) >> 5) << 4

   be the Q16 representation of the Q12 version of the LPC coefficients
   that will eventually be used.  Then for each k from d_LPC-1 down to
   0, if abs(a32_Q16[k][k]) > 65520, the filter is unstable and the
   recurrence stops.  Otherwise, the row k-1 of a32_Q16 is computed from
   row k as



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           rc_Q31[k] = -a32_Q16[k][k] << 15 ,

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

               b1[k] = ilog(div_Q30[k]) - 16 ,

                             (1<<29) - 1
          inv_Qb1[k] = ----------------------- ,
                       div_Q30[k] >> (b1[k]+1)

          err_Q29[k] = (1<<29)
                       - ((div_Q30[k]<<(15-b1[k]))*inv_Qb1[k] >> 16) ,

          mul_Q16[k] = ((inv_Qb1[k] << 16)
                        + (err_Q29[k]*inv_Qb1[k] >> 13)) >> b1[k] ,

               b2[k] = ilog(mul_Q16[k]) - 15 ,

       t_Q16[k-1][n] = a32_Q16[k][n]
                       - ((a32_Q16[k][k-n-1]*rc_Q31[k] >> 32) << 1) ,

     a32_Q16[k-1][n] = ((t_Q16[k-1][n] *
                         (mul_Q16[k] << (16-b2[k]))) >> 32) << b2[k] .

   Here, rc_Q30[k] are the reflection coefficients. div_Q30[k] is the
   denominator for each iteration, and mul_Q16[k] is its multiplicative
   inverse. inv_Qb1[k], which ranges from 16384 to 32767, is a low-
   precision version of that inverse (with b1[k] fractional bits, where
   b1[k] ranges from 3 to 14). err_Q29[k] is the residual error, ranging
   from -32392 to 32763, which is used to improve the accuracy.
   t_Q16[k-1][n], 0 <= n < k, are the numerators for the next row of
   coefficients in the recursion, and a32_Q16[k-1][n] is the final
   version of that row.  Every multiply in this procedure except the one
   used to compute mul_Q16[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_Q16[k][k]) <= 65520 for 0 <= k < d_LPC, then the filter is
   considerd stable.

   On round i, 1 <= i <= 18, if the filter passes this stability check,
   then this procedure stops, and the final LPC coefficients to use for
   reconstruction are

                    a_Q12[k] = (a32_Q17[k] + 16) >> 5 .

   Otherwise, a round of bandwidth expansion is applied using the same
   procedure as in Section 4.2.4.3.4, with



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                       sc_Q16[0] = 65536 - i*(i+9) .

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

4.2.4.4.  Long-Term Prediction (LTP) Parameters

   After the normalized LSF indices and, for 20 ms frames, the LSF
   interpolation index, voiced frames (see Section 4.2.4.1) include
   additional Long-Term Prediction (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.

4.2.4.4.1.  Pitch Lags

   The primary lag index is coded either relative to the primary lag of
   the prior frame or as an absolute index.  Like the quantization
   gains, the first LBRR frame, an LBRR frame where the previous LBRR
   frame was not coded, or the first regular SILK frame in an Opus frame
   all code the pitch lag as an absolute index.  When the prior frame
   was not voiced, this also forces absolute coding.

   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 23 and the low part using the codebook
   corresponding to the current audio bandwidth from Table 24.  The
   final primary pitch lag is then

                lag = lag_high*lag_scale + lag_low + lag_min

   where lag_high is the high part, lag_low is the low part, and
   lag_scale and lag_min are the values from the "Scale" and "Minimum
   Lag" columns of Table 24, 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 23: PDF for High Part of Primary Pitch Lag






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   +------------+------------------------+-------+----------+----------+
   | 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 24: 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 25.  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 previous frame and
   delta_lag_index is the value just decoded.  This allows a per-frame
   change in the pitch lag of -8 to +11 samples.  The decoder does no
   clamping at this point, so this value can fall outside the range of
   2 ms to 18 ms, and the decoder must use this unclamped value when
   using relative coding in the next SILK frame (if any).  However,
   because an Opus frame can use relative coding for at most two
   consecutive SILK frames, integer overflow should not be an issue.

   +-------------------------------------------------------------------+
   | PDF                                                               |
   +-------------------------------------------------------------------+
   | {46, 2, 2, 3, 4, 6, 10, 15, 26, 38, 30, 22, 15, 10, 7, 6, 4, 4,   |
   | 2, 2, 2}/256                                                      |
   +-------------------------------------------------------------------+

                    Table 25: PDF for 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 26 depending on the current frame size
   and audio bandwidth.  Table 27 through Table 30 give the
   corresponding offsets to apply to the primary pitch lag for each
   subframe given the decoded codebook index.



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   +-----------+---------+---------------------------------------------+
   | Audio     | SILK    | PDF                                         |
   | Bandwidth | Frame   |                                             |
   |           | Size    |                                             |
   +-----------+---------+---------------------------------------------+
   | NB        | 10 ms   | {143, 50, 63}/256                           |
   |           |         |                                             |
   | NB        | 20 ms   | {68, 12, 21, 17, 19, 22, 30, 24, 17, 16,    |
   |           |         | 10}/256                                     |
   |           |         |                                             |
   | MB or WB  | 10 ms   | {91, 46, 39, 19, 14, 12, 8, 7, 6, 5, 5,     |
   |           |         | 4}/256                                      |
   |           |         |                                             |
   | MB or WB  | 20 ms   | {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}                  |
   +-----------+---------+---------------------------------------------+

                 Table 26: PDFs for Subframe Pitch Contour

                       +-------+------------------+
                       | Index | Subframe Offsets |
                       +-------+------------------+
                       | 0     |            0,  0 |
                       |       |                  |
                       | 1     |            1,  0 |
                       |       |                  |
                       | 2     |            0,  1 |
                       +-------+------------------+

     Table 27: Codebook Vectors for Subframe Pitch Contour: NB, 10 ms
                                  Frames



















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                       +-------+------------------+
                       | 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 28: 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 29: 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 30: 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() (silk_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 24, respectively.

4.2.5.  LBRR Frames

   LBRR frames, if present, immediately follow the header bits, prior to
   any regular SILK frames.  Each frame whose LBRR flag was set includes
   a separate set of data for each channel.

4.3.  CELT Decoder

   The CELT layer is decoded 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  |



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   |   transient   |              {7, 1}/8             |               |
   |               |                                   |               |
   |     intra     |              {7, 1}/8             |               |
   |               |                                   |               |
   | coarse energy |           Section 4.3.2           |               |
   |               |                                   |               |
   |   tf_change   |           Section 4.3.1           |               |
   |               |                                   |               |
   |   tf_select   |              {1, 1}/2             | Section 4.3.1 |
   |               |                                   |               |
   |     spread    |          {7, 2, 21, 2}/32         |               |
   |               |                                   |               |
   |  dyn. alloc.  |           Section 4.3.3           |               |
   |               |                                   |               |
   |  alloc. trim  |  {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 bit-stream.

                                 Table 31

   The decoder extracts information from the range-coded bit-stream 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.







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4.3.1.  Transient Decoding

   The _transient_ flag encoded in the bit-stream 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
   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: A(z_l,
   z_b)=(1-a*z_l^-1)*(1-z_b^-1)/(1-b*z_b^-1) 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 a=0 b=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 to not suffer
   desynchronization.  We approximate the ideal probability distribution
   of the prediction error using a Laplace distribution with seperate
   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



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   the Laplace-distributed values is implemented in ec_laplace_decode()
   (laplace.c).

4.3.2.2.  Fine energy quantization

   The number of bits assigned to fine energy quantization in each band
   is determined by the bit allocation computation described in
   Section 4.3.3.  Let B_i be the number of fine energy bits for band i;
   the refinement is an integer f in the range [0,2^B_i-1].  The mapping
   between f and the correction applied to the coarse energy is equal to
   (f+1/2)/2^B_i - 1/2.  Fine energy quantization is implemented in
   quant_fine_energy() (quant_bands.c).

   When some bits are left "unused" after all other flags have been
   decoded, these bits are assigned to a "final" step of fine
   allocation.  In effect, these bits are used to add one extra fine
   energy bit per band per channel.  The allocation process determines
   two _priorities_ for the final fine bits.  Any remaining bits are
   first assigned only to bands of priority 0, starting from band 0 and
   going up.  If all bands of priority 0 have received one bit per
   channel, then bands of priority 1 are assigned an extra bit per
   channel, starting from band 0.  If any bit is 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: 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



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   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.

   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 bit-stream 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 also 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



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   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.
   As with other signaled parameters, 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 process of allocation consists of the following steps:
   determining the per-band maximum allocation vector, decoding the
   boosts, decoding the tilt, determining the remaining capacity 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 an shape allocation per-band (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 which 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: bit-stream 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 partially computed form, in order to fit in less memory,
   as a static table (XXX cache.caps).  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
   assign 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()



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   in celt.c.

   The band boosts are represented by a series of binary symbols which
   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 execute on all bands the band boosts have been
   decoded.  This procedure is implemented around line 2352 of celt.c.

   At very low rates it's 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 at around 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



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   5 bias it towards higher frequencies.  Like other signaled
   parameters, signaling of the trim is gated so that it is not included
   if there is insufficient space available in the bitstream.  To decode
   the trim first set the trim value to 5 then 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) then decode the trim value using the
   inverse CDF {127, 126, 124, 119, 109, 87, 41, 19, 9, 4, 2, 0}.

   Stereo parameters

   Anti-collapse reservation

   The allocation computation first begins by setting up some initial
   conditions. 'total' is set to the available remaining 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 assure 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 excessively an 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,
   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.




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   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 number
   of channels, the number MDCT bins in the shortest frame size for this
   mode, the number 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 Decoder

   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
   that is the nearest to the allocated value (rounding down if exactly
   half-way between two values), subject to not exceeding the total
   number of bits available.  For efficiency reasons the search is
   performed against a precomputated allocation table which only permits
   some K values for each N. The number of codebooks entries can be
   computed as explained in Section 5.3.5.2.  The difference between the
   number of bits allocated and the number of bits used is accumulated
   to a _balance_ (initialised to zero) that helps adjusting the
   allocation for the next bands.  One third of the balance is applied
   to the bit allocation of the each band to help achieving 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.

4.3.4.2.  Index Decoding

   The codeword is decoded as a uniformly-distributed integer value by
   decode_pulses() (cwrs.c).  The codeword is converted from a unique
   index in the same way as specified in [PVQ].  The indexing is based



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   on the calculation of V(N,K) (denoted N(L,K) in [PVQ]), which is the
   number of possible combinations of K pulses in N samples.  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 pre-computed
   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 initialise 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 decoding of the codeword from the index is performed as specified
   in [PVQ], as implemented in function decode_pulses() (cwrs.c).

4.3.4.3.  Spreading

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

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, then 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



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   decoded energy.  This is done by denormalise_bands() (bands.c).

4.3.7.  Inverse MDCT

   The inverse MDCT implementation has no special characteristics.  The
   input is N frequency-domain samples and the output is 2*N time-domain
   samples, while scaling by 1/2.  The output is windowed using the same
   window as the encoder.  The IMDCT and windowing are performed by
   mdct_backward (mdct.c).  If a time-domain pre-emphasis window was
   applied in the encoder, the (inverse) time-domain de-emphasis window
   is applied on the IMDCT result.

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.

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/A(z)=1/
   (1-alpha_p*z^-1), where alpha_p=0.8500061035.



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4.3.8.  Packet Loss Concealment (PLC)

   Packet loss concealment (PLC) is an optional decoder-side feature
   which SHOULD be included when transmitting over an unreliable
   channel.  Because PLC is not part of the bit-stream, there are
   several possible ways to implement PLC with different complexity/
   quality trade-offs.  The PLC in the reference implementation 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).

4.4.  Mode Switching

   Switching between the Opus coding modes requires careful
   consideration.  More specifically, the transitions that cannot be
   easily handled are the ones where the lower frequencies have to
   switch between the SILK LP-based model and the CELT transform model.
   If nothing is done, a glitch will occur for these transitions.  On
   the other hand, switching between the SILK-only modes and the hybrid
   mode does not require any special treatment.

   There are two ways to avoid or reduce glitches during the problematic
   mode transitions: with, or without side information.  Only
   transitions with side information are normatively specified.  For
   transitions with no side information, it is RECOMMENDED for the
   decoder to use a concealment technique (e.g. make use of the PLC
   algorithm) to "fill in" the gap or the discontinuity caused by the
   mode transition.  Note that this concealment MUST NOT be applied when
   switching between the SILK mode and the hybrid mode or vice versa.
   Similarly, it MUST NOT be applied when merely changing the bandwidth
   within the same mode.

4.4.1.  Switching Side Information

   Switching with side information involves transmitting in-band a 5-ms
   "redundant" CELT frame within the Opus frame.  This frame is designed
   to fill-in the gap or discontinuity without requiring the decoder to
   conceal it.  For transitons from a CELT-only frame to a SILK-only or
   hybrid frame, the redundant frame is inserted in the frame following
   the transition (i.e. the SILK-only/hybrid frame).  For transitions
   from a SILK-only/hybrid frame to a CELT-only frame, the redundant
   frame is inserted in the first frame.  For all SILK-only and hybrid
   frames (not only those involved in a mode transition), a binary
   symbol of probability 2^-12 needs to be decoded just after the SILK
   part of the bit-stream.  When the symbol value is 1, then the frame
   includes an embedded redundant frame.  The redundant frame always



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   starts and ends on byte boundaries.  For SILK-only frames, the number
   of bytes is simply the number of whole remaining bytes.  For hybrid
   frames, the number of bytes is equal to 2, plus a decoded unsigned
   integer (ec_dec_uint()) between 0 and 255.  For hybrid frames, the
   redundant frame is placed at the end of the frame, after the CELT
   layer of the hybrid frame.  The redundant frame is decoded like any
   other CELT-only frame, with the exception that it does not contain a
   TOC byte.  The bandwidth is instead set to the same bandwidth of the
   current frame (for mediumband frames, the redundant frame is set to
   wideband).

   For CELT-only to SILK-only/hybrid transitions, the first 2.5 ms of
   the redundant frame is used as-is for the reconstructed output.  The
   remaining 2.5 ms is overlapped and added (cross-faded using the
   square of the MDCT power-complemantary window) to the decoded SILK/
   hybrid signal, ensuring a smooth transition.  For SILK-only/hyrid to
   CELT-only transitions, only the second half of the 5-ms decoded
   redundant frame is used.  In that case, only a 2.5-ms cross-fade is
   applied, still using the power-complemantary window.
































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5.  Codec Encoder

   Opus encoder block diagram.


            +----------+    +-------+
            |  sample  |    | SILK  |
         +->|   rate   |--->|encoder|--+
         |  |conversion|    |       |  |
   audio |  +----------+    +-------+  |    +-------+
   ------+                             +--->| Range |
         |  +-------+                       |encoder|---->
         |  | CELT  |                  +--->|       | bit-stream
         +->|encoder|------------------+    +-------+
            |       |
            +-------+


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



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   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 ith symbol in
   the current context.  Then the three-tuple corresponding to the kth
   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 =
   (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 the continuing in the least significant bit



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   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.

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()



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   (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.

5.2.  SILK Encoder

   In the following, we focus on the core encoder and describe its
   components.  For simplicity, we will refer to the core encoder simply
   as the encoder in the remainder of this document.  An overview of the
   encoder is given in Figure 10.


                                                                 +---+
                                  +----------------------------->|   |
           +---------+            |     +---------+              |   |
           |Voice    |            |     |LTP      |              |   |
    +----->|Activity |-----+      +---->|Scaling  |---------+--->|   |
    |      |Detector |  3  |      |     |Control  |<+  12   |    |   |
    |      +---------+     |      |     +---------+ |       |    |   |
    |                      |      |     +---------+ |       |    |   |
    |                      |      |     |Gains    | |  11   |    |   |
    |                      |      |  +->|Processor|-|---+---|--->| R |
    |                      |      |  |  |         | |   |   |    | a |
    |                     \/      |  |  +---------+ |   |   |    | n |
    |                 +---------+ |  |  +---------+ |   |   |    | g |
    |                 |Pitch    | |  |  |LSF      | |   |   |    | e |
    |              +->|Analysis |-+  |  |Quantizer|-|---|---|--->|   |
    |              |  |         |4|  |  |         | | 8 |   |    | E |->
    |              |  +---------+ |  |  +---------+ |   |   |    | n |14
    |              |              |  |   9/\  10|   |   |   |    | c |
    |              |              |  |    |    \/   |   |   |    | o |
    |              |  +---------+ |  |  +----------+|   |   |    | d |
    |              |  |Noise    | +--|->|Prediction|+---|---|--->| e |
    |              +->|Shaping  |-|--+  |Analysis  || 7 |   |    | r |
    |              |  |Analysis |5|  |  |          ||   |   |    |   |
    |              |  +---------+ |  |  +----------+|   |   |    |   |
    |              |              |  |       /\     |   |   |    |   |
    |              |    +---------|--|-------+      |   |   |    |   |
    |              |    |        \/  \/            \/  \/  \/    |   |
    |  +---------+ |    |      +---------+       +------------+  |   |
    |  |High-Pass| |    |      |         |       |Noise       |  |   |
   -+->|Filter   |-+----+----->|Prefilter|------>|Shaping     |->|   |
   1   |         |      2      |         |   6   |Quantization|13|   |
       +---------+             +---------+       +------------+  +---+

   1:  Input speech signal
   2:  High passed input signal



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   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
   14: Range encoded bitstream



                          Encoder block diagram.

                                 Figure 10

5.2.1.  Voice Activity Detection

   The input signal is processed by a VAD (Voice Activity Detector) to
   produce a measure of voice activity, and also spectral tilt and
   signal-to-noise estimates, for each frame.  The VAD uses a sequence
   of half-band filterbanks to split the signal in four subbands: 0 -
   Fs/16, Fs/16 - Fs/8, Fs/8 - Fs/4, and Fs/4 - Fs/2, where Fs is the
   sampling frequency, that is, 8, 12, 16, or 24 kHz.  The lowest
   subband, from 0 - Fs/16 is high-pass filtered with a first-order MA
   (Moving Average) filter (with transfer function H(z) = 1-z^(-1)) to
   reduce the energy at the lowest frequencies.  For each frame, the
   signal energy per subband is computed.  In each subband, a noise
   level estimator tracks the background noise level and an SNR (Signal-
   to-Noise Ratio) 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.




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   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.

5.2.2.  High-Pass Filter

   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 ARMA (Auto
   Regressive Moving Average) filter with a cut-off frequency around
   70 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.

5.2.3.  Pitch Analysis

   The high-passed input signal is processed by the open loop pitch
   estimator shown in Figure 11.




























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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 11

   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 a 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.

5.2.4.  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 fulfil 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  Deemphasizing spectral valleys; by using different coefficients in
      the analysis and synthesis part of the prefilter and noise shaping



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      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: Deemphasized and level matched spectrum
               3: Quantization noise spectrum


           Noise shaping and spectral de-emphasis illustration.

                                 Figure 12

   Figure 12 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
   deemphasized 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 deemphasized signal can be
   described as a filtering operation with a filter


                                                    Wana(z)
               H(z) = G * ( 1 - c_tilt * z^(-1) ) * -------
                                                    Wsyn(z),


   having an adjustment gain G, a first order tilt adjustment filter
   with tilt coefficient c_tilt, and where


                   16                                 d
                   __                                __
    Wana(z) = (1 - \ (a_ana(k) * z^(-k))*(1 - z^(-L) \ b_ana(k)*z^(-k)),
                   /_                                /_
                   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
                   __                                __
    Wsyn(z) = (1 - \ (a_syn(k) * z^(-k))*(1 - z^(-L) \ b_syn(k)*z^(-k)).
                   /_                                /_
                   k=1                               k=-d


   All noise shaping parameters are computed and applied per subframe of
   5 milliseconds.  First, an LPC analysis is performed on a windowed
   signal block of 15 milliseconds.  The signal block has a look-ahead
   of 5 milliseconds 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 16 for best quality or 12 in
   low complexity operation.  The quantization gain is found as the
   square-root of the residual energy from the LPC analysis, multiplied
   by a value inversely proportional to the coding quality control
   parameter and the pitch correlation.

   Next we find the two sets of short-term noise shaping coefficients



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   a_ana(k) and a_syn(k), 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
   origo, using the formulas


                         a_ana(k) = a(k)*g_ana^k, and
                         a_syn(k) = a(k)*g_syn^k,


   where a(k) is the k'th LPC coefficient and the bandwidth expansion
   factors g_ana and g_syn are calculated as


                         g_ana = 0.94 - 0.02*C, and
                         g_syn = 0.94 + 0.02*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
   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 = 0.4, and as
                          c_tilt = 0.04 + 0.06 * C


   for voiced frames, where C again is the coding quality control
   parameter and is between 0 and 1.




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   The adjustment gain G serves to correct any level mismatch between
   original and decoded signal 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
                                ___
                   predGain = ( | | 1 - (r_k)^2 )^(-0.5),
                                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.

5.2.5.  Prefilter

   In the prefilter the input signal is filtered using the spectral
   valley de-emphasis filter coefficients from the noise shaping
   analysis, see Section 5.2.4.  By applying only the noise shaping
   analysis filter to the input signal, it provides the input to the
   noise shaping quantizer.

5.2.6.  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 are described in Section 5.2.6.1 and
   Section 5.2.6.2, respectively.  Inputs to this function include the
   pre-whitened signal from the pitch estimator, see Section 5.2.3.

5.2.6.1.  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



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   four subframes.  The LTP coefficients are used to find an LTP
   residual signal with the simulated output signal as input to obtain
   better modelling of the output signal.  This LTP residual signal is
   the input to an LPC analysis where the LPCs are estimated using Burgs
   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 5.2.7.  After quantization, the
   quantized LSF vector is converted to LPC coefficients and hence by
   using these quantized coefficients the encoder remains fully
   synchronized with the decoder.  The LTP coefficients are quantized
   using a method described in Section 5.2.8.  The quantized LPC and LTP
   coefficients are now used to filter the high-pass filtered input
   signal and measure a residual energy for each of the four subframes.

5.2.6.2.  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 an LTP analysis to be worth-while the cost in
   terms of complexity and rate.  Therefore, the pre-whitened input
   signal is discarded and instead the high-pass filtered input signal
   is used for LPC analysis using Burgs method.  The resulting LPC
   coefficients are converted to an LSF vector, quantized as described
   in the following section and transformed back to obtain quantized LPC
   coefficients.  The quantized LPC coefficients are used to filter the
   high-pass filtered input signal and measure a residual energy for
   each of the four subframes.

5.2.7.  LSF Quantization

   The purpose of quantization in general is to significantly lower the
   bit rate at the cost of some introduced distortion.  A higher rate
   should always result in lower distortion, and lowering the rate will
   generally lead to higher distortion.  A commonly used but generally
   sub-optimal approach is to use a quantization method with a constant
   rate where only the error is minimized when quantizing.

5.2.7.1.  Rate-Distortion Optimization

   Instead, we minimize an objective function that consists of a
   weighted sum of rate and distortion, and use a codebook with an
   associated non-uniform rate table.  Thus, we take into account that
   the probability mass function for selecting the codebook entries are
   by no means guaranteed to be uniform in our scenario.  The advantage
   of this approach is that it ensures that rarely used codebook vector
   centroids, which are modelling statistical outliers in the training
   set can be quantized with a low error but with a relatively high cost



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   in terms of a high rate.  At the same time this approach also
   provides the advantage that frequently used centroids are modelled
   with low error and a relatively low rate.  This approach will lead to
   equal or lower distortion than the fixed rate codebook at any given
   average rate, provided that the data is similar to the data used for
   training the codebook.

5.2.7.2.  Error Mapping

   Instead of minimizing the error in the LSF domain, we map the errors
   to better approximate spectral distortion by applying an individual
   weight to each element in the error vector.  The weight vectors are
   calculated for each input vector using the Inverse Harmonic Mean
   Weighting (IHMW) function proposed by Laroia et al., see
   [laroia-icassp].  Consequently, we solve the following minimization
   problem, i.e.,


         LSF_q = argmin { (LSF - c)' * W * (LSF - c) + mu * rate },
                 c in C


   where LSF_q is the quantized vector, LSF is the input vector to be
   quantized, and c is the quantized LSF vector candidate taken from the
   set C of all possible outcomes of the codebook.

5.2.7.3.  Multi-Stage Vector Codebook

   We arrange the codebook in a multiple stage structure to achieve a
   quantizer that is both memory efficient and highly scalable in terms
   of computational complexity, see e.g. [sinervo-norsig].  In the first
   stage the input is the LSF vector to be quantized, and in any other
   stage s > 1, the input is the quantization error from the previous
   stage, see Figure 13.

















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         Stage 1:           Stage 2:                Stage S:
       +----------+       +----------+            +----------+
       |  c_{1,1} |       |  c_{2,1} |            |  c_{S,1} |
   LSF +----------+ res_1 +----------+  res_{S-1} +----------+
   --->|  c_{1,2} |------>|  c_{2,2} |--> ... --->|  c_{S,2} |--->
       +----------+       +----------+            +----------+ res_S =
           ...                ...                     ...      LSF-LSF_q
       +----------+       +----------+            +----------+
       |c_{1,M1-1}|       |c_{2,M2-1}|            |c_{S,MS-1}|
       +----------+       +----------+            +----------+
       | c_{1,M1} |       | c_{2,M2} |            | c_{S,MS} |
       +----------+       +----------+            +----------+


                Multi-Stage LSF Vector Codebook Structure.

                                 Figure 13

   By storing total of M codebook vectors, i.e.,


                                      S
                                     __
                                 M = \  Ms,
                                     /_
                                     s=1


   where M_s is the number of vectors in stage s, we obtain a total of


                                      S
                                     ___
                                 T = | | Ms
                                     s=1


   possible combinations for generating the quantized vector.  It is for
   example possible to represent 2**36 uniquely combined vectors using
   only 216 vectors in memory, as done in SILK for voiced speech at all
   sample frequencies above 8 kHz.

5.2.7.4.  Survivor Based Codebook Search

   This number of possible combinations is far too high for a full
   search to be carried out for each frame so for all stages but the
   last, i.e., s smaller than S, only the best min( L, Ms ) centroids
   are carried over to stage s+1.  In each stage the objective function,



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   i.e., the weighted sum of accumulated bitrate and distortion, is
   evaluated for each codebook vector entry and the results are sorted.
   Only the best paths and the corresponding quantization errors are
   considered in the next stage.  In the last stage S the single best
   path through the multistage codebook is determined.  By varying the
   maximum number of survivors from each stage to the next L, the
   complexity can be adjusted in real-time at the cost of a potential
   increase when evaluating the objective function for the resulting
   quantized vector.  This approach scales all the way between the two
   extremes, L=1 being a greedy search, and the desirable but infeasible
   full search, L=T/MS.  In fact, a performance almost as good as what
   can be achieved with the infeasible full search can be obtained at a
   substantially lower complexity by using this approach, see e.g.
   [leblanc-tsap].

5.2.7.5.  LSF Stabilization

   If the input is stable, finding the best candidate will usually
   result in the quantized vector also being stable, but due to the
   multi-stage approach it could in theory happen that the best
   quantization candidate is unstable and because of this there is a
   need to explicitly ensure that the quantized vectors are stable.
   Therefore we apply a LSF stabilization method which ensures that the
   LSF parameters are within valid range, increasingly sorted, and have
   minimum distances between each other and the border values that have
   been pre-determined as the 0.01 percentile distance values from a
   large training set.

5.2.7.6.  Off-Line Codebook Training

   The vectors and rate tables for the multi-stage codebook have been
   trained by minimizing the average of the objective function for LSF
   vectors from a large training set.

5.2.8.  LTP Quantization

   For voiced frames, the prediction analysis described in
   Section 5.2.6.1 resulted in four sets (one set per subframe) of five
   LTP coefficients, plus four weighting matrices.  Also, 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



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   measure for a codebook vector cb_i with rate r_i is give by


               RD = u * (b - cb_i)' * W_ltp * (b - cb_i) + r_i,


   where u is a fixed, heuristically-determined parameter balancing the
   distortion and rate.  Which codebook gives the best performance for a
   given LTP vector depends on the weighting matrix for that LTP vector.
   For example, for a low valued W_ltp, it is advantageous to use the
   codebook with 10 vectors as it has a lower average rate.  For a large
   W_ltp, on the other hand, it is often better to use the codebook with
   40 vectors, as it is more likely to contain the best codebook vector.
   The weighting matrix W_ltp depends mostly on two aspects of the input
   signal.  The first is the periodicity of the signal; the more
   periodic the larger W_ltp.  The second is the change in signal energy
   in the current subframe, relative to the signal one pitch lag
   earlier.  A decaying energy leads to a larger W_ltp than an
   increasing energy.  Both aspects do not fluctuate very fast which
   causes the W_ltp matrices for different subframes of one frame often
   to be similar.  As a result, one of the three codebooks typically
   gives good performance for all subframes.  Therefore the codebook
   search for the subframe LTP vectors is constrained to only allow
   codebook vectors to be chosen from the same codebook, resulting in a
   rate reduction.

   To find the best codebook, each of the three vector codebooks is used
   to quantize all subframe LTP vectors and produce a combined weighted
   rate-distortion measure for each vector codebook and 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.

5.2.9.  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
   the noise shaping analysis, and input to a scalar quantizer.  The
   quantization indices of the scalar quantizer represent a signal of



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   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.

5.2.10.  Range Encoder

   Range encoding is a well known method for entropy coding in which a
   bitstream sequence is continually updated with every new symbol,
   based on the probability for that symbol.  It is similar to
   arithmetic coding but rather than being restricted to generating
   binary output symbols, it can generate symbols in any chosen number
   base.  In SILK all side information is range encoded.  Each quantized
   parameter has its own cumulative density function based on histograms
   for the quantization indices obtained by running a training database.

5.2.10.1.  Bitstream Encoding Details

   TBD.

5.3.  CELT Encoder

   Copy from CELT draft.

5.3.1.  Pre-filter

   Inverse of the post-filter

5.3.2.  Forward MDCT

   The MDCT implementation has no special characteristics.  The input is
   a windowed signal (after pre-emphasis) of 2*N samples and the output
   is N frequency-domain samples.  A _low-overlap_ window is used to
   reduce the algorithmic delay.  It is derived from a basic (full
   overlap) window that is the same as the one used in the Vorbis codec:
   W(n)=[sin(pi/2*sin(pi/2*(n+.5)/L))]^2.  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 MDCT is computed in mdct_forward() (mdct.c),
   which includes the windowing operation and a scaling of 2/N.

5.3.3.  Bands and Normalization

   The MDCT output is divided into bands that are designed to match the
   ear's critical bands for the smallest (2.5ms) frame size.  The larger



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   frame sizes use integer multiplies of the 2.5ms 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 *non-
   quantized* 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.

5.3.4.  Energy Envelope Quantization

   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 coarse-fine strategy for encoding the
   energy in the base-2 log domain, as implemented in quant_bands.c

5.3.4.1.  Coarse energy quantization

   The coarse quantization of the energy uses a fixed resolution of 6
   dB.  To minimize the bitrate, prediction is applied both in time
   (using the previous frame) and in frequency (using the previous
   bands).  The prediction using the previous frame can be disabled,
   creating an "intra" frame where the energy is coded without reference
   to prior frames.  An encoder is able to choose the mode used at will
   based on both loss robustness and efficiency considerations.  The 2-D
   z-transform of the prediction filter is: A(z_l, z_b)=(1-a*z_l^-1)*(1-
   z_b^-1)/(1-b*z_b^-1) 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 a=0 b=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 to not suffer desynchronization.
   Identical prediction clamping must be implemented in all encoders and
   decoders.  We approximate the ideal probability distribution of the
   prediction error using a Laplace distribution with seperate
   parameters for each frame size in intra and inter-frame modes.  The
   coarse energy quantization is performed by quant_coarse_energy() and
   quant_coarse_energy() (quant_bands.c).  The encoding of the Laplace-
   distributed values is implemented in ec_laplace_encode() (laplace.c).

5.3.4.2.  Fine energy quantization

   After the coarse energy quantization and encoding, the bit allocation
   is computed (Section 4.3.3) and the number of bits to use for
   refining the energy quantization is determined for each band.  Let



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   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).

   If any bits are unused at the end of the encoding process, these bits
   are used to increase the resolution of the fine energy encoding in
   some bands.  Priority is given to the bands for which the allocation
   (Section 4.3.3) was rounded down.  At the same level of priority,
   lower bands are encoded first.  Refinement bits are added until there
   is no more room for fine energy or until each band has gained an
   additional bit of precision or has the maximum fine energy precision.
   This is implemented in quant_energy_finalise() (quant_bands.c).

5.3.5.  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 5.3.3 directly.  Given a PVQ codevector y, the unit vector X
   is obtained as X = y/||y||, where ||.|| denotes the L2 norm.

5.3.5.1.  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
   tradeoff between quality and complexity.  The method used in the
   reference implementation computes an initial codeword y1 by
   projecting the residual signal R = X - p' onto the codebook pyramid
   of K-1 pulses:

   y0 = round_towards_zero( (K-1) * R / sum(abs(R)))

   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
   R:

   J = -R^T*y / ||y||



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   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 implementors MAY use
   any other search methods.

5.3.5.2.  Index Encoding

   The best PVQ codeword is encoded as a uniformly-distributed integer
   value by encode_pulses() (cwrs.c).  The codeword is converted from a
   unique index in the same way as specified in [PVQ].  The indexing is
   based on the calculation of V(N,K) (denoted N(L,K) in [PVQ]), which
   is the number of possible combinations of K pulses in N samples.

5.3.6.  Stereo support

   When encoding a stereo stream, some parameters are shared across the
   left and right channels, while others are transmitted separately for
   each channel, or jointly encoded.  Only one copy of the flags for the
   features, transients and pitch (pitch period and filter parameters)
   are transmitted.  The coarse and fine energy parameters are
   transmitted separately for each channel.  Both the coarse energy and
   fine energy (including the remaining fine bits at the end of the
   stream) have the left and right bands interleaved in the stream, with
   the left band encoded first.

   The main difference between mono and stereo coding is the PVQ coding
   of the normalized vectors.  In stereo mode, a normalized mid-side
   (M-S) encoding is used.  Let L and R be the normalized vector of a
   certain band for the left and right channels, respectively.  The mid
   and side vectors are computed as M=L+R and S=L-R and no longer have
   unit norm.

   From M and S, an angular parameter theta=2/pi*atan2(||S||, ||M||) is
   computed.  The theta parameter is converted to a Q14 fixed-point
   parameter itheta, which is quantized on a scale from 0 to 1 with an
   interval of 2^-qb, where qb is based the number of bits allocated to
   the band.  From here on, the value of itheta MUST be treated in a
   bit-exact manner since both the encoder and decoder rely on it to
   infer the bit allocation.

   Let m=M/||M|| and s=S/||S||; m and s are separately encoded with the
   PVQ encoder described in Section 5.3.5.  The number of bits allocated
   to m and s depends on the value of itheta.

5.3.7.  Synthesis

   After all the quantization is completed, the quantized energy is used
   along with the quantized normalized band data to resynthesize the



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   MDCT spectrum.  The inverse MDCT (Section 4.3.7) and the weighted
   overlap-add are applied and the signal is stored in the _synthesis
   buffer_.  The encoder MAY omit this step of the processing if it does
   not need the decoded output.

5.3.8.  Variable Bitrate (VBR)

   Each CELT frame can be encoded in a different number of octets,
   making it possible to vary the bitrate at will.  This property can be
   used to implement source-controlled variable bitrate (VBR).  Support
   for VBR is OPTIONAL for the encoder, but a decoder MUST be prepared
   to decode a stream that changes its bitrate dynamically.  The method
   used to vary the bitrate in VBR mode is left to the implementor, as
   long as each frame can be decoded by the reference decoder.





































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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 in
   Appendix B compared to the reference implementation.

   To complement the Opus specification, the "Opus Custom" codec is
   defined to handle special sampling 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, 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 later is available, which is available using the celt_*
   function calls in celt.h.




























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7.  Security Considerations

   The codec needs 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 stream 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 such 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 an implementation avoids them.  For such
   architectures, it is RECOMMENDED that one add very small floating-
   point offsets to prevent significant numbers of denormalized
   operations or to configure the hardware to treat denormals as zero
   (DAZ).  No such issue exists for the fixed-point reference
   implementation.

























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8.  IANA Considerations

   This document has no actions for IANA.
















































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9.  Acknowledgments

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











































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10.  Informative References

   [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.

   [sinervo-norsig]
              Sinervo, U., Nurminen, J., Heikkinen, A., and J. Saarinen,
              "Evaluation of Split and Multistage Techniques in LSF
              Quantization", NORSIG-2001, Norsk symposium i
              signalbehandling, Trondheim, Norge, October 2001.

   [leblanc-tsap]
              LeBlanc, W., Bhattacharya, B., Mahmoud, S., and V.
              Cuperman, "Efficient Search and Design Procedures for
              Robust Multi-Stage VQ of LPC Parameters for 4 kb/s Speech
              Coding", IEEE Transactions on Speech and Audio Processing,
              Vol. 1, No. 4, October 1993.

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

   [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.

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



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              Engineering, Stanford University, May 1976.

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

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

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

   [3]  <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. This implementation
   can be compiled for either floating-point or fixed-point
   architectures.

   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 package,
   but it is easy to substitute any other FFT library.

A.1.  Extracting the source

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

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

   o  tar xzvf opus_source.tar.gz

   o  cd opus_source

   o  make

A.2.  Development Versions

   The current development version of the source code is available in a
   Git repository [3].  Development snapshots are provided at
   <http://opus-codec.org/>.

A.3.  Base64-encoded source code

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Valin, et al.            Expires January 8, 2012              [Page 138]


Internet-Draft           Interactive Audio Codec               July 2011


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Internet-Draft           Interactive Audio Codec               July 2011


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Internet-Draft           Interactive Audio Codec               July 2011


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Internet-Draft           Interactive Audio Codec               July 2011


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Valin, et al.            Expires January 8, 2012              [Page 206]


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Valin, et al.            Expires January 8, 2012              [Page 212]


Internet-Draft           Interactive Audio Codec               July 2011


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Appendix B.  opus_compare.c



   #include <stdio.h>
   #include <stdlib.h>
   #include <math.h>
   #include <string.h>

   #define OPUS_PI (3.14159265F)

   #define OPUS_MIN(_x,_y)      ((_x)<(_y)?(_x):(_y))
   #define OPUS_MAX(_x,_y)      ((_x)>(_y)?(_x):(_y))
   #define OPUS_CLAMP(_a,_b,_c) OPUS_MAX(_a,OPUS_MIN(_b,_c))
   #define OPUS_COSF(_x)        ((float)cos(_x))
   #define OPUS_SINF(_x)        ((float)sin(_x))
   #define OPUS_SQRTF(_x)       ((float)sqrt(_x))
   #define OPUS_LOG10F(_x)      ((float)log10(_x))

   static void *check_alloc(void *_ptr){
     if(_ptr==NULL){
       fprintf(stderr,"Out of memory.\n");
       exit(EXIT_FAILURE);
     }
     return _ptr;
   }

   static void *opus_malloc(size_t _size){
     return check_alloc(malloc(_size));
   }

   static void *opus_realloc(void *_ptr,size_t _size){
     return check_alloc(realloc(_ptr,_size));
   }

   static size_t read_pcm16(float **_samples,FILE *_fin,
                            int _nchannels){
     unsigned char  buf[1024];
     float         *samples;
     size_t         nsamples;
     size_t         csamples;
     size_t         xi;
     size_t         nread;
     samples=NULL;
     nsamples=csamples=0;
     for(;;){
       nread=fread(buf,2*_nchannels,1024/(2*_nchannels),_fin);
       if(nread<=0)break;



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       if(nsamples+nread>csamples){
         do csamples=csamples<<1|1;
         while(nsamples+nread>csamples);
         samples=(float *)opus_realloc(samples,
          _nchannels*csamples*sizeof(*samples));
       }
       for(xi=0;xi<nread;xi++){
         int ci;
         for(ci=0;ci<_nchannels;ci++){
           int s;
           s=buf[2*(xi*_nchannels+ci)+1]<<8|buf[2*(xi*_nchannels+ci)];
           s=((s&0xFFFF)^0x8000)-0x8000;
           samples[(nsamples+xi)*_nchannels+ci]=s;
         }
       }
       nsamples+=nread;
     }
     *_samples=(float *)opus_realloc(samples,
                        _nchannels*nsamples*sizeof(*samples));
     return nsamples;
   }

   static void band_energy(float *_out,const int *_bands,int _nbands,
    const float *_in,int _nchannels,size_t _nframes,int _window_sz,
    int _step){
     float *window;
     float *x;
     float *c;
     float *s;
     size_t xi;
     int    xj;
     window=(float *)opus_malloc((3+_nchannels)*_window_sz
             *sizeof(*window));
     c=window+_window_sz;
     s=c+_window_sz;
     x=s+_window_sz;
     for(xj=0;xj<_window_sz;xj++){
       window[xj]=0.5F-0.5F*OPUS_COSF((2*OPUS_PI/(_window_sz-1))*xj);
     }
     for(xj=0;xj<_window_sz;xj++)
         c[xj]=OPUS_COSF((2*OPUS_PI/_window_sz)*xj);
     for(xj=0;xj<_window_sz;xj++)
         s[xj]=OPUS_SINF((2*OPUS_PI/_window_sz)*xj);
     for(xi=0;xi<_nframes;xi++){
       int ci;
       int xk;
       int bi;
       for(ci=0;ci<_nchannels;ci++){



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         for(xk=0;xk<_window_sz;xk++){
           x[ci*_window_sz+xk]=window[xk]
                              *_in[(xi*_step+xk)*_nchannels+ci];
         }
       }
       for(bi=xj=0;bi<_nbands;bi++){
         float e2;
         e2=0;
         for(;xj<_bands[bi+1];xj++){
           float p;
           p=0;
           for(ci=0;ci<_nchannels;ci++){
             float re;
             float im;
             int   ti;
             ti=0;
             re=im=0;
             for(xk=0;xk<_window_sz;xk++){
               re+=c[ti]*x[ci*_window_sz+xk];
               im-=s[ti]*x[ci*_window_sz+xk];
               ti+=xj;
               if(ti>=_window_sz)ti-=_window_sz;
             }
             p+=OPUS_SQRTF(re*re+im*im);
           }
           p*=(1.0F/_nchannels);
           e2+=p*p;
         }
         _out[xi*_nbands+bi]=e2/(_bands[bi+1]-_bands[bi])+1;
       }
     }
     free(window);
   }

   static int cmp_float(const void *_a,const void *_b){
     float a;
     float b;
     a=*(const float *)_a;
     b=*(const float *)_b;
     return (a>b)-(a<b);
   }

   #define NBANDS (21)

   /*Bands on which we compute the pseudo-NMR (Bark-derived
     CELT bands).*/
   static const int BANDS[NBANDS+1]={
     0,2,4,6,8,10,12,14,16,20,24,28,32,40,48,56,68,80,96,120,156,200



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   };

   /*Per-band NMR threshold.*/
   static const float NMR_THRESH[NBANDS]={
   85113.804F,72443.596F,61659.5F,  52480.746F,44668.359F,38018.940F,
   32359.366F,27542.287F,23442.288F,19952.623F,16982.437F,14454.398F,
   12302.688F,10471.285F, 8912.5094F,7585.7758F,6456.5423F,5495.4087F,
    4677.3514F,3981.0717F,3388.4416F
   };

   /*Noise floor.*/
   static const float NOISE_FLOOR[NBANDS]={
   8.7096359F,7.5857758F,6.6069345F,5.7543994F,5.0118723F,4.3651583F,
   3.8018940F,3.3113112F,2.8840315F,2.5118864F,2.1877616F,1.9054607F,
   1.6595869F,1.4454398F,1.2589254F,1.0964782F,0.95499259F,0.83176377F,
   0.72443596F,0.63095734F,0.54954087F
   };

   #define TEST_WIN_SIZE (480)
   #define TEST_WIN_STEP (TEST_WIN_SIZE>>1)

   int main(int _argc,const char **_argv){
     FILE   *fin1;
     FILE   *fin2;
     float  *x;
     float  *y;
     float  *xb;
     float  *eb;
     float  *nmr;
     float   thresh;
     float   mismatch;
     float   err;
     float   nmr_sum;
     size_t  weight;
     size_t  xlength;
     size_t  ylength;
     size_t  nframes;
     size_t  xi;
     int     bi;
     int     nchannels;
     if(_argc<3||_argc>4){
       fprintf(stderr,"Usage: %s [-s] <file1.sw> <file2.sw>\n",
               _argv[0]);
       return EXIT_FAILURE;
     }
     nchannels=1;
     if(strcmp(_argv[1],"-s")==0)nchannels=2;
     fin1=fopen(_argv[nchannels],"rb");



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     if(fin1==NULL){
       fprintf(stderr,"Error opening '%s'.\n",_argv[nchannels]);
       return EXIT_FAILURE;
     }
     fin2=fopen(_argv[nchannels+1],"rb");
     if(fin2==NULL){
       fprintf(stderr,"Error opening '%s'.\n",_argv[nchannels+1]);
       fclose(fin1);
       return EXIT_FAILURE;
     }
     /*Read in the data and allocate scratch space.*/
     xlength=read_pcm16(&x,fin1,nchannels);
     fclose(fin1);
     ylength=read_pcm16(&y,fin2,nchannels);
     fclose(fin2);
     if(xlength!=ylength){
       fprintf(stderr,"Sample counts do not match (%lu!=%lu).\n",
        (unsigned long)xlength,(unsigned long)ylength);
       return EXIT_FAILURE;
     }
     if(xlength<TEST_WIN_SIZE){
       fprintf(stderr,"Insufficient sample data (%lu<%i).\n",
        (unsigned long)xlength,TEST_WIN_SIZE);
       return EXIT_FAILURE;
     }
     nframes=(xlength-TEST_WIN_SIZE+TEST_WIN_STEP)/TEST_WIN_STEP;
     xb=(float *)opus_malloc(nframes*NBANDS*sizeof(*xb));
     eb=(float *)opus_malloc(nframes*NBANDS*sizeof(*eb));
     nmr=(float *)opus_malloc(nframes*NBANDS*sizeof(*nmr));
     /*Compute the error signal.*/
     for(xi=0;xi<xlength*nchannels;xi++){
       err=x[xi]-y[xi];
       y[xi]=err-OPUS_CLAMP(-1,err,1);
     }
     /*Compute the per-band spectral energy of the original signal
       and the error.*/
     band_energy(xb,BANDS,NBANDS,x,nchannels,nframes,
             TEST_WIN_SIZE,TEST_WIN_STEP);
     free(x);
     band_energy(eb,BANDS,NBANDS,y,nchannels,nframes,
             TEST_WIN_SIZE,TEST_WIN_STEP);
     free(y);
     nmr_sum=0;
     for(xi=0;xi<nframes;xi++){
       /*Frequency masking (low to high): 10 dB/Bark slope.*/
       for(bi=1;bi<NBANDS;bi++)
           xb[xi*NBANDS+bi]+=0.1F*xb[xi*NBANDS+bi-1];
       /*Frequency masking (high to low): 15 dB/Bark slope.*/



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       for(bi=NBANDS-1;bi-->0;)
           xb[xi*NBANDS+bi]+=0.03F*xb[xi*NBANDS+bi+1];
       if(xi>0){
         /*Temporal masking: 5 dB/5ms slope.*/
         for(bi=0;bi<NBANDS;bi++)
             xb[xi*NBANDS+bi]+=0.3F*xb[(xi-1)*NBANDS+bi];
       }
       /*Compute NMR.*/
       for(bi=0;bi<NBANDS;bi++){
         nmr[xi*NBANDS+bi]=xb[xi*NBANDS+bi]/eb[xi*NBANDS+bi];
         nmr_sum+=10*OPUS_LOG10F(nmr[xi*NBANDS+bi]);
       }
     }
     /*Find the 90th percentile of the errors.*/
     memcpy(xb,eb,nframes*NBANDS*sizeof(*xb));
     qsort(xb,nframes*NBANDS,sizeof(*xb),cmp_float);
     thresh=xb[(9*nframes*NBANDS+5)/10];
     free(xb);
     /*Compute the mismatch.*/
     mismatch=0;
     weight=0;
     for(xi=0;xi<nframes;xi++){
       for(bi=0;bi<NBANDS;bi++){
         if(eb[xi*NBANDS+bi]>thresh){
           mismatch+=NMR_THRESH[bi]/nmr[xi*NBANDS+bi];
           weight++;
         }
       }
     }
     free(nmr);
     free(eb);
     printf("Average pseudo-NMR: %3.2f dB\n",nmr_sum/(nframes*NBANDS));
     if(weight<=0){
       err=-100;
       printf("Mismatch level: below noise floor\n");
     }
     else{
       err=10*OPUS_LOG10F(mismatch/weight);
       printf("Weighted mismatch: %3.2f dB\n",err);
     }
     printf("\n");
     if(err<0){
       printf("**Decoder PASSES test (mismatch < 0 dB)\n");
       return EXIT_SUCCESS;
     }
     printf("**Decoder FAILS test (mismatch >= 0 dB)\n");
     return EXIT_FAILURE;
   }



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Authors' Addresses

   Jean-Marc Valin
   Octasic Inc.
   4101, Molson Street
   Montreal, Quebec
   Canada

   Phone: +1 514 282-8858
   Email: jmvalin@jmvalin.ca


   Koen Vos
   Skype Technologies S.A.
   Stadsgarden 6
   Stockholm,   11645
   SE

   Phone: +46 855 921 989
   Email: koen.vos@skype.net


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

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





















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