Last updates for draft -11

[opus.git] / doc / draft-ietf-codec-opus.xml

<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE rfc SYSTEM 'rfc2629.dtd'>
<?rfc toc="yes" symrefs="yes" ?>

<rfc ipr="trust200902" category="std" docName="draft-ietf-codec-opus-11">

<front>
<title abbrev="Interactive Audio Codec">Definition of the Opus Audio Codec</title>


<author initials="JM" surname="Valin" fullname="Jean-Marc Valin">
<organization>Mozilla Corporation</organization>
<address>
<postal>
<street>650 Castro Street</street>
<city>Mountain View</city>
<region>CA</region>
<code>94041</code>
<country>USA</country>
</postal>
<phone>+1 650 903-0800</phone>
<email>jmvalin@jmvalin.ca</email>
</address>
</author>

<author initials="K." surname="Vos" fullname="Koen Vos">
<organization>Skype Technologies S.A.</organization>
<address>
<postal>
<street>Soder Malarstrand 43</street>
<city>Stockholm</city>
<region></region>
<code>11825</code>
<country>SE</country>
</postal>
<phone>+46 73 085 7619</phone>
<email>koen.vos@skype.net</email>
</address>
</author>

<author initials="T." surname="Terriberry" fullname="Timothy B. Terriberry">
<organization>Mozilla Corporation</organization>
<address>
<postal>
<street>650 Castro Street</street>
<city>Mountain View</city>
<region>CA</region>
<code>94041</code>
<country>USA</country>
</postal>
<phone>+1 650 903-0800</phone>
<email>tterriberry@mozilla.com</email>
</address>
</author>

<date day="17" month="February" year="2012" />

<area>General</area>

<workgroup></workgroup>

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

<middle>

<section anchor="introduction" title="Introduction">
<t>
The Opus codec is a real-time interactive audio codec designed to meet the requirements
described in <xref target="requirements"></xref>.
It is composed of a linear
 prediction (LP)-based layer and a Modified Discrete Cosine Transform
 (MDCT)-based layer.
The main idea behind using two layers is that in speech, linear prediction
 techniques (such as CELP) code low frequencies more efficiently than transform
 (e.g., MDCT) domain techniques, while the situation is reversed for music and
 higher speech frequencies.
Thus a codec with both layers available can operate over a wider range than
 either one alone and, by combining them, achieve better quality than either
 one individually.
</t>

<t>
The primary normative part of this specification is provided by the source code
 in <xref target="ref-implementation"></xref>.
Only the decoder portion of this software is normative, though a
 significant amount of code is shared by both the encoder and decoder.
<xref target="conformance"/> provides a decoder conformance test.
The decoder contains a great deal of integer and fixed-point arithmetic which
 must be performed exactly, including all rounding considerations, so any
 useful specification requires domain-specific symbolic language to adequately
 define these operations.
Additionally, any
conflict between the symbolic representation and the included reference
implementation must be resolved. For the practical reasons of compatibility and
testability it would be advantageous to give the reference implementation
priority in any disagreement. The C language is also one of the most
widely understood human-readable symbolic representations for machine
behavior.
For these reasons this RFC uses the reference implementation as the sole
 symbolic representation of the codec.
</t>

<t>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".
</t>

<section anchor="notation" title="Notation and Conventions">
<t>
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 <xref target="rfc2119"></xref>.
</t>
<t>
Even when using floating-point, various operations in the codec require
 bit-exact fixed-point behavior.
The notation "Q&lt;n&gt;", 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.
</t>
<t>
Expressions, where included in the text, follow C operator rules and
 precedence, with the exception that the syntax "x**y" indicates x raised to
 the power y.
The text also makes use of the following functions:
</t>

<section anchor="min" toc="exclude" title="min(x,y)">
<t>
The smallest of two values x and y.
</t>
</section>

<section anchor="max" toc="exclude" title="max(x,y)">
<t>
The largest of two values x and y.
</t>
</section>

<section anchor="clamp" toc="exclude" title="clamp(lo,x,hi)">
<figure align="center">
<artwork align="center"><![CDATA[
clamp(lo,x,hi) = max(lo,min(x,hi))
]]></artwork>
</figure>
<t>
With this definition, if lo&nbsp;&gt;&nbsp;hi, the lower bound is the one that
 is enforced.
</t>
</section>

<section anchor="sign" toc="exclude" title="sign(x)">
<t>
The sign of x, i.e.,
<figure align="center">
<artwork align="center"><![CDATA[
          ( -1,  x < 0 ,
sign(x) = <  0,  x == 0 ,
          (  1,  x > 0 .
]]></artwork>
</figure>
</t>
</section>

<section anchor="log2" toc="exclude" title="log2(f)">
<t>
The base-two logarithm of f.
</t>
</section>

<section anchor="ilog" toc="exclude" title="ilog(n)">
<t>
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.
<figure align="center">
<artwork align="center"><![CDATA[
          ( 0,                 n <= 0,
ilog(n) = <
          ( floor(log2(n))+1,  n > 0
]]></artwork>
</figure>
Examples:
<list style="symbols">
<t>ilog(-1) = 0</t>
<t>ilog(0) = 0</t>
<t>ilog(1) = 1</t>
<t>ilog(2) = 2</t>
<t>ilog(3) = 2</t>
<t>ilog(4) = 3</t>
<t>ilog(7) = 3</t>
</list>
</t>
</section>

</section>

</section>

<section anchor="overview" title="Opus Codec Overview">

<t>
The Opus codec scales from 6&nbsp;kb/s narrowband mono speech to 510&nbsp;kb/s
 fullband stereo music, with algorithmic delays ranging from 5&nbsp;ms to
 65.2&nbsp;ms.
At any given time, either the LP layer, the MDCT layer, or both, may be active.
It can seamlessly switch between all of its various operating modes, giving it
 a great deal of flexibility to adapt to varying content and network
 conditions without renegotiating the current session.
The codec allows input and output of various audio bandwidths, defined as
 follows:
</t>
<texttable anchor="audio-bandwidth">
<ttcol>Abbreviation</ttcol>
<ttcol align="right">Audio Bandwidth</ttcol>
<ttcol align="right">Sample Rate (Effective)</ttcol>
<c>NB (narrowband)</c>       <c>4&nbsp;kHz</c>  <c>8&nbsp;kHz</c>
<c>MB (medium-band)</c>      <c>6&nbsp;kHz</c> <c>12&nbsp;kHz</c>
<c>WB (wideband)</c>         <c>8&nbsp;kHz</c> <c>16&nbsp;kHz</c>
<c>SWB (super-wideband)</c> <c>12&nbsp;kHz</c> <c>24&nbsp;kHz</c>
<c>FB (fullband)</c>        <c>20&nbsp;kHz (*)</c> <c>48&nbsp;kHz</c>
</texttable>
<t>
(*) Although the sampling theorem allows a bandwidth as large as half the
 sampling rate, Opus never codes audio above 20&nbsp;kHz, as that is the
 generally accepted upper limit of human hearing.
</t>

<t>
Opus defines super-wideband (SWB) with an effective sample rate of 24&nbsp;kHz,
 unlike some other audio coding standards that use 32&nbsp;kHz.
This was chosen for a number of reasons.
The band layout in the MDCT layer naturally allows skipping coefficients for
 frequencies over 12&nbsp;kHz, but does not allow cleanly dropping just those
 frequencies over 16&nbsp;kHz.
A sample rate of 24&nbsp;kHz also makes resampling in the MDCT layer easier,
 as 24 evenly divides 48, and when 24&nbsp;kHz is sufficient, it can save
 computation in other processing, such as Acoustic Echo Cancellation (AEC).
Experimental changes to the band layout to allow a 16&nbsp;kHz cutoff
 (32&nbsp;kHz effective sample rate) showed potential quality degradations at
 other sample rates, and at typical bitrates the number of bits saved by using
 such a cutoff instead of coding in fullband (FB) mode is very small.
Therefore, if an application wishes to process a signal sampled at 32&nbsp;kHz,
 it should just use FB.
</t>

<t>
The LP layer is based on the
 <eref target='http://developer.skype.com/silk'>SILK</eref> codec
 <xref target="SILK"></xref>.
It supports NB, MB, or WB audio and frame sizes from 10&nbsp;ms to 60&nbsp;ms,
 and requires an additional 5&nbsp;ms look-ahead for noise shaping estimation.
A small additional delay (up to 1.5 ms) may be required for sampling rate
 conversion.
Like Vorbis and many other modern codecs, SILK is inherently designed for
 variable-bitrate (VBR) coding, though the encoder can also produce
 constant-bitrate (CBR) streams.
The version of SILK used in Opus is substantially modified from, and not
 compatible with, the stand-alone SILK codec previously deployed by Skype.
This document does not serve to define that format, but those interested in the
 original SILK codec should see <xref target="SILK"/> instead.
</t>

<t>
The MDCT layer is based on the
 <eref target='http://www.celt-codec.org/'>CELT</eref>  codec
 <xref target="CELT"></xref>.
It supports NB, WB, SWB, or FB audio and frame sizes from 2.5&nbsp;ms to
 20&nbsp;ms, and requires an additional 2.5&nbsp;ms look-ahead due to the
 overlapping MDCT windows.
The CELT codec is inherently designed for CBR coding, but unlike many CBR
 codecs it is not limited to a set of predetermined rates.
It internally allocates bits to exactly fill any given target budget, and an
 encoder can produce a VBR stream by varying the target on a per-frame basis.
The MDCT layer is not used for speech when the audio bandwidth is WB or less,
 as it is not useful there.
On the other hand, non-speech signals are not always adequately coded using
 linear prediction, so for music only the MDCT layer should be used.
</t>

<t>
A "Hybrid" mode allows the use of both layers simultaneously with a frame size
 of 10&nbsp;or 20&nbsp;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&nbsp;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&nbsp;kHz are discarded, so there is no
 coding redundancy between the two layers.
</t>

<t>
The sample rate (in contrast to the actual audio bandwidth) can be chosen
 independently on the encoder and decoder side, e.g., a fullband signal can be
 decoded as wideband, or vice versa.
This approach ensures a sender and receiver can always interoperate, regardless
 of the capabilities of their actual audio hardware.
Internally, the LP layer always operates at a sample rate of twice the audio
 bandwidth, up to a maximum of 16&nbsp;kHz, which it continues to use for SWB
 and FB.
The decoder simply resamples its output to support different sample rates.
The MDCT layer always operates internally at a sample rate of 48&nbsp;kHz.
Since all the supported sample rates evenly divide this rate, and since the
 the decoder may easily zero out the high frequency portion of the spectrum in
 the frequency domain, it can simply decimate the MDCT layer output to achieve
 the other supported sample rates very cheaply.
</t>

<t>
After conversion to the common, desired output sample rate, the decoder simply
 adds the output from the two layers together.
To compensate for the different look-ahead required by each layer, the CELT
 encoder input is delayed by an additional 2.7&nbsp;ms.
This ensures that low frequencies and high frequencies arrive at the same time.
This extra delay may be reduced by an encoder by using less look-ahead for noise
 shaping or using a simpler resampler in the LP layer, but this will reduce
 quality.
However, the base 2.5&nbsp;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.
</t>

<t>
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.
</t>

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

<section title="Bitrate" toc="exlcude">
<t>
Opus supports all bitrates from 6&nbsp;kb/s to 510&nbsp;kb/s. All other parameters being
equal, higher bitrate results in higher quality. For a frame size of 20&nbsp;ms, these
are the bitrate "sweet spots" for Opus in various configurations:
<list style="symbols">
<t>8-12 kb/s for NB speech,</t>
<t>16-20 kb/s for WB speech,</t>
<t>28-40 kb/s for FB speech,</t>
<t>48-64 kb/s for FB mono music, and</t>
<t>64-128 kb/s for FB stereo music.</t>
</list>
</t>
</section>

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

<section title="Audio Bandwidth" toc="exlcude">
<t>
The audio bandwidths supported by Opus are listed in
 <xref target="audio-bandwidth"/>.
Just like for the number of channels, any decoder can decode audio encoded at
 any bandwidth.
For example, any Opus decoder operating at 8&nbsp;kHz can decode a FB Opus
 frame, and any Opus decoder operating at 48&nbsp;kHz can decode a NB frame.
Similarly, the reference encoder can take a 48&nbsp;kHz input signal and
 encode it as NB.
The higher the audio bandwidth, the higher the required bitrate to achieve
 acceptable quality.
The audio bandwidth can be explicitly specified in real-time, but by default
 the reference encoder attempts to make the best bandwidth decision possible
 given the current bitrate.
</t>
</section>


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

<section title="Complexity" toc="exlcude">
<t>
There are various aspects of the Opus encoding process where trade-offs
can be made between CPU complexity and quality/bitrate. In the reference
encoder, the complexity is selected using an integer from 0 to 10, where
0 is the lowest complexity and 10 is the highest. Examples of
computations for which such trade-offs may occur are:
<list style="symbols">
<t>The order of the pitch analysis whitening filter,</t>
<t>The order of the short-term noise shaping filter,</t>
<t>The number of states in delayed decision quantization of the
residual signal, and</t>
<t>The use of certain bit-stream features such as variable time-frequency
resolution and the pitch post-filter.</t>
</list>
</t>
</section>

<section title="Packet Loss Resilience" toc="exlcude">
<t>
Audio codecs often exploit inter-frame correlations to reduce the
bitrate at a cost in error propagation: after losing one packet
several packets need to be received before the decoder is able to
accurately reconstruct the speech signal.  The extent to which Opus
exploits inter-frame dependencies can be adjusted on the fly to
choose a trade-off between bitrate and amount of error propagation.
</t>
</section>

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

<section title="Constant/Variable Bitrate" toc="exlcude">
<t>
Opus is more efficient when operating with variable bitrate (VBR), which is
the default. However, in some (rare) applications, constant bitrate (CBR)
is required. There are two main reasons to operate in CBR mode:
<list style="symbols">
<t>When the transport only supports a fixed size for each compressed frame</t>
<t>When security is important <spanx style="emph">and</spanx> the input audio
not a normal conversation but is highly constrained (e.g. yes/no, recorded prompts)
<xref target="SRTP-VBR"></xref> </t>
</list>

When low-latency transmission is required over a relatively slow connection, then
constrained VBR can also be used. This uses VBR in a way that simulates a
"bit reservoir" and is equivalent to what MP3 and AAC call CBR (i.e. not true
CBR due to the bit reservoir).
</t>
</section>

<section title="Discontinuous Transmission (DTX)" toc="exlcude">
<t>
   Discontinuous Transmission (DTX) reduces the bitrate during silence
   or background noise.  When DTX is enabled, only one frame is encoded
   every 400 milliseconds.
</t>
</section>

</section>

</section>

<section anchor="modes" title="Internal Framing">

<t>
The Opus encoder produces "packets", which are each a contiguous set of bytes
 meant to be transmitted as a single unit.
The packets described here do not include such things as IP, UDP, or RTP
 headers which are normally found in a transport-layer packet.
A single packet may contain multiple audio frames, so long as they share a
 common set of parameters, including the operating mode, audio bandwidth, frame
 size, and channel count (mono vs. stereo).
This section describes the possible combinations of these parameters and the
 internal framing used to pack multiple frames into a single packet.
This framing is not self-delimiting.
Instead, it assumes that a higher layer (such as UDP or RTP or Ogg or Matroska)
 will communicate the length, in bytes, of the packet, and it uses this
 information to reduce the framing overhead in the packet itself.
A decoder implementation MUST support the framing described in this section.
An alternative, self-delimiting variant of the framing is described in
 <xref target="self-delimiting-framing"/>.
Support for that variant is OPTIONAL.
</t>

<section anchor="toc_byte" title="The TOC Byte">
<t>
An Opus packet begins with a single-byte table-of-contents (TOC) header that
 signals which of the various modes and configurations a given packet uses.
It is composed of a frame count code, "c", a stereo flag, "s", and a
 configuration number, "config", arranged as illustrated in
 <xref target="toc_byte_fig"/>.
A description of each of these fields follows.
</t>

<figure anchor="toc_byte_fig" title="The TOC byte">
<artwork align="center"><![CDATA[
 0
 0 1 2 3 4 5 6 7
+-+-+-+-+-+-+-+-+
| c |s| config  |
+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
The top five bits of the TOC byte, labeled "config", encode one of 32 possible
 configurations of operating mode, audio bandwidth, and frame size.
As described, the LP layer and MDCT layer can be combined in three possible
 operating modes:
<list style="numbers">
<t>An LP-only mode for use in low bitrate connections with an audio bandwidth
 of WB or less,</t>
<t>A Hybrid (LP+MDCT) mode for SWB or FB speech at medium bitrates, and</t>
<t>An MDCT-only mode for very low delay speech transmission as well as music
 transmission (NB to FB).</t>
</list>
The 32 possible configurations each identify which one of these operating modes
 the packet uses, as well as the audio bandwidth and the frame size.
<xref target="config_bits"/> lists the parameters for each configuration.
</t>
<texttable anchor="config_bits" title="TOC Byte Configuration Parameters">
<ttcol>Configuration Number(s)</ttcol>
<ttcol>Mode</ttcol>
<ttcol>Bandwidth</ttcol>
<ttcol>Frame Sizes</ttcol>
<c>0...3</c>   <c>SILK-only</c> <c>NB</c>  <c>10, 20, 40, 60&nbsp;ms</c>
<c>4...7</c>   <c>SILK-only</c> <c>MB</c>  <c>10, 20, 40, 60&nbsp;ms</c>
<c>8...11</c>  <c>SILK-only</c> <c>WB</c>  <c>10, 20, 40, 60&nbsp;ms</c>
<c>12...13</c> <c>Hybrid</c>    <c>SWB</c> <c>10, 20&nbsp;ms</c>
<c>14...15</c> <c>Hybrid</c>    <c>FB</c>  <c>10, 20&nbsp;ms</c>
<c>16...19</c> <c>CELT-only</c> <c>NB</c>  <c>2.5, 5, 10, 20&nbsp;ms</c>
<c>20...23</c> <c>CELT-only</c> <c>WB</c>  <c>2.5, 5, 10, 20&nbsp;ms</c>
<c>24...27</c> <c>CELT-only</c> <c>SWB</c> <c>2.5, 5, 10, 20&nbsp;ms</c>
<c>28...31</c> <c>CELT-only</c> <c>FB</c>  <c>2.5, 5, 10, 20&nbsp;ms</c>
</texttable>
<t>
The configuration numbers in each range (e.g., 0...3 for NB SILK-only)
 correspond to the various choices of frame size, in the same order.
For example, configuration 0 has a 10&nbsp;ms frame size and configuration 3
 has a 60&nbsp;ms frame size.
</t>

<t>
One additional bit, labeled "s", signals mono vs. stereo, with 0 indicating
 mono and 1 indicating stereo.
</t>

<t>
The remaining two bits of the TOC byte, labeled "c", code the number of frames
 per packet (codes 0 to 3) as follows:
<list style="symbols">
<t>0:    1 frame in the packet</t>
<t>1:    2 frames in the packet, each with equal compressed size</t>
<t>2:    2 frames in the packet, with different compressed sizes</t>
<t>3:    an arbitrary number of frames in the packet</t>
</list>
This draft refers to a packet as a code 0 packet, code 1 packet, etc., based on
 the value of "c".
</t>

<t>
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.
</t>
</section>

<section title="Frame Packing">

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

<section anchor="frame-length-coding" title="Frame Length Coding">
<t>
When a packet contains multiple VBR frames (i.e., code 2 or 3), the compressed
 length of one or more of these frames is indicated with a one- or two-byte
 sequence, with the meaning of the first byte as follows:
<list style="symbols">
<t>0:          No frame (discontinuous transmission (DTX) or lost packet)</t>
<t>1...251:    Length of the frame in bytes</t>
<t>252...255:  A second byte is needed. The total length is (len[1]*4)+len[0]</t>
</list>
</t>

<t>
The special length 0 indicates that no frame is available, either because it
 was dropped during transmission by some intermediary or because the encoder
 chose not to transmit it.
A length of 0 is valid for any Opus frame in any mode.
</t>

<t>
The maximum representable length is 255*4+255=1275&nbsp;bytes.
For 20&nbsp;ms frames, this represents a bitrate of 510&nbsp;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.
</t>

<t>
No length is transmitted for the last frame in a VBR packet, or for any of the
 frames in a CBR packet, as it can be inferred from the total size of the
 packet and the size of all other data in the packet.
However, the length of any individual frame MUST NOT exceed 1275&nbsp;bytes, to
 allow for repacketization by gateways, conference bridges, or other software.
</t>
</section>

<section title="Code 0: One Frame in the Packet">

<t>
For code&nbsp;0 packets, the TOC byte is immediately followed by N-1&nbsp;bytes
 of compressed data for a single frame (where N is the size of the packet),
 as illustrated in <xref target="code0_packet"/>.
</t>
<figure anchor="code0_packet" title="A Code 0 Packet" align="center">
<artwork align="center"><![CDATA[
 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)...          :
:                                                               |
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
</section>

<section title="Code 1: Two Frames in the Packet, Each with Equal Compressed Size">
<t>
For code 1 packets, the TOC byte is immediately followed by the
 (N-1)/2&nbsp;bytes of compressed data for the first frame, followed by
 (N-1)/2&nbsp;bytes of compressed data for the second frame, as illustrated in
 <xref target="code1_packet"/>.
The number of payload bytes available for compressed data, N-1, MUST be even
 for all code 1 packets.
</t>
<figure anchor="code1_packet" title="A Code 1 Packet" align="center">
<artwork align="center"><![CDATA[
 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)...             |
:                                               +-+-+-+-+-+-+-+-+
|                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
</section>

<section title="Code 2: Two Frames in the Packet, with Different Compressed Sizes">
<t>
For code 2 packets, the TOC byte is followed by a one- or two-byte sequence
 indicating the length of the first frame (marked N1 in the figure below),
 followed by N1 bytes of compressed data for the first frame.
The remaining N-N1-2 or N-N1-3&nbsp;bytes are the compressed data for the
 second frame.
This is illustrated in <xref target="code2_packet"/>.
A code 2 packet MUST contain enough bytes to represent a valid length.
For example, a 1-byte code 2 packet is always invalid, and a 2-byte code 2
 packet whose second byte is in the range 252...255 is also invalid.
The length of the first frame, N1, MUST also be no larger than the size of the
 payload remaining after decoding that length for all code 2 packets.
This makes, for example, a 2-byte code 2 packet with a second byte in the range
 1...251 invalid as well (the only valid 2-byte code 2 packet is one where the
 length of both frames is zero).
</t>
<figure anchor="code2_packet" title="A Code 2 Packet" align="center">
<artwork align="center"><![CDATA[
 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...                     :
:                                                               |
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
</section>

<section title="Code 3: An Arbitrary Number of Frames in the Packet">
<t>
Code 3 packets may encode an arbitrary number of frames, as well as additional
 padding, called "Opus padding" to indicate that this padding is added at the
 Opus layer, rather than at the transport layer.
Code 3 packets MUST have at least 2 bytes.
The TOC byte is followed by a byte encoding the number of frames in the packet
 in bits 0 to 5 (marked "M" in the figure below), with bit 6 indicating whether
 or not Opus padding is inserted (marked "p" in the figure below), and bit 7
 indicating VBR (marked "v" in the figure below).
M MUST NOT be zero, and the audio duration contained within a packet MUST NOT
 exceed 120&nbsp;ms.
This limits the maximum frame count for any frame size to 48 (for 2.5&nbsp;ms
 frames), with lower limits for longer frame sizes.
<xref target="frame_count_byte"/> illustrates the layout of the frame count
 byte.
</t>
<figure anchor="frame_count_byte" title="The frame count byte">
<artwork align="center"><![CDATA[
 0
 0 1 2 3 4 5 6 7
+-+-+-+-+-+-+-+-+
|     M     |p|v|
+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
<t>
When Opus padding is used, the number of bytes of padding is encoded in the
 bytes following the frame count byte.
Values from 0...254 indicate that 0...254&nbsp;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&nbsp;bytes,
 plus the padding value encoded in the next byte.
There MUST be at least one more byte in the packet in this case.
By using the value 255 multiple times, it is possible to create a packet of any
 specific, desired size.
The additional padding bytes appear at the end of the packet, and MUST be set
 to zero by the encoder to avoid creating a covert channel.
The decoder MUST accept any value for the padding bytes, however.
Let P be the total amount of padding, including both the trailing padding bytes
 themselves and the header bytes used to indicate how many trailing bytes there
 are.
Then P MUST be no more than N-2.
</t>
<t>
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 a non-negative integer multiple of M.
The compressed data for all M frames then follows, each of size
 (N-2-P)/M&nbsp;bytes, as illustrated in <xref target="code3cbr_packet"/>.
</t>

<figure anchor="code3cbr_packet" title="A CBR Code 3 Packet" align="center">
<artwork align="center"><![CDATA[
 0                   1                   2                   3
 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|1|1|s| config  |     M     |p|0|  Padding length (Optional)    :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:            Compressed frame 1 ((N-2-P)/M bytes)...            :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:            Compressed frame 2 ((N-2-P)/M bytes)...            :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:                              ...                              :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:            Compressed frame M ((N-2-P)/M bytes)...            :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
:                  Opus Padding (Optional)...                   |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
In the VBR case, the (optional) padding length is followed by M-1 frame
 lengths (indicated by "N1" to "N[M-1]" in the figure below), each encoded in a
 one- or two-byte sequence as described above.
The packet MUST contain enough data for the M-1 lengths after removing the
 (optional) padding, and the sum of these lengths MUST be no larger than the
 number of bytes remaining in the packet after decoding them.
The compressed data for all M frames follows, each frame consisting of the
 indicated number of bytes, with the final frame consuming any remaining bytes
 before the final padding, as illustrated in <xref target="code3cbr_packet"/>.
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.
</t>

<figure anchor="code3vbr_packet" title="A VBR Code 3 Packet" align="center">
<artwork align="center"><![CDATA[
 0                   1                   2                   3
 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|1|1|s| config  |     M     |p|1| Padding length (Optional)     :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
: N1 (1-2 bytes): N2 (1-2 bytes):     ...       :     N[M-1]    |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:               Compressed frame 1 (N1 bytes)...                :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:               Compressed frame 2 (N2 bytes)...                :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:                              ...                              :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
:                     Compressed frame M...                     :
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
:                  Opus Padding (Optional)...                   |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
</section>
</section>

<section anchor="examples" title="Examples">
<t>
Simplest case, one NB mono 20&nbsp;ms SILK frame:
</t>

<figure>
<artwork><![CDATA[
 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...              :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
Two FB mono 5&nbsp;ms CELT frames of the same compressed size:
</t>

<figure>
<artwork><![CDATA[
 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...              :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
Two FB mono 20&nbsp;ms Hybrid frames of different compressed size:
</t>

<figure>
<artwork><![CDATA[
 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...                      :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
Four FB stereo 20&nbsp;ms CELT frames of the same compressed size:
</t>

<figure>
<artwork><![CDATA[
 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...       :
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>
</section>

<section title="Extending Opus">
<t>
A receiver MUST NOT process packets which violate any of the rules above as
 normal Opus packets.
They are reserved for future applications, such as in-band headers (containing
 metadata, etc.).
These constraints are summarized here for reference:
<list style="symbols">
<t>Packets are at least one byte.</t>
<t>No implicit frame length is larger than 1275 bytes.</t>
<t>Code 1 packets have an odd total length, N, so that (N-1)/2 is an
 integer.</t>
<t>Code 2 packets have enough bytes after the TOC for a valid frame length, and
 that length is no larger than the number of bytes remaining in the packet.</t>
<t>Code 3 packets contain at least one frame, but no more than 120&nbsp;ms of
 audio total.</t>
<t>The length of a CBR code 3 packet, N, is at least two bytes, the size of the
 padding, P (including both the padding length bytes in the header and the
 trailing padding bytes) is no more than N-2, and the frame count, M, satisfies
 the constraint that (N-2-P) is a non-negative integer multiple of M.</t>
<t>VBR code 3 packets are large enough to contain all the header bytes (TOC
 byte, frame count byte, any padding length bytes, and any frame length bytes),
 plus the length of the first M-1 frames, plus any trailing padding bytes.</t>
</list>
</t>
</section>

</section>

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

<section anchor="range-decoder" title="Range Decoder">
<t>
Opus uses an entropy coder based on <xref target="range-coding"></xref>,
which is itself a rediscovery of the FIFO arithmetic code introduced by <xref target="coding-thesis"></xref>.
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.
</t>
<t>
Symbols may also be coded as "raw bits" packed directly into the bitstream,
 bypassing the range coder.
These are packed backwards starting at the end of the frame, as illustrated in
 <xref target="rawbits-example"/>.
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.
</t>

<figure anchor="rawbits-example" title="Illustrative example of packing range
 coder and raw bits data">
<artwork align="center"><![CDATA[
               0               1               2               3
 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Range coder data (packed MSB to LSB) ->                       :
+                                                               +
:                                                               :
+     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
:     | <- Boundary occurs at an arbitrary bit position         :
+-+-+-+                                                         +
:                          <- Raw bits data (packed LSB to MSB) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
]]></artwork>
</figure>

<t>
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.
</t>
<t>
Suppose there is a context with n symbols, identified with an index that ranges
 from 0 to n-1.
The parameters needed to encode or decode symbol k in this context are
 represented by a three-tuple (fl[k],&nbsp;fh[k],&nbsp;ft), with
 0&nbsp;&lt;=&nbsp;fl[k]&nbsp;&lt;&nbsp;fh[k]&nbsp;&lt;=&nbsp;ft&nbsp;&lt;=&nbsp;65535.
The values of this tuple are derived from the probability model for the
 symbol, represented by traditional "frequency counts".
Because Opus uses static contexts these are not updated as symbols are decoded.
Let f[i] be the frequency of symbol i.
Then the three-tuple corresponding to symbol k is given by
</t>
<figure align="center">
<artwork align="center"><![CDATA[
        k-1                                   n-1
        __                                    __
fl[k] = \  f[i],  fh[k] = fl[k] + f[k],  ft = \  f[i]
        /_                                    /_
        i=0                                   i=0
]]></artwork>
</figure>
<t>
The range decoder extracts the symbols and integers encoded using the range
 encoder in <xref target="range-encoder"/>.
The range decoder maintains an internal state vector composed of the two-tuple
 (val,&nbsp;rng), representing the difference between the high end of the
 current range and the actual coded value, minus one, and the size of the
 current range, respectively.
Both val and rng are 32-bit unsigned integer values.
The decoder initializes rng to 128 and initializes val to 127 minus the top 7
 bits of the first input octet.
It saves the remaining bit for use in the renormalization procedure described
 in <xref target="range-decoder-renorm"/>, which the decoder invokes
 immediately after initialization to read additional bits and establish the
 invariant that rng&nbsp;&gt;&nbsp;2**23.
</t>

<section anchor="decoding-symbols" title="Decoding Symbols">
<t>
Decoding a symbol is a two-step process.
The first step determines a 16-bit unsigned value fs, which lies within the
 range of some symbol in the current context.
The second step updates the range decoder state with the three-tuple
 (fl[k],&nbsp;fh[k],&nbsp;ft) corresponding to that symbol.
</t>
<t>
The first step is implemented by ec_decode() (entdec.c), which computes
<figure align="center">
<artwork align="center"><![CDATA[
               val
fs = ft - min(------ + 1, ft) .
              rng/ft
]]></artwork>
</figure>
The divisions here are exact integer division.
</t>
<t>
The decoder then identifies the symbol in the current context corresponding to
 fs; i.e., the value of k whose three-tuple (fl[k],&nbsp;fh[k],&nbsp;ft)
 satisfies fl[k]&nbsp;&lt;=&nbsp;fs&nbsp;&lt;&nbsp;fh[k].
It uses this tuple to update val according to
<figure align="center">
<artwork align="center"><![CDATA[
            rng
val = val - --- * (ft - fh[k]) .
            ft
]]></artwork>
</figure>
If fl[k] is greater than zero, then the decoder updates rng using
<figure align="center">
<artwork align="center"><![CDATA[
      rng
rng = --- * (fh[k] - fl[k]) .
      ft
]]></artwork>
</figure>
Otherwise, it updates rng using
<figure align="center">
<artwork align="center"><![CDATA[
            rng
rng = rng - --- * (ft - fh[k]) .
            ft
]]></artwork>
</figure>
</t>
<t>
Using a special case for the first symbol (rather than the last symbol, as is
 commonly done in other arithmetic coders) ensures that all the truncation
 error from the finite precision arithmetic accumulates in symbol 0.
This makes the cost of coding a 0 slightly smaller, on average, than its
 estimated probability indicates and makes the cost of coding any other symbol
 slightly larger.
When contexts are designed so that 0 is the most probable symbol, which is
 often the case, this strategy minimizes the inefficiency introduced by the
 finite precision.
It also makes some of the special-case decoding routines in
 <xref target="decoding-alternate"/> particularly simple.
</t>
<t>
After the updates, implemented by ec_dec_update() (entdec.c), the decoder
 normalizes the range using the procedure in the next section, and returns the
 index k.
</t>

<section anchor="range-decoder-renorm" title="Renormalization">
<t>
To normalize the range, the decoder repeats the following process, implemented
 by ec_dec_normalize() (entdec.c), until rng&nbsp;&gt;&nbsp;2**23.
If rng is already greater than 2**23, the entire process is skipped.
First, it sets rng to (rng&lt;&lt;8).
Then it reads the next octet of the payload and combines it with the left-over
 bit buffered from the previous octet to form the 8-bit value sym.
It takes the left-over bit as the high bit (bit 7) of sym, and the top 7 bits
 of the octet it just read as the other 7 bits of sym.
The remaining bit in the octet just read is buffered for use in the next
 iteration.
If no more input octets remain, it uses zero bits instead.
Then, it sets
<figure align="center">
<artwork align="center"><![CDATA[
val = ((val<<8) + (255-sym)) & 0x7FFFFFFF .
]]></artwork>
</figure>
</t>
<t>
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 <xref target="finalize-example"/>.
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.
<xref target="encoder-finalizing"/> 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.
</t>

<figure anchor="finalize-example" title="Illustrative example of raw bits
 overlapping range coder data">
<artwork align="center"><![CDATA[
               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
      +-------------------------------------------------------...
]]></artwork>
</figure>
</section>
</section>

<section anchor="decoding-alternate" title="Alternate Decoding Methods">
<t>
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.
</t>
<section anchor="ec_decode_bin" title="ec_decode_bin()">
<t>
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&nbsp;=&nbsp;(1&lt;&lt;ftb), but avoids one of the divisions.
</t>
</section>
<section anchor="ec_dec_bit_logp" title="ec_dec_bit_logp()">
<t>
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&nbsp;=&nbsp;(1&lt;&lt;logp), followed by ec_dec_update() with
 the 3-tuple (fl[k]&nbsp;=&nbsp;0,
 fh[k]&nbsp;=&nbsp;(1&lt;&lt;logp)&nbsp;-&nbsp;1,
 ft&nbsp;=&nbsp;(1&lt;&lt;logp)) if the returned value
 of fs is less than (1&lt;&lt;logp)&nbsp;-&nbsp;1 (a "0" was decoded), and with
 (fl[k]&nbsp;=&nbsp;(1&lt;&lt;logp)&nbsp;-&nbsp;1,
 fh[k]&nbsp;=&nbsp;ft&nbsp;=&nbsp;(1&lt;&lt;logp)) otherwise (a "1" was
 decoded).
The implementation requires no multiplications or divisions.
</t>
</section>
<section anchor="ec_dec_icdf" title="ec_dec_icdf()">
<t>
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&lt;&lt;ftb) is equivalent to ft.
idcf[k], on the other hand, stores (1&lt;&lt;ftb)-fh[k], which is equal to
 (1&lt;&lt;ftb)&nbsp;-&nbsp;fl[k+1].
fl[0] is assumed to be 0, and the table is terminated by a value of 0 (where
 fh[k]&nbsp;==&nbsp;ft).
</t>
<t>
The function is mathematically equivalent to calling ec_decode() with
 ft&nbsp;=&nbsp;(1&lt;&lt;ftb), using the returned value fs to search the table
 for the first entry where fs&nbsp;&lt;&nbsp;(1&lt;&lt;ftb)-icdf[k], and
 calling ec_dec_update() with
 fl[k]&nbsp;=&nbsp;(1&lt;&lt;ftb)&nbsp;-&nbsp;icdf[k-1] (or 0
 if k&nbsp;==&nbsp;0), fh[k]&nbsp;=&nbsp;(1&lt;&lt;ftb)&nbsp;-&nbsp;idcf[k],
 and ft&nbsp;=&nbsp;(1&lt;&lt;ftb).
Combining the search with the update allows the division to be replaced by a
 series of multiplications (which are usually much cheaper), and using an
 inverse CDF allows the use of an ftb as large as 8 in an 8-bit table without
 any special cases.
This is the primary interface with the range decoder in the SILK layer, though
 it is used in a few places in the CELT layer as well.
</t>
<t>
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&nbsp;=&nbsp;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.
</t>
</section>
</section>

<section anchor="decoding-bits" title="Decoding Raw Bits">
<t>
The raw bits used by the CELT layer are packed at the end of the packet, with
 the least significant bit of the first value packed in the least significant
 bit of the last byte, filling up to the most significant bit in the last byte,
 continuing on to the least significant bit of the penultimate byte, and so on.
The reference implementation reads them using ec_dec_bits() (entdec.c).
Because the range decoder must read several bytes ahead in the stream, as
 described in <xref target="range-decoder-renorm"/>, 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.
</t>
</section>

<section anchor="ec_dec_uint" title="Decoding Uniformly Distributed Integers">
<t>
The function ec_dec_uint() (entdec.c) decodes one of ft equiprobable values in
 the range 0 to (ft&nbsp;-&nbsp;1), inclusive, each with a frequency of 1,
 where ft may be as large as (2**32&nbsp;-&nbsp;1).
Because ec_decode() is limited to a total frequency of (2**16&nbsp;-&nbsp;1),
 it splits up the value into a range coded symbol representing up to 8 of the
 high bits, and, if necessary, raw bits representing the remainder of the
 value.
The limit of 8 bits in the range coded symbol is a trade-off between
 implementation complexity, modeling error (since the symbols no longer truly
 have equal coding cost), and rounding error introduced by the range coder
 itself (which gets larger as more bits are included).
Using raw bits reduces the maximum number of divisions required in the worst
 case, but means that it may be possible to decode a value outside the range
 0 to (ft&nbsp;-&nbsp;1), inclusive.
</t>

<t>
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&nbsp;-&nbsp;1), inclusive.
Let ftb&nbsp;=&nbsp;ilog(ft&nbsp;-&nbsp;1), i.e., the number of bits required
 to store (ft&nbsp;-&nbsp;1) in two's complement notation.
If ftb is 8 or less, then t is decoded with t&nbsp;=&nbsp;ec_decode(ft), and
 the range coder state is updated using the three-tuple (t, t&nbsp;+&nbsp;1,
 ft).
</t>
<t>
If ftb is greater than 8, then the top 8 bits of t are decoded using
<figure align="center">
<artwork align="center"><![CDATA[
t = ec_decode(((ft - 1) >> (ftb - 8)) + 1) ,
]]></artwork>
</figure>
 the decoder state is updated using the three-tuple
 (t, t&nbsp;+&nbsp;1,
 ((ft&nbsp;-&nbsp;1)&nbsp;&gt;&gt;&nbsp;(ftb&nbsp;-&nbsp;8))&nbsp;+&nbsp;1),
 and the remaining bits are decoded as raw bits, setting
<figure align="center">
<artwork align="center"><![CDATA[
t = (t << (ftb - 8)) | ec_dec_bits(ftb - 8) .
]]></artwork>
</figure>
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 the error has occurred.
</t>

</section>

<section anchor="decoder-tell" title="Current Bit Usage">
<t>
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.
</t>
<t>
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().
</t>
<t>
However, this error is bounded, and periodic calls to ec_tell() or
 ec_tell_frac() at precisely defined points in the decoding process prevent it
 from accumulating.
For a range coder symbol that requires a whole number of bits (i.e.,
 for which ft/(fh[k]&nbsp;-&nbsp;fl[k]) is a power of two), where there are at
 least p 1/8th bits available, decoding the symbol will never cause ec_tell() or
 ec_tell_frac() to exceed the size of the frame ("bust the budget").
In this case the return value of ec_tell_frac() will only advance by more than
 p 1/8th bits if there was an additional, fractional number of bits remaining,
 and it will never advance beyond the next whole-bit boundary, which is safe,
 since frames always contain a whole number of bits.
However, when p is not a whole number of bits, an extra 1/8th bit is required
 to ensure that decoding the symbol will not bust the budget.
</t>
<t>
The reference implementation keeps track of the total number of whole bits that
 have been processed by the decoder so far in the variable nbits_total,
 including the (possibly fractional) number of bits that are currently
 buffered, but not consumed, inside the range coder.
nbits_total is initialized to 9 just before the initial range renormalization
 process completes (or equivalently, it can be initialized to 33 after the
 first renormalization).
The extra two bits over the actual amount buffered by the range coder
 guarantees that it is an upper bound and that there is enough room for the
 encoder to terminate the stream.
Each iteration through the range coder's renormalization loop increases
 nbits_total by 8.
Reading raw bits increases nbits_total by the number of raw bits read.
</t>

<section anchor="ec_tell" title="ec_tell()">
<t>
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).
</t>
<t>
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.
</t>
</section>

<section anchor="ec_tell_frac" title="ec_tell_frac()">
<t>
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
<figure align="center">
<artwork align="center">
<![CDATA[
r_Q15 = rng >> (l-16) ,
]]></artwork>
</figure>
 so that 32768 &lt;= r_Q15 &lt; 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
<figure align="center">
<artwork align="center">
<![CDATA[
r_Q15 = (r_Q15*r_Q15) >> 15 .
]]></artwork>
</figure>
Then add the 16th bit of r_Q15 to l via
<figure align="center">
<artwork align="center">
<![CDATA[
l = 2*l + (r_Q15 >> 16) .
]]></artwork>
</figure>
Finally, if this bit was a 1, reduce r_Q15 by a factor of two via
<figure align="center">
<artwork align="center">
<![CDATA[
r_Q15 = r_Q15 >> 1 ,
]]></artwork>
</figure>
 so that it once again lies in the range 32768 &lt;= r_Q15 &lt; 65536.
</t>
<t>
This procedure is repeated three times to extend l to 1/8th bit precision.
ec_tell_frac() then returns (nbits_total*8 - l).
</t>
</section>

</section>

</section>

<section anchor="silk_decoder_outline" title="SILK Decoder">
<t>
The decoder's LP layer uses a modified version of the SILK codec (herein simply
 called "SILK"), which runs a decoded excitation signal through adaptive
 long-term and short-term prediction synthesis filters.
It runs at NB, MB, and WB sample rates internally.
When used in a SWB or FB Hybrid frame, the LP layer itself still only runs in
 WB.
</t>

<section title="SILK Decoder Modules">
<t>
An overview of the decoder is given in <xref target="silk_decoder_figure"/>.
</t>
<figure align="center" anchor="silk_decoder_figure" title="SILK Decoder">
<artwork align="center">
<![CDATA[
   +---------+    +------------+
-->| Range   |--->| Decode     |---------------------------+
 1 | Decoder | 2  | Parameters |----------+       5        |
   +---------+    +------------+     4    |                |
                       3 |                |                |
                        \/               \/               \/
                  +------------+   +------------+   +------------+
                  | Generate   |-->| LTP        |-->| LPC        |
                  | Excitation |   | Synthesis  |   | Synthesis  |
                  +------------+   +------------+   +------------+
                                          ^                |
                                          |                |
                      +-------------------+----------------+
                      |                                      6
                      |   +------------+   +-------------+
                      +-->| Stereo     |-->| Sample Rate |-->
                          | Unmixing   | 7 | Conversion  | 8
                          +------------+   +-------------+

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

<t>
The decoder feeds the bitstream (1) to the range decoder from
 <xref target="range-decoder"/>, and then decodes the parameters in it (2)
 using the procedures detailed in
 Sections&nbsp;<xref format="counter" target="silk_header_bits"/>
 through&nbsp;<xref format="counter" target="silk_signs"/>.
These parameters (3, 4, 5) are used to generate an excitation signal (see
 <xref target="silk_excitation_reconstruction"/>), which is fed to an optional
 long-term prediction (LTP) filter (voiced frames only, see
 <xref target="silk_ltp_synthesis"/>) and then a short-term prediction filter
 (see <xref target="silk_lpc_synthesis"/>), producing the decoded signal (6).
For stereo streams, the mid-side representation is converted to separate left
 and right channels (7).
The result is finally resampled to the desired output sample rate (e.g.,
 48&nbsp;kHz) so that the resampled signal (8) can be mixed with the CELT
 layer.
</t>

</section>

<section anchor="silk_layer_organization" title="LP Layer Organization">

<t>
Internally, the LP layer of a single Opus frame is composed of either a single
 10&nbsp;ms regular SILK frame or between one and three 20&nbsp;ms regular SILK
 frames.
A stereo Opus frame may double the number of regular SILK frames (up to a total
 of six), since it includes separate frames for a mid channel and, optionally,
 a side channel.
Optional Low Bit-Rate Redundancy (LBRR) frames, which are reduced-bitrate
 encodings of previous SILK frames, may be included to aid in recovery from
 packet loss.
If present, these appear before the regular SILK frames.
They are in most respects identical to regular, active SILK frames, except that
 they are usually encoded with a lower bitrate.
This draft uses "SILK frame" to refer to either one and "regular SILK frame" if
 it needs to draw a distinction between the two.
</t>
<t>
Logically, each SILK frame is in turn composed of either two or four 5&nbsp;ms
 subframes.
Various parameters, such as the quantization gain of the excitation and the
 pitch lag and filter coefficients can vary on a subframe-by-subframe basis.
Physically, the parameters for each subframe are interleaved in the bitstream,
 as described in the relevant sections for each parameter.
</t>
<t>
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).
</t>

<t>
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&nbsp;ms frames.
</t>

<t>
<xref target="silk_symbols"/> summarizes the overall grouping of the contents of
 the LP layer.
Figures&nbsp;<xref format="counter" target="silk_mono_60ms_frame"/>
 and&nbsp;<xref format="counter" target="silk_stereo_60ms_frame"/> illustrate
 the ordering of the various SILK frames for a 60&nbsp;ms Opus frame, for both
 mono and stereo, respectively.
</t>

<texttable anchor="silk_symbols"
 title="Organization of the SILK layer of an Opus frame">
<ttcol align="center">Symbol(s)</ttcol>
<ttcol align="center">PDF(s)</ttcol>
<ttcol align="center">Condition</ttcol>

<c>VAD flags</c>
<c>{1, 1}/2</c>
<c/>

<c>LBRR flag</c>
<c>{1, 1}/2</c>
<c/>

<c>Per-frame LBRR flags</c>
<c><xref target="silk_lbrr_flag_pdfs"/></c>
<c><xref target="silk_lbrr_flags"/></c>

<c>LBRR Frame(s)</c>
<c><xref target="silk_frame"/></c>
<c><xref target="silk_lbrr_flags"/></c>

<c>Regular SILK Frame(s)</c>
<c><xref target="silk_frame"/></c>
<c/>

</texttable>

<figure align="center" anchor="silk_mono_60ms_frame"
 title="A 60&nbsp;ms Mono Frame">
<artwork align="center"><![CDATA[
+---------------------------------+
|            VAD Flags            |
+---------------------------------+
|            LBRR Flag            |
+---------------------------------+
| Per-Frame LBRR Flags (Optional) |
+---------------------------------+
|     LBRR Frame 1 (Optional)     |
+---------------------------------+
|     LBRR Frame 2 (Optional)     |
+---------------------------------+
|     LBRR Frame 3 (Optional)     |
+---------------------------------+
|      Regular SILK Frame 1       |
+---------------------------------+
|      Regular SILK Frame 2       |
+---------------------------------+
|      Regular SILK Frame 3       |
+---------------------------------+
]]></artwork>
</figure>

<figure align="center" anchor="silk_stereo_60ms_frame"
 title="A 60&nbsp;ms Stereo Frame">
<artwork align="center"><![CDATA[
+---------------------------------------+
|             Mid VAD Flags             |
+---------------------------------------+
|             Mid LBRR Flag             |
+---------------------------------------+
|             Side VAD Flags            |
+---------------------------------------+
|             Side LBRR Flag            |
+---------------------------------------+
|  Mid Per-Frame LBRR Flags (Optional)  |
+---------------------------------------+
| Side Per-Frame LBRR Flags (Optional)  |
+---------------------------------------+
|     Mid LBRR Frame 1 (Optional)       |
+---------------------------------------+
|     Side LBRR Frame 1 (Optional)      |
+---------------------------------------+
|     Mid LBRR Frame 2 (Optional)       |
+---------------------------------------+
|     Side LBRR Frame 2 (Optional)      |
+---------------------------------------+
|     Mid LBRR Frame 3 (Optional)       |
+---------------------------------------+
|     Side LBRR Frame 3 (Optional)      |
+---------------------------------------+
|      Mid Regular SILK Frame 1         |
+---------------------------------------+
| Side Regular SILK Frame 1 (Optional)  |
+---------------------------------------+
|      Mid Regular SILK Frame 2         |
+---------------------------------------+
| Side Regular SILK Frame 2 (Optional)  |
+---------------------------------------+
|      Mid Regular SILK Frame 3         |
+---------------------------------------+
| Side Regular SILK Frame 3 (Optional)  |
+---------------------------------------+
]]></artwork>
</figure>

</section>

<section anchor="silk_header_bits" title="Header Bits">
<t>
The LP layer begins with two to eight header bits, decoded in silk_Decode()
 (dec_API.c).
These consist of one Voice Activity Detection (VAD) bit per frame (up to 3),
 followed by a single flag indicating the presence of LBRR frames.
For a stereo packet, these first flags correspond to the mid channel, and a
 second set of flags is included for the side channel.
</t>
<t>
Because these are the first symbols decoded by the range coder and because they
 are coded as binary values with uniform probability, they can be extracted
 directly from the most significant bits of the first byte of compressed data.
Thus, a receiver can determine if an Opus frame contains any active SILK frames
 without the overhead of using the range decoder.
</t>
</section>

<section anchor="silk_lbrr_flags" title="Per-Frame LBRR Flags">
<t>
For Opus frames longer than 20&nbsp;ms, a set of LBRR flags is
 decoded for each channel that has its LBRR flag set.
Each set contains one flag per 20&nbsp;ms SILK frame.
40&nbsp;ms Opus frames use the 2-frame LBRR flag PDF from
 <xref target="silk_lbrr_flag_pdfs"/>, and 60&nbsp;ms Opus frames use the
 3-frame LBRR flag PDF.
For each channel, the resulting 2- or 3-bit integer contains the corresponding
 LBRR flag for each frame, packed in order from the LSB to the MSB.
</t>

<texttable anchor="silk_lbrr_flag_pdfs" title="LBRR Flag PDFs">
<ttcol>Frame Size</ttcol>
<ttcol>PDF</ttcol>
<c>40&nbsp;ms</c> <c>{0, 53, 53, 150}/256</c>
<c>60&nbsp;ms</c> <c>{0, 41, 20, 29, 41, 15, 28, 82}/256</c>
</texttable>

<t>
A 10&nbsp;or 20&nbsp;ms Opus frame does not contain any per-frame LBRR flags,
 as there may be at most one LBRR frame per channel.
The global LBRR flag in the header bits (see <xref target="silk_header_bits"/>)
 is already sufficient to indicate the presence of that single LBRR frame.
</t>

</section>

<section anchor="silk_lbrr_frames" title="LBRR Frames">
<t>
The LBRR frames, if present, contain an encoded representation of the signal
 immediately prior to the current Opus frame as if it were encoded with the
 current mode, frame size, audio bandwidth, and channel count, even if those
 differ from the prior Opus frame.
When one of these parameters changes from one Opus frame to the next, this
 implies that the LBRR frames of the current Opus frame may not be simple
 drop-in replacements for the contents of the previous Opus frame.
</t>

<t>
For example, when switching from 20&nbsp;ms to 60&nbsp;ms, the 60&nbsp;ms Opus
 frame may contain LBRR frames covering up to three prior 20&nbsp;ms Opus
 frames, even if those frames already contained LBRR frames covering some of
 the same time periods.
When switching from 20&nbsp;ms to 10&nbsp;ms, the 10&nbsp;ms Opus frame can
 contain an LBRR frame covering at most half the prior 20&nbsp;ms Opus frame,
 potentially leaving a hole that needs to be concealed from even a single
 packet loss.
When switching from mono to stereo, the LBRR frames in the first stereo Opus
 frame MAY contain a non-trivial side channel.
</t>

<t>
In order to properly produce LBRR frames under all conditions, an encoder might
 need to buffer up to 60&nbsp;ms of audio and re-encode it during these
 transitions.
However, the reference implementation opts to disable LBRR frames at the
 transition point for simplicity.
</t>

<t>
The LBRR frames immediately follow the LBRR flags, prior to any regular SILK
 frames.
<xref target="silk_frame"/> describes their exact contents.
LBRR frames do not include their own separate VAD flags.
LBRR frames are only meant to be transmitted for active speech, thus all LBRR
 frames are treated as active.
</t>

<t>
In a stereo Opus frame longer than 20&nbsp;ms, although the per-frame LBRR
 flags for the mid channel are coded as a unit before the per-frame LBRR flags
 for the side channel, the LBRR frames themselves are interleaved.
The decoder parses an LBRR frame for the mid channel of a given 20&nbsp;ms
 interval (if present) and then immediately parses the corresponding LBRR
 frame for the side channel (if present), before proceeding to the next
 20&nbsp;ms interval.
</t>
</section>

<section anchor="silk_regular_frames" title="Regular SILK Frames">
<t>
The regular SILK frame(s) follow the LBRR frames (if any).
<xref target="silk_frame"/> describes their contents, as well.
Unlike the LBRR frames, a regular SILK frame is coded for each time interval in
 an Opus frame, even if the corresponding VAD flags are unset.
For stereo Opus frames longer than 20&nbsp;ms, the regular mid and side SILK
 frames for each 20&nbsp;ms interval are interleaved, just as with the LBRR
 frames.
The side frame may be skipped by coding an appropriate flag, as detailed in
 <xref target="silk_mid_only_flag"/>.
</t>
</section>

<section anchor="silk_frame" title="SILK Frame Contents">
<t>
Each SILK frame includes a set of side information that encodes
<list style="symbols">
<t>The frame type and quantization type (<xref target="silk_frame_type"/>),</t>
<t>Quantization gains (<xref target="silk_gains"/>),</t>
<t>Short-term prediction filter coefficients (<xref target="silk_nlsfs"/>),</t>
<t>An LSF interpolation weight (<xref target="silk_nlsf_interpolation"/>),</t>
<t>
Long-term prediction filter lags and gains (<xref target="silk_ltp_params"/>),
 and
</t>
<t>A linear congruential generator (LCG) seed (<xref target="silk_seed"/>).</t>
</list>
The quantized excitation signal (see <xref target="silk_excitation"/>) follows
 these at the end of the frame.
<xref target="silk_frame_symbols"/> details the overall organization of a
 SILK frame.
</t>

<texttable anchor="silk_frame_symbols"
 title="Order of the symbols in an individual SILK frame">
<ttcol align="center">Symbol(s)</ttcol>
<ttcol align="center">PDF(s)</ttcol>
<ttcol align="center">Condition</ttcol>

<c>Stereo Prediction Weights</c>
<c><xref target="silk_stereo_pred_pdfs"/></c>
<c><xref target="silk_stereo_pred"/></c>

<c>Mid-only Flag</c>
<c><xref target="silk_mid_only_pdf"/></c>
<c><xref target="silk_mid_only_flag"/></c>

<c>Frame Type</c>
<c><xref target="silk_frame_type"/></c>
<c/>

<c>Subframe Gains</c>
<c><xref target="silk_gains"/></c>
<c/>

<c>Normalized LSF Stage 1 Index</c>
<c><xref target="silk_nlsf_stage1_pdfs"/></c>
<c/>

<c>Normalized LSF Stage 2 Residual</c>
<c><xref target="silk_nlsf_stage2"/></c>
<c/>

<c>Normalized LSF Interpolation Weight</c>
<c><xref target="silk_nlsf_interp_pdf"/></c>
<c>20&nbsp;ms frame</c>

<c>Primary Pitch Lag</c>
<c><xref target="silk_ltp_lags"/></c>
<c>Voiced frame</c>

<c>Subframe Pitch Contour</c>
<c><xref target="silk_pitch_contour_pdfs"/></c>
<c>Voiced frame</c>

<c>Periodicity Index</c>
<c><xref target="silk_perindex_pdf"/></c>
<c>Voiced frame</c>

<c>LTP Filter</c>
<c><xref target="silk_ltp_filter_pdfs"/></c>
<c>Voiced frame</c>

<c>LTP Scaling</c>
<c><xref target="silk_ltp_scaling_pdf"/></c>
<c><xref target="silk_ltp_scaling"/></c>

<c>LCG Seed</c>
<c><xref target="silk_seed_pdf"/></c>
<c/>

<c>Excitation Rate Level</c>
<c><xref target="silk_rate_level_pdfs"/></c>
<c/>

<c>Excitation Pulse Counts</c>
<c><xref target="silk_pulse_count_pdfs"/></c>
<c/>

<c>Excitation Pulse Locations</c>
<c><xref target="silk_pulse_locations"/></c>
<c>Non-zero pulse count</c>

<c>Excitation LSBs</c>
<c><xref target="silk_shell_lsb_pdf"/></c>
<c><xref target="silk_pulse_counts"/></c>

<c>Excitation Signs</c>
<c><xref target="silk_sign_pdfs"/></c>
<c/>

</texttable>

<section anchor="silk_stereo_pred" toc="include"
 title="Stereo Prediction Weights">
<t>
A SILK frame corresponding to the mid channel of a stereo Opus frame begins
 with a pair of side channel prediction weights, designed such that zeros
 indicate normal mid-side coupling.
Since these weights can change on every frame, the first portion of each frame
 linearly interpolates between the previous weights and the current ones, using
 zeros for the previous weights if none are available.
These prediction weights are never included in a mono Opus frame, and the
 previous weights are reset to zeros on any transition from mono to stereo.
They are also not included in an LBRR frame for the side channel, even if the
 LBRR flags indicate the corresponding mid channel was not coded.
In that case, the previous weights are used, again substituting in zeros if no
 previous weights are available since the last decoder reset
 (see <xref target="decoder-reset"/>).
</t>

<t>
To summarize, these weights are coded if and only if
<list style="symbols">
<t>This is a stereo Opus frame (<xref target="toc_byte"/>), and</t>
<t>The current SILK frame corresponds to the mid channel.</t>
</list>
</t>

<t>
The prediction weights are coded in three separate pieces, which are decoded
 by silk_stereo_decode_pred() (decode_stereo_pred.c).
The first piece jointly codes the high-order part of a table index for both
 weights.
The second piece codes the low-order part of each table index.
The third piece codes an offset used to linearly interpolate between table
 indices.
The details are as follows.
</t>

<t>
Let n be an index decoded with the 25-element stage-1 PDF in
 <xref target="silk_stereo_pred_pdfs"/>.
Then let i0 and i1 be indices decoded with the stage-2 and stage-3 PDFs in
 <xref target="silk_stereo_pred_pdfs"/>, respectively, and let i2 and i3
 be two more indices decoded with the stage-2 and stage-3 PDFs, all in that
 order.
</t>

<texttable anchor="silk_stereo_pred_pdfs" title="Stereo Weight PDFs">
<ttcol align="left">Stage</ttcol>
<ttcol align="left">PDF</ttcol>
<c>Stage 1</c>
<c>{7,  2,  1,  1,  1,
   10, 24,  8,  1,  1,
    3, 23, 92, 23,  3,
    1,  1,  8, 24, 10,
    1,  1,  1,  2,  7}/256</c>

<c>Stage 2</c>
<c>{85, 86, 85}/256</c>

<c>Stage 3</c>
<c>{51, 51, 52, 51, 51}/256</c>
</texttable>

<t>
Then use n, i0, and i2 to form two table indices, wi0 and wi1, according to
<figure align="center">
<artwork align="center"><![CDATA[
wi0 = i0 + 3*(n/5)
wi1 = i2 + 3*(n%5)
]]></artwork>
</figure>
 where the division is exact integer division.
The range of these indices is 0 to 14, inclusive.
Let w[i] be the i'th weight from <xref target="silk_stereo_weights_table"/>.
Then the two prediction weights, w0_Q13 and w1_Q13, are
<figure align="center">
<artwork align="center"><![CDATA[
w1_Q13 = w_Q13[wi1]
         + ((w_Q13[wi1+1] - w_Q13[wi1])*6554) >> 16)*(2*i3 + 1)

w0_Q13 = w_Q13[wi0]
         + ((w_Q13[wi0+1] - w_Q13[wi0])*6554) >> 16)*(2*i1 + 1)
         - w1_Q13
]]></artwork>
</figure>
N.b., w1_Q13 is computed first here, because w0_Q13 depends on it.
</t>

<texttable anchor="silk_stereo_weights_table"
 title="Stereo Weight Table">
<ttcol align="left">Index</ttcol>
<ttcol align="right">Weight (Q13)</ttcol>
 <c>0</c> <c>-13732</c>
 <c>1</c> <c>-10050</c>
 <c>2</c>  <c>-8266</c>
 <c>3</c>  <c>-7526</c>
 <c>4</c>  <c>-6500</c>
 <c>5</c>  <c>-5000</c>
 <c>6</c>  <c>-2950</c>
 <c>7</c>   <c>-820</c>
 <c>8</c>    <c>820</c>
 <c>9</c>   <c>2950</c>
<c>10</c>   <c>5000</c>
<c>11</c>   <c>6500</c>
<c>12</c>   <c>7526</c>
<c>13</c>   <c>8266</c>
<c>14</c>  <c>10050</c>
<c>15</c>  <c>13732</c>
</texttable>

</section>

<section anchor="silk_mid_only_flag" toc="include" title="Mid-only Flag">
<t>
A flag appears after the stereo prediction weights that indicates if only the
 mid channel is coded for this time interval.
It appears only when
<list style="symbols">
<t>This is a stereo Opus frame (see <xref target="toc_byte"/>),</t>
<t>The current SILK frame corresponds to the mid channel, and</t>
<t>Either
<list style="symbols">
<t>This is a regular SILK frame where the VAD flags
 (see <xref target="silk_header_bits"/>) indicate that the corresponding side
 channel is not active.</t>
<t>
This is an LBRR frame where the LBRR flags
 (see <xref target="silk_header_bits"/> and <xref target="silk_lbrr_flags"/>)
 indicate that the corresponding side channel is not coded.
</t>
</list>
</t>
</list>
It is omitted when there are no stereo weights, for all of the same reasons.
It is also omitted for a regular SILK frame when the VAD flag of the
 corresponding side channel frame is set (indicating it is active).
The side channel must be coded in this case, making the mid-only flag
 redundant.
It is also omitted for an LBRR frame when the corresponding LBRR flags
 indicate the side channel is coded.
</t>

<t>
When the flag is present, the decoder reads a single value using the PDF in
 <xref target="silk_mid_only_pdf"/>, as implemented in
 silk_stereo_decode_mid_only() (decode_stereo_pred.c).
If the flag is set, then there is no corresponding SILK frame for the side
 channel, the entire decoding process for the side channel is skipped, and
 zeros are fed to the stereo unmixing process (see
 <xref target="silk_stereo_unmixing"/>) instead.
As stated above, LBRR frames still include this flag when the LBRR flag
 indicates that the side channel is not coded.
In that case, if this flag is zero (indicating that there should be a side
 channel), then Packet Loss Concealment (PLC, see
 <xref target="Packet Loss Concealment"/>) SHOULD be invoked to recover a
 side channel signal.
</t>

<texttable anchor="silk_mid_only_pdf" title="Mid-only Flag PDF">
<ttcol align="left">PDF</ttcol>
<c>{192, 64}/256</c>
</texttable>

</section>

<section anchor="silk_frame_type" toc="include" title="Frame Type">
<t>
Each SILK frame contains a single "frame type" symbol that jointly codes the
 signal type and quantization offset type of the corresponding frame.
If the current frame is a regular SILK frame whose VAD bit was not set (an
 "inactive" frame), then the frame type symbol takes on a value of either 0 or
 1 and is decoded using the first PDF in <xref target="silk_frame_type_pdfs"/>.
If the frame is an LBRR frame or a regular SILK frame whose VAD flag was set
 (an "active" frame), then the value of the symbol may range from 2 to 5,
 inclusive, and is decoded using the second PDF in
 <xref target="silk_frame_type_pdfs"/>.
<xref target="silk_frame_type_table"/> translates between the value of the
 frame type symbol and the corresponding signal type and quantization offset
 type.
</t>

<texttable anchor="silk_frame_type_pdfs" title="Frame Type PDFs">
<ttcol>VAD Flag</ttcol>
<ttcol>PDF</ttcol>
<c>Inactive</c> <c>{26, 230, 0, 0, 0, 0}/256</c>
<c>Active</c>   <c>{0, 0, 24, 74, 148, 10}/256</c>
</texttable>

<texttable anchor="silk_frame_type_table"
 title="Signal Type and Quantization Offset Type from Frame Type">
<ttcol>Frame Type</ttcol>
<ttcol>Signal Type</ttcol>
<ttcol align="right">Quantization Offset Type</ttcol>
<c>0</c> <c>Inactive</c> <c>Low</c>
<c>1</c> <c>Inactive</c> <c>High</c>
<c>2</c> <c>Unvoiced</c> <c>Low</c>
<c>3</c> <c>Unvoiced</c> <c>High</c>
<c>4</c> <c>Voiced</c>   <c>Low</c>
<c>5</c> <c>Voiced</c>   <c>High</c>
</texttable>

</section>

<section anchor="silk_gains" toc="include" title="Subframe Gains">
<t>
A separate quantization gain is coded for each 5&nbsp;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&nbsp;bits on a
 log scale, giving them a resolution of approximately 1.369&nbsp;dB and a range
 of approximately 1.94&nbsp;dB to 88.21&nbsp;dB.
</t>
<t>
The subframe gains are either coded independently, or relative to the gain from
 the most recent coded subframe in the same channel.
Independent coding is used if and only if
<list style="symbols">
<t>
This is the first subframe in the current SILK frame, and
</t>
<t>Either
<list style="symbols">
<t>
This is the first SILK frame of its type (LBRR or regular) for this channel in
 the current Opus frame, or
 </t>
<t>
The previous SILK frame of the same type (LBRR or regular) for this channel in
 the same Opus frame was not coded.
</t>
</list>
</t>
</list>
</t>

<t>
In an independently coded subframe gain, the 3 most significant bits of the
 quantization gain are decoded using a PDF selected from
 <xref target="silk_independent_gain_msb_pdfs"/> based on the decoded signal
 type (see <xref target="silk_frame_type"/>).
</t>

<texttable anchor="silk_independent_gain_msb_pdfs"
 title="PDFs for Independent Quantization Gain MSB Coding">
<ttcol align="left">Signal Type</ttcol>
<ttcol align="left">PDF</ttcol>
<c>Inactive</c> <c>{32, 112, 68, 29, 12,  1,  1, 1}/256</c>
<c>Unvoiced</c>  <c>{2,  17, 45, 60, 62, 47, 19, 4}/256</c>
<c>Voiced</c>    <c>{1,   3, 26, 71, 94, 50,  9, 2}/256</c>
</texttable>

<t>
The 3 least significant bits are decoded using a uniform PDF:
</t>
<texttable anchor="silk_independent_gain_lsb_pdf"
 title="PDF for Independent Quantization Gain LSB Coding">
<ttcol align="left">PDF</ttcol>
<c>{32, 32, 32, 32, 32, 32, 32, 32}/256</c>
</texttable>

<t>
These 6 bits are combined to form a gain index between 0 and 63.
When the gain for the previous subframe is available, then the current gain is
 limited as follows:
<figure align="center">
<artwork align="center"><![CDATA[
log_gain = max(gain_index, previous_log_gain - 16) .
]]></artwork>
</figure>
This may help some implementations limit the change in precision of their
 internal LTP history.
The indices which this clamp applies to cannot simply be removed from the
 codebook, because the previous gain index will not be available after packet
 loss.
This step is skipped after a decoder reset, and in the side channel if the
 previous frame in the side channel was not coded, since there is no previous
 gain index.
It MAY also be skipped after packet loss.
</t>

<t>
For subframes which do not have an independent gain (including the first
 subframe of frames not listed as using independent coding above), the
 quantization gain is coded relative to the gain from the previous subframe (in
 the same channel).
The PDF in <xref target="silk_delta_gain_pdf"/> yields a delta gain index
 between 0 and 40, inclusive.
</t>
<texttable anchor="silk_delta_gain_pdf"
 title="PDF for Delta Quantization Gain Coding">
<ttcol align="left">PDF</ttcol>
<c>{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</c>
</texttable>
<t>
The following formula translates this index into a quantization gain for the
 current subframe using the gain from the previous subframe:
<figure align="center">
<artwork align="center"><![CDATA[
log_gain = clamp(0, max(2*gain_index - 16,
                   previous_log_gain + gain_index - 4), 63) .
]]></artwork>
</figure>
</t>
<t>
silk_gains_dequant() (gain_quant.c) dequantizes log_gain for the k'th subframe
 and converts it into a linear Q16 scale factor via
<figure align="center">
<artwork align="center"><![CDATA[
gain_Q16[k] = silk_log2lin((0x1D1C71*log_gain>>16) + 2090)
]]></artwork>
</figure>
</t>
<t>
The function silk_log2lin() (log2lin.c) computes an approximation of
 2**(inLog_Q7/128.0), where inLog_Q7 is its Q7 input.
Let i = inLog_Q7&gt;&gt;7 be the integer part of inLogQ7 and
 f = inLog_Q7&amp;127 be the fractional part.
Then
<figure align="center">
<artwork align="center"><![CDATA[
(1<<i) + ((-174*f*(128-f)>>16)+f)*((1<<i)>>7)
]]></artwork>
</figure>
 yields the approximate exponential.
The final Q16 gain values lies between 81920 and 1686110208, inclusive
 (representing scale factors of 1.25 to 25728, respectively).
</t>
</section>

<section anchor="silk_nlsfs" toc="include" title="Normalized Line Spectral
 Frequency (LSF) and Linear Predictive Coding (LPC) Coefficients">
<t>
A set of normalized Line Spectral Frequency (LSF) coefficients follow the
 quantization gains in the bitstream, and represent the Linear Predictive
 Coding (LPC) coefficients for the current SILK frame.
Once decoded, the normalized LSFs form an increasing list of Q15 values between
 0 and 1.
These represent the interleaved zeros on the unit circle between 0 and pi
 (hence "normalized") in the standard decomposition of the LPC filter into a
 symmetric part and an anti-symmetric part (P and Q in
 <xref target="silk_nlsf2lpc"/>).
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.
</t>
<t>
The normalized LSFs are coded using a two-stage vector quantizer (VQ)
 (<xref target="silk_nlsf_stage1"/> and <xref target="silk_nlsf_stage2"/>).
NB and MB frames use an order-10 predictor, while WB frames use an order-16
 predictor, and thus have different sets of tables.
After reconstructing the normalized LSFs
 (<xref target="silk_nlsf_reconstruction"/>), the decoder runs them through a
 stabilization process (<xref target="silk_nlsf_stabilization"/>), interpolates
 them between frames (<xref target="silk_nlsf_interpolation"/>), converts them
 back into LPC coefficients (<xref target="silk_nlsf2lpc"/>), and then runs
 them through further processes to limit the range of the coefficients
 (<xref target="silk_lpc_range_limit"/>) and the gain of the filter
 (<xref target="silk_lpc_gain_limit"/>).
All of this is necessary to ensure the reconstruction process is stable.
</t>

<section anchor="silk_nlsf_stage1" title="Stage 1 Normalized LSF Decoding">
<t>
The first VQ stage uses a 32-element codebook, coded with one of the PDFs in
 <xref target="silk_nlsf_stage1_pdfs"/>, 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
 <xref target="silk_nlsf_nbmb_codebook"/> and
 <xref target="silk_nlsf_wb_codebook"/>, but they are not needed until the last
 stages of reconstructing the LSF coefficients.
</t>

<texttable anchor="silk_nlsf_stage1_pdfs"
 title="PDFs for Normalized LSF Index Stage-1 Decoding">
<ttcol align="left">Audio Bandwidth</ttcol>
<ttcol align="left">Signal Type</ttcol>
<ttcol align="left">PDF</ttcol>
<c>NB or MB</c> <c>Inactive or unvoiced</c>
<c>
{44, 34, 30, 19, 21, 12, 11,  3,
  3,  2, 16,  2,  2,  1,  5,  2,
  1,  3,  3,  1,  1,  2,  2,  2,
  3,  1,  9,  9,  2,  7,  2,  1}/256
</c>
<c>NB or MB</c> <c>Voiced</c>
<c>
{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
</c>
<c>WB</c> <c>Inactive or unvoiced</c>
<c>
{31, 21,  3, 17,  1,  8, 17,  4,
  1, 18, 16,  4,  2,  3,  1, 10,
  1,  3, 16, 11, 16,  2,  2,  3,
  2, 11,  1,  4,  9,  8,  7,  3}/256
</c>
<c>WB</c> <c>Voiced</c>
<c>
{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
</c>
</texttable>

</section>

<section anchor="silk_nlsf_stage2" title="Stage 2 Normalized LSF Decoding">
<t>
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
 <xref target="silk_nlsf_stage2_nbmb_pdfs"/>, and the 8 (i...p) for WB frames
 given in <xref target="silk_nlsf_stage2_wb_pdfs"/>.
Which PDF is used for which coefficient is driven by the index, I1,
 decoded in the first stage.
<xref target="silk_nlsf_nbmb_stage2_cb_sel"/> lists the letter of the
 corresponding PDF for each normalized LSF coefficient for NB and MB, and
 <xref target="silk_nlsf_wb_stage2_cb_sel"/> lists the same information for WB.
</t>

<texttable anchor="silk_nlsf_stage2_nbmb_pdfs"
 title="PDFs for NB/MB Normalized LSF Index Stage-2 Decoding">
<ttcol align="left">Codebook</ttcol>
<ttcol align="left">PDF</ttcol>
<c>a</c> <c>{1,   1,   1,  15, 224,  11,   1,   1,   1}/256</c>
<c>b</c> <c>{1,   1,   2,  34, 183,  32,   1,   1,   1}/256</c>
<c>c</c> <c>{1,   1,   4,  42, 149,  55,   2,   1,   1}/256</c>
<c>d</c> <c>{1,   1,   8,  52, 123,  61,   8,   1,   1}/256</c>
<c>e</c> <c>{1,   3,  16,  53, 101,  74,   6,   1,   1}/256</c>
<c>f</c> <c>{1,   3,  17,  55,  90,  73,  15,   1,   1}/256</c>
<c>g</c> <c>{1,   7,  24,  53,  74,  67,  26,   3,   1}/256</c>
<c>h</c> <c>{1,   1,  18,  63,  78,  58,  30,   6,   1}/256</c>
</texttable>

<texttable anchor="silk_nlsf_stage2_wb_pdfs"
 title="PDFs for WB Normalized LSF Index Stage-2 Decoding">
<ttcol align="left">Codebook</ttcol>
<ttcol align="left">PDF</ttcol>
<c>i</c> <c>{1,   1,   1,   9, 232,   9,   1,   1,   1}/256</c>
<c>j</c> <c>{1,   1,   2,  28, 186,  35,   1,   1,   1}/256</c>
<c>k</c> <c>{1,   1,   3,  42, 152,  53,   2,   1,   1}/256</c>
<c>l</c> <c>{1,   1,  10,  49, 126,  65,   2,   1,   1}/256</c>
<c>m</c> <c>{1,   4,  19,  48, 100,  77,   5,   1,   1}/256</c>
<c>n</c> <c>{1,   1,  14,  54, 100,  72,  12,   1,   1}/256</c>
<c>o</c> <c>{1,   1,  15,  61,  87,  61,  25,   4,   1}/256</c>
<c>p</c> <c>{1,   7,  21,  50,  77,  81,  17,   1,   1}/256</c>
</texttable>

<texttable anchor="silk_nlsf_nbmb_stage2_cb_sel"
 title="Codebook Selection for NB/MB Normalized LSF Index Stage 2 Decoding">
<ttcol>I1</ttcol>
<ttcol>Coefficient</ttcol>
<c/>
<c><spanx style="vbare">0&nbsp;1&nbsp;2&nbsp;3&nbsp;4&nbsp;5&nbsp;6&nbsp;7&nbsp;8&nbsp;9</spanx></c>
<c> 0</c>
<c><spanx style="vbare">a&nbsp;a&nbsp;a&nbsp;a&nbsp;a&nbsp;a&nbsp;a&nbsp;a&nbsp;a&nbsp;a</spanx></c>
<c> 1</c>
<c><spanx style="vbare">b&nbsp;d&nbsp;b&nbsp;c&nbsp;c&nbsp;b&nbsp;c&nbsp;b&nbsp;b&nbsp;b</spanx></c>
<c> 2</c>
<c><spanx style="vbare">c&nbsp;b&nbsp;b&nbsp;b&nbsp;b&nbsp;b&nbsp;b&nbsp;b&nbsp;b&nbsp;b</spanx></c>
<c> 3</c>
<c><spanx style="vbare">b&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;b&nbsp;c&nbsp;b&nbsp;b&nbsp;b</spanx></c>
<c> 4</c>
<c><spanx style="vbare">c&nbsp;d&nbsp;d&nbsp;d&nbsp;d&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c</spanx></c>
<c> 5</c>
<c><spanx style="vbare">a&nbsp;f&nbsp;d&nbsp;d&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;b&nbsp;b</spanx></c>
<c> g</c>
<c><spanx style="vbare">a&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;b</spanx></c>
<c> 7</c>
<c><spanx style="vbare">c&nbsp;d&nbsp;g&nbsp;e&nbsp;e&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f</spanx></c>
<c> 8</c>
<c><spanx style="vbare">c&nbsp;e&nbsp;f&nbsp;f&nbsp;e&nbsp;f&nbsp;e&nbsp;g&nbsp;e&nbsp;e</spanx></c>
<c> 9</c>
<c><spanx style="vbare">c&nbsp;e&nbsp;e&nbsp;h&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f&nbsp;e</spanx></c>
<c>10</c>
<c><spanx style="vbare">e&nbsp;d&nbsp;d&nbsp;d&nbsp;c&nbsp;d&nbsp;c&nbsp;c&nbsp;c&nbsp;c</spanx></c>
<c>11</c>
<c><spanx style="vbare">b&nbsp;f&nbsp;f&nbsp;g&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f&nbsp;f</spanx></c>
<c>12</c>
<c><spanx style="vbare">c&nbsp;h&nbsp;e&nbsp;g&nbsp;f&nbsp;f&nbsp;f&nbsp;f&nbsp;f&nbsp;f</spanx></c>
<c>13</c>
<c><spanx style="vbare">c&nbsp;h&nbsp;f&nbsp;f&nbsp;f&nbsp;f&nbsp;f&nbsp;g&nbsp;f&nbsp;e</spanx></c>
<c>14</c>
<c><spanx style="vbare">d&nbsp;d&nbsp;f&nbsp;e&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;e&nbsp;e</spanx></c>
<c>15</c>
<c><spanx style="vbare">c&nbsp;d&nbsp;d&nbsp;f&nbsp;f&nbsp;e&nbsp;e&nbsp;e&nbsp;e&nbsp;e</spanx></c>
<c>16</c>
<c><spanx style="vbare">c&nbsp;e&nbsp;e&nbsp;g&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f&nbsp;f</spanx></c>
<c>17</c>
<c><spanx style="vbare">c&nbsp;f&nbsp;e&nbsp;g&nbsp;f&nbsp;f&nbsp;f&nbsp;e&nbsp;f&nbsp;e</spanx></c>
<c>18</c>
<c><spanx style="vbare">c&nbsp;h&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f&nbsp;f</spanx></c>
<c>19</c>
<c><spanx style="vbare">c&nbsp;f&nbsp;e&nbsp;g&nbsp;h&nbsp;g&nbsp;f&nbsp;g&nbsp;f&nbsp;e</spanx></c>
<c>20</c>
<c><spanx style="vbare">d&nbsp;g&nbsp;h&nbsp;e&nbsp;g&nbsp;f&nbsp;f&nbsp;g&nbsp;e&nbsp;f</spanx></c>
<c>21</c>
<c><spanx style="vbare">c&nbsp;h&nbsp;g&nbsp;e&nbsp;e&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;f</spanx></c>
<c>22</c>
<c><spanx style="vbare">e&nbsp;f&nbsp;f&nbsp;e&nbsp;g&nbsp;g&nbsp;f&nbsp;g&nbsp;f&nbsp;e</spanx></c>
<c>23</c>
<c><spanx style="vbare">c&nbsp;f&nbsp;f&nbsp;g&nbsp;f&nbsp;g&nbsp;e&nbsp;g&nbsp;e&nbsp;e</spanx></c>
<c>24</c>
<c><spanx style="vbare">e&nbsp;f&nbsp;f&nbsp;f&nbsp;d&nbsp;h&nbsp;e&nbsp;f&nbsp;f&nbsp;e</spanx></c>
<c>25</c>
<c><spanx style="vbare">c&nbsp;d&nbsp;e&nbsp;f&nbsp;f&nbsp;g&nbsp;e&nbsp;f&nbsp;f&nbsp;e</spanx></c>
<c>26</c>
<c><spanx style="vbare">c&nbsp;d&nbsp;c&nbsp;d&nbsp;d&nbsp;e&nbsp;c&nbsp;d&nbsp;d&nbsp;d</spanx></c>
<c>27</c>
<c><spanx style="vbare">b&nbsp;b&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;c&nbsp;d&nbsp;c&nbsp;c</spanx></c>
<c>28</c>
<c><spanx style="vbare">e&nbsp;f&nbsp;f&nbsp;g&nbsp;g&nbsp;g&nbsp;f&nbsp;g&nbsp;e&nbsp;f</spanx></c>
<c>29</c>
<c><spanx style="vbare">d&nbsp;f&nbsp;f&nbsp;e&nbsp;e&nbsp;e&nbsp;e&nbsp;d&nbsp;d&nbsp;c</spanx></c>
<c>30</c>
<c><spanx style="vbare">c&nbsp;f&nbsp;d&nbsp;h&nbsp;f&nbsp;f&nbsp;e&nbsp;e&nbsp;f&nbsp;e</spanx></c>
<c>31</c>
<c><spanx style="vbare">e&nbsp;e&nbsp;f&nbsp;e&nbsp;f&nbsp;g&nbsp;f&nbsp;g&nbsp;f&nbsp;e</spanx></c>
</texttable>

<texttable anchor="silk_nlsf_wb_stage2_cb_sel"
 title="Codebook Selection for WB Normalized LSF Index Stage 2 Decoding">
<ttcol>I1</ttcol>
<ttcol>Coefficient</ttcol>
<c/>
<c><spanx style="vbare">0&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;&nbsp;3&nbsp;&nbsp;4&nbsp;&nbsp;5&nbsp;&nbsp;6&nbsp;&nbsp;7&nbsp;&nbsp;8&nbsp;&nbsp;9&nbsp;10&nbsp;11&nbsp;12&nbsp;13&nbsp;14&nbsp;15</spanx></c>
<c> 0</c>
<c><spanx style="vbare">i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
<c> 1</c>
<c><spanx style="vbare">k&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;l</spanx></c>
<c> 2</c>
<c><spanx style="vbare">k&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;p&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;k&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l</spanx></c>
<c> 3</c>
<c><spanx style="vbare">i&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;j</spanx></c>
<c> 4</c>
<c><spanx style="vbare">i&nbsp;&nbsp;o&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;p&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l</spanx></c>
<c> 5</c>
<c><spanx style="vbare">i&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;m</spanx></c>
<c> 6</c>
<c><spanx style="vbare">i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
<c> 7</c>
<c><spanx style="vbare">i&nbsp;&nbsp;k&nbsp;&nbsp;o&nbsp;&nbsp;l&nbsp;&nbsp;p&nbsp;&nbsp;k&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;l</spanx></c>
<c> 8</c>
<c><spanx style="vbare">i&nbsp;&nbsp;o&nbsp;&nbsp;k&nbsp;&nbsp;o&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;o&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l</spanx></c>
<c> 9</c>
<c><spanx style="vbare">k&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
<c>10</c>
<c><spanx style="vbare">i&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;j</spanx></c>
<c>11</c>
<c><spanx style="vbare">k&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;l</spanx></c>
<c>12</c>
<c><spanx style="vbare">k&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;l</spanx></c>
<c>13</c>
<c><spanx style="vbare">l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;m</spanx></c>
<c>14</c>
<c><spanx style="vbare">i&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;p&nbsp;&nbsp;n&nbsp;&nbsp;k&nbsp;&nbsp;o&nbsp;&nbsp;n&nbsp;&nbsp;p&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l</spanx></c>
<c>15</c>
<c><spanx style="vbare">i&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;j&nbsp;&nbsp;i</spanx></c>
<c>16</c>
<c><spanx style="vbare">j&nbsp;&nbsp;o&nbsp;&nbsp;n&nbsp;&nbsp;p&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;m</spanx></c>
<c>17</c>
<c><spanx style="vbare">j&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m</spanx></c>
<c>18</c>
<c><spanx style="vbare">k&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;m</spanx></c>
<c>19</c>
<c><spanx style="vbare">i&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
<c>20</c>
<c><spanx style="vbare">l&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;m</spanx></c>
<c>21</c>
<c><spanx style="vbare">k&nbsp;&nbsp;o&nbsp;&nbsp;l&nbsp;&nbsp;p&nbsp;&nbsp;p&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;l</spanx></c>
<c>22</c>
<c><spanx style="vbare">k&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;o&nbsp;&nbsp;o&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;m</spanx></c>
<c>23</c>
<c><spanx style="vbare">j&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;j</spanx></c>
<c>24</c>
<c><spanx style="vbare">k&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;o&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;p&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l</spanx></c>
<c>25</c>
<c><spanx style="vbare">i&nbsp;&nbsp;o&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
<c>26</c>
<c><spanx style="vbare">i&nbsp;&nbsp;o&nbsp;&nbsp;o&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;k&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;p&nbsp;&nbsp;p&nbsp;&nbsp;m&nbsp;&nbsp;m&nbsp;&nbsp;m</spanx></c>
<c>27</c>
<c><spanx style="vbare">l&nbsp;&nbsp;l&nbsp;&nbsp;p&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;l</spanx></c>
<c>28</c>
<c><spanx style="vbare">i&nbsp;&nbsp;i&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;j</spanx></c>
<c>29</c>
<c><spanx style="vbare">i&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;j</spanx></c>
<c>30</c>
<c><spanx style="vbare">l&nbsp;&nbsp;n&nbsp;&nbsp;n&nbsp;&nbsp;m&nbsp;&nbsp;p&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;i&nbsp;&nbsp;j&nbsp;&nbsp;i</spanx></c>
<c>31</c>
<c><spanx style="vbare">k&nbsp;&nbsp;l&nbsp;&nbsp;n&nbsp;&nbsp;l&nbsp;&nbsp;m&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;l&nbsp;&nbsp;k&nbsp;&nbsp;j&nbsp;&nbsp;k&nbsp;&nbsp;o&nbsp;&nbsp;m&nbsp;&nbsp;i&nbsp;&nbsp;i&nbsp;&nbsp;i</spanx></c>
</texttable>

<t>
Decoding the second stage residual proceeds as follows.
For each coefficient, the decoder reads a symbol using the PDF corresponding to
 I1 from either <xref target="silk_nlsf_nbmb_stage2_cb_sel"/> or
 <xref target="silk_nlsf_wb_stage2_cb_sel"/>, and subtracts 4 from the result
 to give an index in the range -4 to 4, inclusive.
If the index is either -4 or 4, it reads a second symbol using the PDF in
 <xref target="silk_nlsf_ext_pdf"/>, 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.
</t>

<texttable anchor="silk_nlsf_ext_pdf"
 title="PDF for Normalized LSF Index Extension Decoding">
<ttcol align="left">PDF</ttcol>
<c>{156, 60, 24,  9,  4,  2,  1}/256</c>
</texttable>

<t>
The decoded indices from both stages are translated back into normalized LSF
 coefficients in silk_NLSF_decode() (NLSF_decode.c).
The stage-2 indices represent residuals after both the first stage of the VQ
 and a separate backwards-prediction step.
The backwards prediction process in the encoder subtracts a prediction from
 each residual formed by a multiple of the coefficient that follows it.
The decoder must undo this process.
<xref target="silk_nlsf_pred_weights"/> 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.
</t>

<texttable anchor="silk_nlsf_pred_weights"
 title="Prediction Weights for Normalized LSF Decoding">
<ttcol align="left">Coefficient</ttcol>
<ttcol align="right">A</ttcol>
<ttcol align="right">B</ttcol>
<ttcol align="right">C</ttcol>
<ttcol align="right">D</ttcol>
 <c>0</c> <c>179</c> <c>116</c> <c>175</c>  <c>68</c>
 <c>1</c> <c>138</c>  <c>67</c> <c>148</c>  <c>62</c>
 <c>2</c> <c>140</c>  <c>82</c> <c>160</c>  <c>66</c>
 <c>3</c> <c>148</c>  <c>59</c> <c>176</c>  <c>60</c>
 <c>4</c> <c>151</c>  <c>92</c> <c>178</c>  <c>72</c>
 <c>5</c> <c>149</c>  <c>72</c> <c>173</c> <c>117</c>
 <c>6</c> <c>153</c> <c>100</c> <c>174</c>  <c>85</c>
 <c>7</c> <c>151</c>  <c>89</c> <c>164</c>  <c>90</c>
 <c>8</c> <c>163</c>  <c>92</c> <c>177</c> <c>118</c>
 <c>9</c> <c/>        <c/>      <c>174</c> <c>136</c>
<c>10</c> <c/>        <c/>      <c>196</c> <c>151</c>
<c>11</c> <c/>        <c/>      <c>182</c> <c>142</c>
<c>12</c> <c/>        <c/>      <c>198</c> <c>160</c>
<c>13</c> <c/>        <c/>      <c>192</c> <c>142</c>
<c>14</c> <c/>        <c/>      <c>182</c> <c>155</c>
</texttable>

<t>
The prediction is undone using the procedure implemented in
 silk_NLSF_residual_dequant() (NLSF_decode.c), which is as follows.
Each coefficient selects its prediction weight from one of the two lists based
 on the stage-1 index, I1.
<xref target="silk_nlsf_nbmb_weight_sel"/> gives the selections for each
 coefficient for NB and MB, and <xref target="silk_nlsf_wb_weight_sel"/> 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&nbsp;&lt;=&nbsp;k&nbsp;&lt;&nbsp;d_LPC-1.
Then, the stage-2 residual for each coefficient is computed via
<figure align="center">
<artwork align="center"><![CDATA[
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) ,
]]></artwork>
</figure>
 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).
</t>

<texttable anchor="silk_nlsf_nbmb_weight_sel"
 title="Prediction Weight Selection for NB/MB Normalized LSF Decoding">
<ttcol>I1</ttcol>
<ttcol>Coefficient</ttcol>
<c/>
<c><spanx style="vbare">0&nbsp;1&nbsp;2&nbsp;3&nbsp;4&nbsp;5&nbsp;6&nbsp;7&nbsp;8</spanx></c>
<c> 0</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c> 1</c>
<c><spanx style="vbare">B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c> 2</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c> 3</c>
<c><spanx style="vbare">B&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c> 4</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c> 5</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c> 6</c>
<c><spanx style="vbare">B&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c> 7</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;A</spanx></c>
<c> 8</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A&nbsp;B&nbsp;B</spanx></c>
<c> 9</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;B</spanx></c>
<c>10</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c>11</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>12</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>13</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>14</c>
<c><spanx style="vbare">B&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B</spanx></c>
<c>15</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c>16</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c>17</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B</spanx></c>
<c>18</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>19</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c>20</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;B&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c>21</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>22</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B</spanx></c>
<c>23</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;B&nbsp;B</spanx></c>
<c>24</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B</spanx></c>
<c>25</c>
<c><spanx style="vbare">A&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;B&nbsp;A</spanx></c>
<c>26</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c>27</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c>28</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A</spanx></c>
<c>29</c>
<c><spanx style="vbare">B&nbsp;A&nbsp;A&nbsp;B&nbsp;A&nbsp;A&nbsp;A&nbsp;A&nbsp;A</spanx></c>
<c>30</c>
<c><spanx style="vbare">A&nbsp;A&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;A&nbsp;B</spanx></c>
<c>31</c>
<c><spanx style="vbare">B&nbsp;A&nbsp;B&nbsp;B&nbsp;A&nbsp;B&nbsp;B&nbsp;B&nbsp;B</spanx></c>
</texttable>

<texttable anchor="silk_nlsf_wb_weight_sel"
 title="Prediction Weight Selection for WB Normalized LSF Decoding">
<ttcol>I1</ttcol>
<ttcol>Coefficient</ttcol>
<c/>
<c><spanx style="vbare">0&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;&nbsp;3&nbsp;&nbsp;4&nbsp;&nbsp;5&nbsp;&nbsp;6&nbsp;&nbsp;7&nbsp;&nbsp;8&nbsp;&nbsp;9&nbsp;10&nbsp;11&nbsp;12&nbsp;13&nbsp;14</spanx></c>
<c> 0</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c> 1</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c> 2</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c> 3</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c> 4</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
<c> 5</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c> 6</c>
<c><spanx style="vbare">D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
<c> 7</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c> 8</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D</spanx></c>
<c> 9</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c>10</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>11</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>12</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>13</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>14</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D</spanx></c>
<c>15</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
<c>16</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>17</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>18</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c>19</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>20</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>21</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
<c>22</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>23</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
<c>24</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D</spanx></c>
<c>25</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c>26</c>
<c><spanx style="vbare">C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D</spanx></c>
<c>27</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D</spanx></c>
<c>28</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c>29</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D</spanx></c>
<c>30</c>
<c><spanx style="vbare">D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;C</spanx></c>
<c>31</c>
<c><spanx style="vbare">C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C&nbsp;&nbsp;C&nbsp;&nbsp;D&nbsp;&nbsp;C</spanx></c>
</texttable>

</section>

<section anchor="silk_nlsf_reconstruction"
 title="Reconstructing the Normalized LSF Coefficients">
<t>
Once the stage-1 index I1 and the stage-2 residual res_Q10[] have been decoded,
 the final normalized LSF coefficients can be reconstructed.
</t>
<t>
The spectral distortion introduced by the quantization of each LSF coefficient
 varies, so the stage-2 residual is weighted accordingly, using the
 low-complexity Inverse Harmonic Mean Weighting (IHMW) function proposed in
 <xref target="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
 <xref target="silk_nlsf_nbmb_codebook"/> or
 <xref target="silk_nlsf_wb_codebook"/>.
Then for 0&nbsp;&lt;=&nbsp;k&nbsp;&lt;&nbsp;d_LPC the following expression
 computes the square of the weight as a Q18 value:
<figure align="center">
<artwork align="center">
<![CDATA[
w2_Q18[k] = (1024/(cb1_Q8[k] - cb1_Q8[k-1])
             + 1024/(cb1_Q8[k+1] - cb1_Q8[k])) << 16 ,
]]>
</artwork>
</figure>
 where cb1_Q8[-1]&nbsp;=&nbsp;0 and cb1_Q8[d_LPC]&nbsp;=&nbsp;256, and the
 division is exact integer division.
This is reduced to an unsquared, Q9 value using the following square-root
 approximation:
<figure align="center">
<artwork align="center"><![CDATA[
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)
]]></artwork>
</figure>
The cb1_Q8[] vector completely determines these weights, and they may be
 tabulated and stored as 13-bit unsigned values (with a range of 1819 to 5227,
 inclusive) to avoid computing them when decoding.
The reference implementation already requires code to compute these weights on
 unquantized coefficients in the encoder, in silk_NLSF_VQ_weights_laroia()
 (NLSF_VQ_weights_laroia.c) and its callers, so it reuses that code in the
 decoder instead of using a pre-computed table to reduce the amount of ROM
 required.
</t>

<texttable anchor="silk_nlsf_nbmb_codebook"
           title="Codebook Vectors for NB/MB Normalized LSF Stage 1 Decoding">
<ttcol>I1</ttcol>
<ttcol>Codebook (Q8)</ttcol>
<c/>
<c><spanx style="vbare">&nbsp;0&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;3&nbsp;&nbsp;&nbsp;4&nbsp;&nbsp;&nbsp;5&nbsp;&nbsp;&nbsp;6&nbsp;&nbsp;&nbsp;7&nbsp;&nbsp;&nbsp;8&nbsp;&nbsp;&nbsp;9</spanx></c>
<c>0</c>
<c><spanx style="vbare">12&nbsp;&nbsp;35&nbsp;&nbsp;60&nbsp;&nbsp;83&nbsp;108&nbsp;132&nbsp;157&nbsp;180&nbsp;206&nbsp;228</spanx></c>
<c>1</c>
<c><spanx style="vbare">15&nbsp;&nbsp;32&nbsp;&nbsp;55&nbsp;&nbsp;77&nbsp;101&nbsp;125&nbsp;151&nbsp;175&nbsp;201&nbsp;225</spanx></c>
<c>2</c>
<c><spanx style="vbare">19&nbsp;&nbsp;42&nbsp;&nbsp;66&nbsp;&nbsp;89&nbsp;114&nbsp;137&nbsp;162&nbsp;184&nbsp;209&nbsp;230</spanx></c>
<c>3</c>
<c><spanx style="vbare">12&nbsp;&nbsp;25&nbsp;&nbsp;50&nbsp;&nbsp;72&nbsp;&nbsp;97&nbsp;120&nbsp;147&nbsp;172&nbsp;200&nbsp;223</spanx></c>
<c>4</c>
<c><spanx style="vbare">26&nbsp;&nbsp;44&nbsp;&nbsp;69&nbsp;&nbsp;90&nbsp;114&nbsp;135&nbsp;159&nbsp;180&nbsp;205&nbsp;225</spanx></c>
<c>5</c>
<c><spanx style="vbare">13&nbsp;&nbsp;22&nbsp;&nbsp;53&nbsp;&nbsp;80&nbsp;106&nbsp;130&nbsp;156&nbsp;180&nbsp;205&nbsp;228</spanx></c>
<c>6</c>
<c><spanx style="vbare">15&nbsp;&nbsp;25&nbsp;&nbsp;44&nbsp;&nbsp;64&nbsp;&nbsp;90&nbsp;115&nbsp;142&nbsp;168&nbsp;196&nbsp;222</spanx></c>
<c>7</c>
<c><spanx style="vbare">19&nbsp;&nbsp;24&nbsp;&nbsp;62&nbsp;&nbsp;82&nbsp;100&nbsp;120&nbsp;145&nbsp;168&nbsp;190&nbsp;214</spanx></c>
<c>8</c>
<c><spanx style="vbare">22&nbsp;&nbsp;31&nbsp;&nbsp;50&nbsp;&nbsp;79&nbsp;103&nbsp;120&nbsp;151&nbsp;170&nbsp;203&nbsp;227</spanx></c>
<c>9</c>
<c><spanx style="vbare">21&nbsp;&nbsp;29&nbsp;&nbsp;45&nbsp;&nbsp;65&nbsp;106&nbsp;124&nbsp;150&nbsp;171&nbsp;196&nbsp;224</spanx></c>
<c>10</c>
<c><spanx style="vbare">30&nbsp;&nbsp;49&nbsp;&nbsp;75&nbsp;&nbsp;97&nbsp;121&nbsp;142&nbsp;165&nbsp;186&nbsp;209&nbsp;229</spanx></c>
<c>11</c>
<c><spanx style="vbare">19&nbsp;&nbsp;25&nbsp;&nbsp;52&nbsp;&nbsp;70&nbsp;&nbsp;93&nbsp;116&nbsp;143&nbsp;166&nbsp;192&nbsp;219</spanx></c>
<c>12</c>
<c><spanx style="vbare">26&nbsp;&nbsp;34&nbsp;&nbsp;62&nbsp;&nbsp;75&nbsp;&nbsp;97&nbsp;118&nbsp;145&nbsp;167&nbsp;194&nbsp;217</spanx></c>
<c>13</c>
<c><spanx style="vbare">25&nbsp;&nbsp;33&nbsp;&nbsp;56&nbsp;&nbsp;70&nbsp;&nbsp;91&nbsp;113&nbsp;143&nbsp;165&nbsp;196&nbsp;223</spanx></c>
<c>14</c>
<c><spanx style="vbare">21&nbsp;&nbsp;34&nbsp;&nbsp;51&nbsp;&nbsp;72&nbsp;&nbsp;97&nbsp;117&nbsp;145&nbsp;171&nbsp;196&nbsp;222</spanx></c>
<c>15</c>
<c><spanx style="vbare">20&nbsp;&nbsp;29&nbsp;&nbsp;50&nbsp;&nbsp;67&nbsp;&nbsp;90&nbsp;117&nbsp;144&nbsp;168&nbsp;197&nbsp;221</spanx></c>
<c>16</c>
<c><spanx style="vbare">22&nbsp;&nbsp;31&nbsp;&nbsp;48&nbsp;&nbsp;66&nbsp;&nbsp;95&nbsp;117&nbsp;146&nbsp;168&nbsp;196&nbsp;222</spanx></c>
<c>17</c>
<c><spanx style="vbare">24&nbsp;&nbsp;33&nbsp;&nbsp;51&nbsp;&nbsp;77&nbsp;116&nbsp;134&nbsp;158&nbsp;180&nbsp;200&nbsp;224</spanx></c>
<c>18</c>
<c><spanx style="vbare">21&nbsp;&nbsp;28&nbsp;&nbsp;70&nbsp;&nbsp;87&nbsp;106&nbsp;124&nbsp;149&nbsp;170&nbsp;194&nbsp;217</spanx></c>
<c>19</c>
<c><spanx style="vbare">26&nbsp;&nbsp;33&nbsp;&nbsp;53&nbsp;&nbsp;64&nbsp;&nbsp;83&nbsp;117&nbsp;152&nbsp;173&nbsp;204&nbsp;225</spanx></c>
<c>20</c>
<c><spanx style="vbare">27&nbsp;&nbsp;34&nbsp;&nbsp;65&nbsp;&nbsp;95&nbsp;108&nbsp;129&nbsp;155&nbsp;174&nbsp;210&nbsp;225</spanx></c>
<c>21</c>
<c><spanx style="vbare">20&nbsp;&nbsp;26&nbsp;&nbsp;72&nbsp;&nbsp;99&nbsp;113&nbsp;131&nbsp;154&nbsp;176&nbsp;200&nbsp;219</spanx></c>
<c>22</c>
<c><spanx style="vbare">34&nbsp;&nbsp;43&nbsp;&nbsp;61&nbsp;&nbsp;78&nbsp;&nbsp;93&nbsp;114&nbsp;155&nbsp;177&nbsp;205&nbsp;229</spanx></c>
<c>23</c>
<c><spanx style="vbare">23&nbsp;&nbsp;29&nbsp;&nbsp;54&nbsp;&nbsp;97&nbsp;124&nbsp;138&nbsp;163&nbsp;179&nbsp;209&nbsp;229</spanx></c>
<c>24</c>
<c><spanx style="vbare">30&nbsp;&nbsp;38&nbsp;&nbsp;56&nbsp;&nbsp;89&nbsp;118&nbsp;129&nbsp;158&nbsp;178&nbsp;200&nbsp;231</spanx></c>
<c>25</c>
<c><spanx style="vbare">21&nbsp;&nbsp;29&nbsp;&nbsp;49&nbsp;&nbsp;63&nbsp;&nbsp;85&nbsp;111&nbsp;142&nbsp;163&nbsp;193&nbsp;222</spanx></c>
<c>26</c>
<c><spanx style="vbare">27&nbsp;&nbsp;48&nbsp;&nbsp;77&nbsp;103&nbsp;133&nbsp;158&nbsp;179&nbsp;196&nbsp;215&nbsp;232</spanx></c>
<c>27</c>
<c><spanx style="vbare">29&nbsp;&nbsp;47&nbsp;&nbsp;74&nbsp;&nbsp;99&nbsp;124&nbsp;151&nbsp;176&nbsp;198&nbsp;220&nbsp;237</spanx></c>
<c>28</c>
<c><spanx style="vbare">33&nbsp;&nbsp;42&nbsp;&nbsp;61&nbsp;&nbsp;76&nbsp;&nbsp;93&nbsp;121&nbsp;155&nbsp;174&nbsp;207&nbsp;225</spanx></c>
<c>29</c>
<c><spanx style="vbare">29&nbsp;&nbsp;53&nbsp;&nbsp;87&nbsp;112&nbsp;136&nbsp;154&nbsp;170&nbsp;188&nbsp;208&nbsp;227</spanx></c>
<c>30</c>
<c><spanx style="vbare">24&nbsp;&nbsp;30&nbsp;&nbsp;52&nbsp;&nbsp;84&nbsp;131&nbsp;150&nbsp;166&nbsp;186&nbsp;203&nbsp;229</spanx></c>
<c>31</c>
<c><spanx style="vbare">37&nbsp;&nbsp;48&nbsp;&nbsp;64&nbsp;&nbsp;84&nbsp;104&nbsp;118&nbsp;156&nbsp;177&nbsp;201&nbsp;230</spanx></c>
</texttable>

<texttable anchor="silk_nlsf_wb_codebook"
           title="Codebook Vectors for WB Normalized LSF Stage 1 Decoding">
<ttcol>I1</ttcol>
<ttcol>Codebook (Q8)</ttcol>
<c/>
<c><spanx style="vbare">&nbsp;0&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;&nbsp;3&nbsp;&nbsp;4&nbsp;&nbsp;&nbsp;5&nbsp;&nbsp;&nbsp;6&nbsp;&nbsp;&nbsp;7&nbsp;&nbsp;&nbsp;8&nbsp;&nbsp;&nbsp;9&nbsp;&nbsp;10&nbsp;&nbsp;11&nbsp;&nbsp;12&nbsp;&nbsp;13&nbsp;&nbsp;14&nbsp;&nbsp;15</spanx></c>
<c>0</c>
<c><spanx style="vbare">&nbsp;7&nbsp;23&nbsp;38&nbsp;54&nbsp;69&nbsp;&nbsp;85&nbsp;100&nbsp;116&nbsp;131&nbsp;147&nbsp;162&nbsp;178&nbsp;193&nbsp;208&nbsp;223&nbsp;239</spanx></c>
<c>1</c>
<c><spanx style="vbare">13&nbsp;25&nbsp;41&nbsp;55&nbsp;69&nbsp;&nbsp;83&nbsp;&nbsp;98&nbsp;112&nbsp;127&nbsp;142&nbsp;157&nbsp;171&nbsp;187&nbsp;203&nbsp;220&nbsp;236</spanx></c>
<c>2</c>
<c><spanx style="vbare">15&nbsp;21&nbsp;34&nbsp;51&nbsp;61&nbsp;&nbsp;78&nbsp;&nbsp;92&nbsp;106&nbsp;126&nbsp;136&nbsp;152&nbsp;167&nbsp;185&nbsp;205&nbsp;225&nbsp;240</spanx></c>
<c>3</c>
<c><spanx style="vbare">10&nbsp;21&nbsp;36&nbsp;50&nbsp;63&nbsp;&nbsp;79&nbsp;&nbsp;95&nbsp;110&nbsp;126&nbsp;141&nbsp;157&nbsp;173&nbsp;189&nbsp;205&nbsp;221&nbsp;237</spanx></c>
<c>4</c>
<c><spanx style="vbare">17&nbsp;20&nbsp;37&nbsp;51&nbsp;59&nbsp;&nbsp;78&nbsp;&nbsp;89&nbsp;107&nbsp;123&nbsp;134&nbsp;150&nbsp;164&nbsp;184&nbsp;205&nbsp;224&nbsp;240</spanx></c>
<c>5</c>
<c><spanx style="vbare">10&nbsp;15&nbsp;32&nbsp;51&nbsp;67&nbsp;&nbsp;81&nbsp;&nbsp;96&nbsp;112&nbsp;129&nbsp;142&nbsp;158&nbsp;173&nbsp;189&nbsp;204&nbsp;220&nbsp;236</spanx></c>
<c>6</c>
<c><spanx style="vbare">&nbsp;8&nbsp;21&nbsp;37&nbsp;51&nbsp;65&nbsp;&nbsp;79&nbsp;&nbsp;98&nbsp;113&nbsp;126&nbsp;138&nbsp;155&nbsp;168&nbsp;179&nbsp;192&nbsp;209&nbsp;218</spanx></c>
<c>7</c>
<c><spanx style="vbare">12&nbsp;15&nbsp;34&nbsp;55&nbsp;63&nbsp;&nbsp;78&nbsp;&nbsp;87&nbsp;108&nbsp;118&nbsp;131&nbsp;148&nbsp;167&nbsp;185&nbsp;203&nbsp;219&nbsp;236</spanx></c>
<c>8</c>
<c><spanx style="vbare">16&nbsp;19&nbsp;32&nbsp;36&nbsp;56&nbsp;&nbsp;79&nbsp;&nbsp;91&nbsp;108&nbsp;118&nbsp;136&nbsp;154&nbsp;171&nbsp;186&nbsp;204&nbsp;220&nbsp;237</spanx></c>
<c>9</c>
<c><spanx style="vbare">11&nbsp;28&nbsp;43&nbsp;58&nbsp;74&nbsp;&nbsp;89&nbsp;105&nbsp;120&nbsp;135&nbsp;150&nbsp;165&nbsp;180&nbsp;196&nbsp;211&nbsp;226&nbsp;241</spanx></c>
<c>10</c>
<c><spanx style="vbare">&nbsp;6&nbsp;16&nbsp;33&nbsp;46&nbsp;60&nbsp;&nbsp;75&nbsp;&nbsp;92&nbsp;107&nbsp;123&nbsp;137&nbsp;156&nbsp;169&nbsp;185&nbsp;199&nbsp;214&nbsp;225</spanx></c>
<c>11</c>
<c><spanx style="vbare">11&nbsp;19&nbsp;30&nbsp;44&nbsp;57&nbsp;&nbsp;74&nbsp;&nbsp;89&nbsp;105&nbsp;121&nbsp;135&nbsp;152&nbsp;169&nbsp;186&nbsp;202&nbsp;218&nbsp;234</spanx></c>
<c>12</c>
<c><spanx style="vbare">12&nbsp;19&nbsp;29&nbsp;46&nbsp;57&nbsp;&nbsp;71&nbsp;&nbsp;88&nbsp;100&nbsp;120&nbsp;132&nbsp;148&nbsp;165&nbsp;182&nbsp;199&nbsp;216&nbsp;233</spanx></c>
<c>13</c>
<c><spanx style="vbare">17&nbsp;23&nbsp;35&nbsp;46&nbsp;56&nbsp;&nbsp;77&nbsp;&nbsp;92&nbsp;106&nbsp;123&nbsp;134&nbsp;152&nbsp;167&nbsp;185&nbsp;204&nbsp;222&nbsp;237</spanx></c>
<c>14</c>
<c><spanx style="vbare">14&nbsp;17&nbsp;45&nbsp;53&nbsp;63&nbsp;&nbsp;75&nbsp;&nbsp;89&nbsp;107&nbsp;115&nbsp;132&nbsp;151&nbsp;171&nbsp;188&nbsp;206&nbsp;221&nbsp;240</spanx></c>
<c>15</c>
<c><spanx style="vbare">&nbsp;9&nbsp;16&nbsp;29&nbsp;40&nbsp;56&nbsp;&nbsp;71&nbsp;&nbsp;88&nbsp;103&nbsp;119&nbsp;137&nbsp;154&nbsp;171&nbsp;189&nbsp;205&nbsp;222&nbsp;237</spanx></c>
<c>16</c>
<c><spanx style="vbare">16&nbsp;19&nbsp;36&nbsp;48&nbsp;57&nbsp;&nbsp;76&nbsp;&nbsp;87&nbsp;105&nbsp;118&nbsp;132&nbsp;150&nbsp;167&nbsp;185&nbsp;202&nbsp;218&nbsp;236</spanx></c>
<c>17</c>
<c><spanx style="vbare">12&nbsp;17&nbsp;29&nbsp;54&nbsp;71&nbsp;&nbsp;81&nbsp;&nbsp;94&nbsp;104&nbsp;126&nbsp;136&nbsp;149&nbsp;164&nbsp;182&nbsp;201&nbsp;221&nbsp;237</spanx></c>
<c>18</c>
<c><spanx style="vbare">15&nbsp;28&nbsp;47&nbsp;62&nbsp;79&nbsp;&nbsp;97&nbsp;115&nbsp;129&nbsp;142&nbsp;155&nbsp;168&nbsp;180&nbsp;194&nbsp;208&nbsp;223&nbsp;238</spanx></c>
<c>19</c>
<c><spanx style="vbare">&nbsp;8&nbsp;14&nbsp;30&nbsp;45&nbsp;62&nbsp;&nbsp;78&nbsp;&nbsp;94&nbsp;111&nbsp;127&nbsp;143&nbsp;159&nbsp;175&nbsp;192&nbsp;207&nbsp;223&nbsp;239</spanx></c>
<c>20</c>
<c><spanx style="vbare">17&nbsp;30&nbsp;49&nbsp;62&nbsp;79&nbsp;&nbsp;92&nbsp;107&nbsp;119&nbsp;132&nbsp;145&nbsp;160&nbsp;174&nbsp;190&nbsp;204&nbsp;220&nbsp;235</spanx></c>
<c>21</c>
<c><spanx style="vbare">14&nbsp;19&nbsp;36&nbsp;45&nbsp;61&nbsp;&nbsp;76&nbsp;&nbsp;91&nbsp;108&nbsp;121&nbsp;138&nbsp;154&nbsp;172&nbsp;189&nbsp;205&nbsp;222&nbsp;238</spanx></c>
<c>22</c>
<c><spanx style="vbare">12&nbsp;18&nbsp;31&nbsp;45&nbsp;60&nbsp;&nbsp;76&nbsp;&nbsp;91&nbsp;107&nbsp;123&nbsp;138&nbsp;154&nbsp;171&nbsp;187&nbsp;204&nbsp;221&nbsp;236</spanx></c>
<c>23</c>
<c><spanx style="vbare">13&nbsp;17&nbsp;31&nbsp;43&nbsp;53&nbsp;&nbsp;70&nbsp;&nbsp;83&nbsp;103&nbsp;114&nbsp;131&nbsp;149&nbsp;167&nbsp;185&nbsp;203&nbsp;220&nbsp;237</spanx></c>
<c>24</c>
<c><spanx style="vbare">17&nbsp;22&nbsp;35&nbsp;42&nbsp;58&nbsp;&nbsp;78&nbsp;&nbsp;93&nbsp;110&nbsp;125&nbsp;139&nbsp;155&nbsp;170&nbsp;188&nbsp;206&nbsp;224&nbsp;240</spanx></c>
<c>25</c>
<c><spanx style="vbare">&nbsp;8&nbsp;15&nbsp;34&nbsp;50&nbsp;67&nbsp;&nbsp;83&nbsp;&nbsp;99&nbsp;115&nbsp;131&nbsp;146&nbsp;162&nbsp;178&nbsp;193&nbsp;209&nbsp;224&nbsp;239</spanx></c>
<c>26</c>
<c><spanx style="vbare">13&nbsp;16&nbsp;41&nbsp;66&nbsp;73&nbsp;&nbsp;86&nbsp;&nbsp;95&nbsp;111&nbsp;128&nbsp;137&nbsp;150&nbsp;163&nbsp;183&nbsp;206&nbsp;225&nbsp;241</spanx></c>
<c>27</c>
<c><spanx style="vbare">17&nbsp;25&nbsp;37&nbsp;52&nbsp;63&nbsp;&nbsp;75&nbsp;&nbsp;92&nbsp;102&nbsp;119&nbsp;132&nbsp;144&nbsp;160&nbsp;175&nbsp;191&nbsp;212&nbsp;231</spanx></c>
<c>28</c>
<c><spanx style="vbare">19&nbsp;31&nbsp;49&nbsp;65&nbsp;83&nbsp;100&nbsp;117&nbsp;133&nbsp;147&nbsp;161&nbsp;174&nbsp;187&nbsp;200&nbsp;213&nbsp;227&nbsp;242</spanx></c>
<c>29</c>
<c><spanx style="vbare">18&nbsp;31&nbsp;52&nbsp;68&nbsp;88&nbsp;103&nbsp;117&nbsp;126&nbsp;138&nbsp;149&nbsp;163&nbsp;177&nbsp;192&nbsp;207&nbsp;223&nbsp;239</spanx></c>
<c>30</c>
<c><spanx style="vbare">16&nbsp;29&nbsp;47&nbsp;61&nbsp;76&nbsp;&nbsp;90&nbsp;106&nbsp;119&nbsp;133&nbsp;147&nbsp;161&nbsp;176&nbsp;193&nbsp;209&nbsp;224&nbsp;240</spanx></c>
<c>31</c>
<c><spanx style="vbare">15&nbsp;21&nbsp;35&nbsp;50&nbsp;61&nbsp;&nbsp;73&nbsp;&nbsp;86&nbsp;&nbsp;97&nbsp;110&nbsp;119&nbsp;129&nbsp;141&nbsp;175&nbsp;198&nbsp;218&nbsp;237</spanx></c>
</texttable>

<t>
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
<figure align="center">
<artwork align="center"><![CDATA[
NLSF_Q15[k] = clamp(0,
               (cb1_Q8[k]<<7) + (res_Q10[k]<<14)/w_Q9[k], 32767) ,
]]></artwork>
</figure>
 where the division is exact integer division.
However, nothing in either the reconstruction process or the
 quantization process in the encoder thus far guarantees that the coefficients
 are monotonically increasing and separated well enough to ensure a stable
 filter.
When using the reference encoder, roughly 2% of frames violate this constraint.
The next section describes a stabilization procedure used to make these
 guarantees.
</t>

</section>

<section anchor="silk_nlsf_stabilization" title="Normalized LSF Stabilization">
<t>
The normalized LSF stabilization procedure is implemented in
 silk_NLSF_stabilize() (NLSF_stabilize.c).
This process ensures that consecutive values of the normalized LSF
 coefficients, NLSF_Q15[], are spaced some minimum distance apart
 (predetermined to be the 0.01 percentile of a large training set).
<xref target="silk_nlsf_min_spacing"/> 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.
</t>

<texttable anchor="silk_nlsf_min_spacing"
           title="Minimum Spacing for Normalized LSF Coefficients">
<ttcol>Coefficient</ttcol>
<ttcol align="right">NB and MB</ttcol>
<ttcol align="right">WB</ttcol>
 <c>0</c> <c>250</c> <c>100</c>
 <c>1</c>   <c>3</c>   <c>3</c>
 <c>2</c>   <c>6</c>  <c>40</c>
 <c>3</c>   <c>3</c>   <c>3</c>
 <c>4</c>   <c>3</c>   <c>3</c>
 <c>5</c>   <c>3</c>   <c>3</c>
 <c>6</c>   <c>4</c>   <c>5</c>
 <c>7</c>   <c>3</c>  <c>14</c>
 <c>8</c>   <c>3</c>  <c>14</c>
 <c>9</c>   <c>3</c>  <c>10</c>
<c>10</c> <c>461</c>  <c>11</c>
<c>11</c>       <c/>   <c>3</c>
<c>12</c>       <c/>   <c>8</c>
<c>13</c>       <c/>   <c>9</c>
<c>14</c>       <c/>   <c>7</c>
<c>15</c>       <c/>   <c>3</c>
<c>16</c>       <c/> <c>347</c>
</texttable>

<t>
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.
</t>
<t>
Let NDeltaMin_Q15[k] be the minimum required spacing for the current audio
 bandwidth from <xref target="silk_nlsf_min_spacing"/>.
First, the procedure finds the index i where
 NLSF_Q15[i]&nbsp;-&nbsp;NLSF_Q15[i-1]&nbsp;-&nbsp;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&nbsp;==&nbsp;0, it sets NLSF_Q15[0] to NDeltaMin_Q15[0], and if
 i&nbsp;==&nbsp;d_LPC, it sets NLSF_Q15[d_LPC-1] to
 (32768&nbsp;-&nbsp;NDeltaMin_Q15[d_LPC]).
For all other values of i, both NLSF_Q15[i-1] and NLSF_Q15[i] are updated as
 follows:
<figure align="center">
<artwork align="center"><![CDATA[
                                      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] .
]]></artwork>
</figure>
Then the procedure repeats again, until it has either executed 20 times or
 has stopped because the coefficients satisfy all the constraints.
</t>
<t>
After the 20th repetition of the above procedure, the following fallback
 procedure executes once.
First, the values of NLSF_Q15[k] for 0&nbsp;&lt;=&nbsp;k&nbsp;&lt;&nbsp;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
<figure align="center">
<artwork align="center"><![CDATA[
max(NLSF_Q15[k], NLSF_Q15[k-1] + NDeltaMin_Q15[k]) .
]]></artwork>
</figure>
Next, for each value of k from d_LPC-1 down to 0, NLSF_Q15[k] is set to
<figure align="center">
<artwork align="center"><![CDATA[
min(NLSF_Q15[k], NLSF_Q15[k+1] - NDeltaMin_Q15[k+1]) .
]]></artwork>
</figure>
</t>

</section>

<section anchor="silk_nlsf_interpolation" title="Normalized LSF Interpolation">
<t>
For 20&nbsp;ms SILK frames, the first half of the frame (i.e., the first two
 subframes) may use normalized LSF coefficients that are interpolated between
 the decoded LSFs for the most recent coded frame (in the same channel) and the
 current frame.
A Q2 interpolation factor follows the LSF coefficient indices in the bitstream,
 which is decoded using the PDF in <xref target="silk_nlsf_interp_pdf"/>.
This happens in silk_decode_indices() (decode_indices.c).
After either
<list style="symbols">
<t>An uncoded regular SILK frame in the side channel, or</t>
<t>A decoder reset (see <xref target="decoder-reset"/>),</t>
</list>
 the decoder still decodes this factor, but ignores its value and always uses
 4 instead.
For 10&nbsp;ms SILK frames, this factor is not stored at all.
</t>

<texttable anchor="silk_nlsf_interp_pdf"
           title="PDF for Normalized LSF Interpolation Index">
<ttcol>PDF</ttcol>
<c>{13, 22, 29, 11, 181}/256</c>
</texttable>

<t>
Let n2_Q15[k] be the normalized LSF coefficients decoded by the procedure in
 <xref target="silk_nlsfs"/>, 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&nbsp;ms
 frame, n1_Q15[k], are
<figure align="center">
<artwork align="center"><![CDATA[
n1_Q15[k] = n0_Q15[k] + (w_Q2*(n2_Q15[k] - n0_Q15[k]) >> 2) .
]]></artwork>
</figure>
This interpolation is performed in silk_decode_parameters()
 (decode_parameters.c).
</t>
</section>

<section anchor="silk_nlsf2lpc"
 title="Converting Normalized LSFs to LPC Coefficients">
<t>
Any LPC filter A(z) can be split into a symmetric part P(z) and an
 anti-symmetric part Q(z) such that
<figure align="center">
<artwork align="center"><![CDATA[
          d_LPC
           __         -k   1
A(z) = 1 - \  a[k] * z   = - * (P(z) + Q(z))
           /_              2
           k=1
]]></artwork>
</figure>
with
<figure align="center">
<artwork align="center"><![CDATA[
               -d_LPC-1      -1
P(z) = A(z) + z         * A(z  )

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

<texttable anchor="silk_nlsf_orderings"
           title="LSF Ordering for Polynomial Evaluation">
<ttcol>Coefficient</ttcol>
<ttcol align="right">NB and MB</ttcol>
<ttcol align="right">WB</ttcol>
 <c>0</c>  <c>0</c>  <c>0</c>
 <c>1</c>  <c>9</c> <c>15</c>
 <c>2</c>  <c>6</c>  <c>8</c>
 <c>3</c>  <c>3</c>  <c>7</c>
 <c>4</c>  <c>4</c>  <c>4</c>
 <c>5</c>  <c>5</c> <c>11</c>
 <c>6</c>  <c>8</c> <c>12</c>
 <c>7</c>  <c>1</c>  <c>3</c>
 <c>8</c>  <c>2</c>  <c>2</c>
 <c>9</c>  <c>7</c> <c>13</c>
<c>10</c>      <c/> <c>10</c>
<c>11</c>      <c/>  <c>5</c>
<c>12</c>      <c/>  <c>6</c>
<c>13</c>      <c/>  <c>9</c>
<c>14</c>      <c/> <c>14</c>
<c>15</c>      <c/>  <c>1</c>
</texttable>

<t>
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&nbsp;=&nbsp;(n[k]&nbsp;&gt;&gt;&nbsp;8) be the integer index and
 f&nbsp;=&nbsp;(n[k]&nbsp;&amp;&nbsp;255) be the fractional part of a given
 coefficient.
Then the re-ordered, approximated cosine, c_Q17[ordering[k]], is
<figure align="center">
<artwork align="center"><![CDATA[
c_Q17[ordering[k]] = (cos_Q12[i]*256
                      + (cos_Q12[i+1]-cos_Q12[i])*f + 4) >> 3 ,
]]></artwork>
</figure>
 where ordering[k] is the k'th entry of the column of
 <xref target="silk_nlsf_orderings"/> corresponding to the current audio
 bandwidth and cos_Q12[i] is the i'th entry of <xref target="silk_cos_table"/>.
</t>

<texttable anchor="silk_cos_table"
           title="Q12 Cosine Table for LSF Conversion">
<ttcol align="right">i</ttcol>
<ttcol align="right">+0</ttcol>
<ttcol align="right">+1</ttcol>
<ttcol align="right">+2</ttcol>
<ttcol align="right">+3</ttcol>
<c>0</c>
 <c>4096</c> <c>4095</c> <c>4091</c> <c>4085</c>
<c>4</c>
 <c>4076</c> <c>4065</c> <c>4052</c> <c>4036</c>
<c>8</c>
 <c>4017</c> <c>3997</c> <c>3973</c> <c>3948</c>
<c>12</c>
 <c>3920</c> <c>3889</c> <c>3857</c> <c>3822</c>
<c>16</c>
 <c>3784</c> <c>3745</c> <c>3703</c> <c>3659</c>
<c>20</c>
 <c>3613</c> <c>3564</c> <c>3513</c> <c>3461</c>
<c>24</c>
 <c>3406</c> <c>3349</c> <c>3290</c> <c>3229</c>
<c>28</c>
 <c>3166</c> <c>3102</c> <c>3035</c> <c>2967</c>
<c>32</c>
 <c>2896</c> <c>2824</c> <c>2751</c> <c>2676</c>
<c>36</c>
 <c>2599</c> <c>2520</c> <c>2440</c> <c>2359</c>
<c>40</c>
 <c>2276</c> <c>2191</c> <c>2106</c> <c>2019</c>
<c>44</c>
 <c>1931</c> <c>1842</c> <c>1751</c> <c>1660</c>
<c>48</c>
 <c>1568</c> <c>1474</c> <c>1380</c> <c>1285</c>
<c>52</c>
 <c>1189</c> <c>1093</c>  <c>995</c>  <c>897</c>
<c>56</c>
  <c>799</c>  <c>700</c>  <c>601</c>  <c>501</c>
<c>60</c>
  <c>401</c>  <c>301</c>  <c>201</c>  <c>101</c>
<c>64</c>
    <c>0</c> <c>-101</c> <c>-201</c> <c>-301</c>
<c>68</c>
 <c>-401</c> <c>-501</c> <c>-601</c> <c>-700</c>
<c>72</c>
 <c>-799</c> <c>-897</c> <c>-995</c> <c>-1093</c>
<c>76</c>
<c>-1189</c><c>-1285</c><c>-1380</c><c>-1474</c>
<c>80</c>
<c>-1568</c><c>-1660</c><c>-1751</c><c>-1842</c>
<c>84</c>
<c>-1931</c><c>-2019</c><c>-2106</c><c>-2191</c>
<c>88</c>
<c>-2276</c><c>-2359</c><c>-2440</c><c>-2520</c>
<c>92</c>
<c>-2599</c><c>-2676</c><c>-2751</c><c>-2824</c>
<c>96</c>
<c>-2896</c><c>-2967</c><c>-3035</c><c>-3102</c>
<c>100</c>
<c>-3166</c><c>-3229</c><c>-3290</c><c>-3349</c>
<c>104</c>
<c>-3406</c><c>-3461</c><c>-3513</c><c>-3564</c>
<c>108</c>
<c>-3613</c><c>-3659</c><c>-3703</c><c>-3745</c>
<c>112</c>
<c>-3784</c><c>-3822</c><c>-3857</c><c>-3889</c>
<c>116</c>
<c>-3920</c><c>-3948</c><c>-3973</c><c>-3997</c>
<c>120</c>
<c>-4017</c><c>-4036</c><c>-4052</c><c>-4065</c>
<c>124</c>
<c>-4076</c><c>-4085</c><c>-4091</c><c>-4095</c>
<c>128</c>
<c>-4096</c>        <c/>        <c/>        <c/>
</texttable>

<t>
Given the list of cosine values, silk_NLSF2A_find_poly() (NLSF2A.c)
 computes the coefficients of P and Q, described here via a simple recurrence.
Let p_Q16[k][j] and q_Q16[k][j] be the coefficients of the products of the
 first (k+1) root pairs for P and Q, with j indexing the coefficient number.
Only the first (k+2) coefficients are needed, as the products are symmetric.
Let p_Q16[0][0]&nbsp;=&nbsp;q_Q16[0][0]&nbsp;=&nbsp;1&lt;&lt;16,
 p_Q16[0][1]&nbsp;=&nbsp;-c_Q17[0], q_Q16[0][1]&nbsp;=&nbsp;-c_Q17[1], and
 d2&nbsp;=&nbsp;d_LPC/2.
As boundary conditions, assume
 p_Q16[k][j]&nbsp;=&nbsp;q_Q16[k][j]&nbsp;=&nbsp;0 for all
 j&nbsp;&lt;&nbsp;0.
Also, assume p_Q16[k][k+2]&nbsp;=&nbsp;p_Q16[k][k] and
 q_Q16[k][k+2]&nbsp;=&nbsp;q_Q16[k][k] (because of the symmetry).
Then, for 0&nbsp;&lt;&nbsp;k&nbsp;&lt;&nbsp;d2 and 0&nbsp;&lt;=&nbsp;j&nbsp;&lt;=&nbsp;k+1,
<figure align="center">
<artwork align="center"><![CDATA[
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) .
]]></artwork>
</figure>
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.
</t>
<t>
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&nbsp;&lt;=&nbsp;k&nbsp;&lt;&nbsp;d2:
<figure align="center">
<artwork align="center"><![CDATA[
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])) .
]]></artwork>
</figure>
The sum and difference of two terms from each of the p_Q16 and q_Q16
 coefficient lists reflect the (1&nbsp;+&nbsp;z**-1) and
 (1&nbsp;-&nbsp;z**-1) factors of P and Q, respectively.
The promotion of the expression from Q16 to Q17 implicitly scales the result
 by 1/2.
</t>
</section>

<section anchor="silk_lpc_range_limit"
 title="Limiting the Range of the LPC Coefficients">
<t>
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.
</t>
<t>
For each round, the process first finds the index k such that abs(a32_Q17[k])
 is largest, breaking ties by choosing the lowest value of k.
Then, it computes the corresponding Q12 precision value, maxabs_Q12, subject to
 an upper bound to avoid overflow in subsequent computations:
<figure align="center">
<artwork align="center"><![CDATA[
maxabs_Q12 = min((maxabs_Q17 + 16) >> 5, 163838) .
]]></artwork>
</figure>
If this is larger than 32767, the procedure derives the chirp factor,
 sc_Q16[0], to use in the bandwidth expansion as
<figure align="center">
<artwork align="center"><![CDATA[
                    (maxabs_Q12 - 32767) << 14
sc_Q16[0] = 65470 - -------------------------- ,
                    (maxabs_Q12 * (k+1)) >> 2
]]></artwork>
</figure>
 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&nbsp;&gt;&nbsp;0 when maxabs_Q12 is much greater than 32767, still slightly
 too large.
</t>
<t>
silk_bwexpander_32() (bwexpander_32.c) performs the bandwidth expansion (again,
 only when maxabs_Q12 is greater than 32767) using the following recurrence:
<figure align="center">
<artwork align="center"><![CDATA[
 a32_Q17[k] = (a32_Q17[k]*sc_Q16[k]) >> 16

sc_Q16[k+1] = (sc_Q16[0]*sc_Q16[k] + 32768) >> 16
]]></artwork>
</figure>
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.
</t>
<t>
After 10 rounds of bandwidth expansion are performed, they are simply saturated
 to 16 bits:
<figure align="center">
<artwork align="center"><![CDATA[
a32_Q17[k] = clamp(-32768, (a32_Q17[k] + 16) >> 5, 32767) << 5 .
]]></artwork>
</figure>
Because this performs the actual saturation in the Q12 domain, but converts the
 coefficients back to the Q17 domain for the purposes of prediction gain
 limiting, this step must be performed after the 10th round of bandwidth
 expansion, regardless of whether or not the Q12 version of any coefficient
 still overflows a 16-bit integer.
This saturation is not performed if maxabs_Q12 drops to 32767 or less prior to
 the 10th round.
</t>
</section>

<section anchor="silk_lpc_gain_limit"
 title="Limiting the Prediction Gain of the LPC Filter">
<t>
The prediction gain of an LPC synthesis filter is the square-root of the output
 energy when the filter is excited by a unit-energy impulse.
Even if the Q12 coefficients would fit, the resulting filter may still have a
 significant gain (especially for voiced sounds), making the filter unstable.
silk_NLSF2A() applies up to 18 additional rounds of bandwidth expansion to
 limit the prediction gain.
Instead of controlling the amount of bandwidth expansion using the prediction
 gain itself (which may diverge to infinity for an unstable filter),
 silk_NLSF2A() uses silk_LPC_inverse_pred_gain_QA() (LPC_inv_pred_gain.c) to
 compute the reflection coefficients associated with the filter.
The filter is stable if and only if the magnitude of these coefficients is
 sufficiently less than one.
The reflection coefficients, rc[k], can be computed using a simple Levinson
 recurrence, initialized with the LPC coefficients
 a[d_LPC-1][n]&nbsp;=&nbsp;a[n], and then updated via
<figure align="center">
<artwork align="center"><![CDATA[
    rc[k] = -a[k][k] ,

            a[k][n] - a[k][k-n-1]*rc[k]
a[k-1][n] = --------------------------- .
                             2
                    1 - rc[k]
]]></artwork>
</figure>
</t>
<t>
However, silk_LPC_inverse_pred_gain_QA() approximates this using fixed-point
 arithmetic to guarantee reproducible results across platforms and
 implementations.
Since small changes in the coefficients can make a stable filter unstable, it
 takes the real Q12 coefficients that will be used during reconstruction as
 input.
Thus, let
<figure align="center">
<artwork align="center"><![CDATA[
a32_Q12[n] = (a32_Q17[n] + 16) >> 5
]]></artwork>
</figure>
 be the Q12 version of the LPC coefficients that will eventually be used.
As a simple initial check, the decoder computes the DC response as
<figure align="center">
<artwork align="center"><![CDATA[
        d_PLC-1
          __
DC_resp = \   a32_Q12[n]
          /_
          n=0
]]></artwork>
</figure>
 and if DC_resp&nbsp;&gt;&nbsp;4096, the filter is unstable.
</t>
<t>
Increasing the precision of these Q12 coefficients to Q24 for intermediate
 computations allows more accurate computation of the reflection coefficients,
 so the decoder initializes the recurrence via
<figure align="center">
<artwork align="center"><![CDATA[
a32_Q24[d_LPC-1][n] = a32_Q12[n] << 12 .
]]></artwork>
</figure>
Then for each k from d_LPC-1 down to 0, if
 abs(a32_Q24[k][k])&nbsp;&gt;&nbsp;16773022, the filter is unstable and the
 recurrence stops.
Otherwise, row k-1 of a32_Q24 is computed from row k as
<figure align="center">
<artwork align="center"><![CDATA[
      rc_Q31[k] = -a32_Q24[k][k] << 7 ,

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

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

          b2[k] = b1[k] - 16 ,

                        (1<<29) - 1
     inv_Qb2[k] = ----------------------- ,
                  div_Q30[k] >> (b2[k]+1)

     err_Q29[k] = (1<<29)
                  - ((div_Q30[k]<<(15-b2[k]))*inv_Qb2[k] >> 16) ,