Definition of the Opus Audio Codec

The Opus decoder consists of two main blocks: the SILK decoder and the CELT decoder. At any given time, one or both of the SILK and CELT decoders may be active. The output of the Opus decode is the sum of the outputs from the SILK and CELT decoders with proper sample rate conversion and delay compensation on the SILK side, and optional decimation (when decoding to sample rates less than 48 kHz) on the CELT side, as illustrated in the block diagram below.

| Decoder |--->| Rate |----+ Bit- +---------+ | | | | Conversion | v stream | Range |---+ +---------+ +------------+ /---\ Audio ------->| Decoder | | + |------> | |---+ +---------+ +------------+ \---/ +---------+ | | CELT | | Decimation | ^ +->| Decoder |--->| (Optional) |----+ | | | | +---------+ +------------+ ]]>

Opus uses an entropy coder based on , which is itself a rediscovery of the FIFO arithmetic code introduced by . It is very similar to arithmetic encoding, except that encoding is done with digits in any base instead of with bits, so it is faster when using larger bases (i.e., an octet). All of the calculations in the range coder must use bit-exact integer arithmetic. Symbols may also be coded as "raw bits" packed directly into the bitstream, bypassing the range coder. These are packed backwards starting at the end of the frame, as illustrated in . 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.

: + + : : + +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ : | <- Boundary occurs at an arbitrary bit position : +-+-+-+ + : <- Raw bits data (packed LSb to MSb) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ ]]> Each symbol coded by the range coder is drawn from a finite alphabet and coded in a separate "context", which describes the size of the alphabet and the relative frequency of each symbol in that alphabet. Suppose there is a context with n symbols, identified with an index that ranges from 0 to n-1. The parameters needed to encode or decode a symbol in this context are represented by a three-tuple (fl[k], fh[k], ft), with 0 <= fl[k] < fh[k] <= ft <= 65535. The values of this tuple are derived from the probability model for the symbol, represented by traditional "frequency counts". Because Opus uses static contexts these are not updated as symbols are decoded. Let f[i] be the frequency of symbol i. Then the three-tuple corresponding to symbol k is given by

The range decoder extracts the symbols and integers encoded using the range encoder in . The range decoder maintains an internal state vector composed of the two-tuple (val,rng), representing the difference between the high end of the current range and the actual coded value, minus one, and the size of the current range, respectively. Both val and rng are 32-bit unsigned integer values. The decoder initializes rng to 128 and initializes val to 127 minus the top 7 bits of the first input octet. The remaining bit is saved for use in the renormalization procedure described in , which the decoder invokes immediately after initialization to read additional bits and establish the invariant that rng > 2**23.

Decoding a symbol is a two-step process. The first step determines a 16-bit unsigned value fs, which lies within the range of some symbol in the current context. The second step updates the range decoder state with the three-tuple (fl[k], fh[k], ft) corresponding to that symbol. The first step is implemented by ec_decode() (entdec.c), which computes

The divisions here are exact integer division. The decoder then identifies the symbol in the current context corresponding to fs; i.e., the value of k whose three-tuple (fl[k], fh[k], ft) satisfies fl[k] <= fs < fh[k]. It uses this tuple to update val according to

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

Otherwise, it updates rng using

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 particularly simple. After the updates, implemented by ec_dec_update() (entdec.c), the decoder normalizes the range using the procedure in the next section, and returns the index k.

To normalize the range, the decoder repeats the following process, implemented by ec_dec_normalize() (entdec.c), until rng > 2**23. If rng is already greater than 2**23, the entire process is skipped. First, it sets rng to (rng<<8). Then it reads the next octet of the payload and combines it with the left-over bit buffered from the previous octet to form the 8-bit value sym. It takes the left-over bit as the high bit (bit 7) of sym, and the top 7 bits of the octet it just read as the other 7 bits of sym. The remaining bit in the octet just read is buffered for use in the next iteration. If no more input octets remain, it uses zero bits instead. Then, it sets

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 . 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. 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.

| : +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ ^ ^ | End of data buffered by the range coder | ...-----------------------------------------------+ | | End of data consumed by raw bits +-------------------------------------------------------... ]]>

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.

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

The next is ec_dec_bit_logp() (entdec.c), which decodes a single binary symbol, replacing both the ec_decode() and ec_dec_update() steps. The context is described by a single parameter, logp, which is the absolute value of the base-2 logarithm of the probability of a "1". It is mathematically equivalent to calling ec_decode() with ft = (1<<logp), followed by ec_dec_update() with the 3-tuple (fl[k] = 0, fh[k] = (1<<logp)-1, ft = (1<<logp)) if the returned value of fs is less than (1<<logp)-1 (a "0" was decoded), and with (fl[k] = (1<<logp)-1, fh[k] = ft = (1<<logp)) otherwise (a "1" was decoded). The implementation requires no multiplications or divisions.

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

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 , 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.

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

The bit allocation routines in the CELT decoder need a conservative upper bound on the number of bits that have been used from the current frame thus far, including both range coder bits and raw bits. This drives allocation decisions that must match those made in the encoder. The upper bound is computed in the reference implementation to whole-bit precision by the function ec_tell() (entcode.h) and to fractional 1/8th bit precision by the function ec_tell_frac() (entcode.c). Like all operations in the range coder, it must be implemented in a bit-exact manner, and must produce exactly the same value returned by the same functions in the encoder after encoding the same symbols. ec_tell() is guaranteed to return ceil(ec_tell_frac()/8.0). In various places the codec will check to ensure there is enough room to contain a symbol before attempting to decode it. In practice, although the number of bits used so far is an upper bound, decoding a symbol whose probability model suggests it has a worst-case cost of p 1/8th bits may actually advance the return value of ec_tell_frac() by p-1, p, or p+1 1/8th bits, due to approximation error in that upper bound, truncation error in the range coder, and for large values of ft, modeling error in ec_dec_uint(). However, this error is bounded, and periodic calls to ec_tell() or ec_tell_frac() at precisely defined points in the decoding process prevent it from accumulating. For a range coder symbol that requires a whole number of bits (i.e., for which ft/(fh[k]-fl[k]) is a power of two), where there are at least p 1/8th bits available, decoding the symbol will never cause ec_tell() or ec_tell_frac() to exceed the size of the frame ("bust the budget"). In this case the return value of ec_tell_frac() will only advance by more than p 1/8th bits if there was an additional, fractional number of bits remaining, and it will never advance beyond the next whole-bit boundary, which is safe, since frames always contain a whole number of bits. However, when p is not a whole number of bits, an extra 1/8th bit is required to ensure that decoding the symbol will not bust the budget. The reference implementation keeps track of the total number of whole bits that have been processed by the decoder so far in the variable nbits_total, including the (possibly fractional) number of bits that are currently buffered, but not consumed, inside the range coder. nbits_total is initialized to 33 just after the initial range renormalization process completes (or equivalently, it can be initialized to 9 before the first renormalization). The extra two bits over the actual amount buffered by the range coder guarantees that it is an upper bound and that there is enough room for the encoder to terminate the stream. Each iteration through the range coder's renormalization loop increases nbits_total by 8. Reading raw bits increases nbits_total by the number of raw bits read.

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

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

> (l-16) , ]]> so that 32768 <= r_Q15 < 65536, an unsigned Q15 value representing the fractional part of rng. Then the following procedure can be used to add one bit of precision to l. First, update

> 15 . ]]> Then add the 16th bit of r_Q15 to l via

> 16) . ]]> Finally, if this bit was a 1, reduce r_Q15 by a factor of two via

> 1 , ]]> so that it once again lies in the range 32768 <= r_Q15 < 65536. This procedure is repeated three times to extend l to 1/8th bit precision. ec_tell_frac() then returns (nbits_total*8 - l).

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

An overview of the decoder is given in .

| Range |--->| Decode |---------------------------+ 1 | Decoder | 2 | Parameters |----------+ 5 | +---------+ +------------+ 4 | | 3 | | | \/ \/ \/ +------------+ +------------+ +------------+ | Generate |-->| LTP |-->| LPC | | Excitation | | Synthesis | | Synthesis | +------------+ +------------+ +------------+ ^ | | | +-------------------+----------------+ | 6 | +------------+ +-------------+ +-->| Stereo |-->| Sample Rate |--> 8 | Unmixing | 7 | Conversion | 8 +------------+ +-------------+ 1: Range encoded bitstream 2: Coded parameters 3: Pulses and gains 4: Pitch lags and LTP coefficients 5: LPC coefficients 6: Decoded signal (mono or mid-side stereo) 7: Unmixed signal (mono or left-right stereo) 8: Resampled signal ]]> Decoder block diagram. The decoder feeds the bitstream (1) to the range decoder from , and then decodes the parameters in it (2) using the procedures detailed in Sections through . These parameters (3, 4, 5) are used to generate an excitation signal (see ), which is fed to an optional long-term prediction (LTP) filter (voiced frames only, see ) and then a short-term prediction filter (see ), producing the decoded signal (6). For stereo streams, the mid-side representation is converted to separate left and right channels (7). The result is finally resampled to the desired output sample rate (e.g., 48 kHz) so that the resampled signal (8) can be mixed with the CELT layer.

Internally, the LP layer of a single Opus frame is composed of either a single 10 ms regular SILK frame or between one and three 20 ms regular SILK frames. A stereo Opus frame may double the number of regular SILK frames (up to a total of six), since it includes separate frames for a mid channel and, optionally, a side channel. Optional Low Bit-Rate Redundancy (LBRR) frames, which are reduced-bitrate encodings of previous SILK frames, may be included to aid in recovery from packet loss. If present, these appear before the regular SILK frames. They are in most respects identical to regular, active SILK frames, except that they are usually encoded with a lower bitrate. This draft uses "SILK frame" to refer to either one and "regular SILK frame" if it needs to draw a distinction between the two. Logically, each SILK frame is in turn composed of either two or four 5 ms subframes. Various parameters, such as the quantization gain of the excitation and the pitch lag and filter coefficients can vary on a subframe-by-subframe basis. Physically, the parameters for each subframe are interleaved in the bitstream, as described in the relevant sections for each parameter. All of these frames and subframes are decoded from the same range coder, with no padding between them. Thus packing multiple SILK frames in a single Opus frame saves, on average, half a byte per SILK frame. It also allows some parameters to be predicted from prior SILK frames in the same Opus frame, since this does not degrade packet loss robustness (beyond any penalty for merely using fewer, larger packets to store multiple frames). Stereo support in SILK uses a variant of mid-side coding, allowing a mono decoder to simply decode the mid channel. However, the data for the two channels is interleaved, so a mono decoder must still unpack the data for the side channel. It would be required to do so anyway for hybrid Opus frames, or to support decoding individual 20 ms frames. summarizes the overal grouping of the contents of the LP layer. Figures and illustrate the ordering of the various SILK frames for a 60&nbps;ms Opus frame, for both mono and stereo, respectively. Symbol(s) PDF(s) Condition VAD flags {1, 1}/2 LBRR flag {1, 1}/2 Per-frame LBRR flags LBRR Frame(s) Regular SILK Frame(s) Organization of the SILK layer of an Opus frame.

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. 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.

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

The LBRR frames, if present, contain an encoded representation of the signal immediately prior to the current Opus frame as if it were encoded with the current mode, frame size, audio bandwidth, and channel count, even if those differ from the prior Opus frame. When one of these parameters changes from one Opus frame to the next, this implies that the LBRR frames of the current Opus frame may not be simple drop-in replacements for the contents of the previous Opus frame. For example, when switching from 20 ms to 60 ms, the 60 ms Opus frame may contain LBRR frames covering up to three prior 20 ms Opus frames, even if those frames already contained LBRR frames covering some of the same time periods. When switching from 20 ms to 10 ms, the 10 ms Opus frame can contain an LBRR frame covering at most half the prior 20 ms Opus frame, potentially leaving a hole that needs to be concealed from even a single packet loss. When switching from mono to stereo, the LBRR frames in the first stereo Opus frame MAY contain a non-trivial side channel. In order to properly produce LBRR frames under all conditions, an encoder might need to buffer up to 60 ms of audio and re-encode it during these transitions. However, the reference implmentation opts to disable LBRR frames at the transition point for simplicity. The LBRR frames immediately follow the LBRR flags, prior to any regular SILK frames. describes their exact contents. LBRR frames do not include their own separate VAD flags. LBRR frames are only meant to be transmitted for active speech, thus all LBRR frames are treated as active. In a stereo Opus frame longer than 20 ms, although the per-frame LBRR flags for the mid channel are coded as a unit before the per-frame LBRR flags for the side channel, the LBRR frames themselves are interleaved. The decoder parses an LBRR frame for the mid channel of a given 20 ms interval (if present) and then immediately parses the corresponding LBRR frame for the side channel (if present), before proceeding to the next 20 ms interval.

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

Each SILK frame includes a set of side information that encodes The frame type and quantization type (), Quantization gains (), Short-term prediction filter coefficients (), An LSF interpolation weight (), Long-term prediction filter lags and gains (), and A linear congruential generator (LCG) seed (). The quantized excitation signal (see ) follows these at the end of the frame. details the overall organization of a SILK frame. Symbol(s) PDF(s) Condition Stereo Prediction Weights Mid-only Flag Frame Type Subframe Gains Normalized LSF Stage 1 Index Normalized LSF Stage 2 Residual Normalized LSF Interpolation Weight Primary Pitch Lag Voiced frame Subframe Pitch Contour Voiced frame Periodicity Index Voiced frame LTP Filter Voiced frame LTP Scaling LCG Seed Excitation Rate Level Excitation Pulse Counts Excitation Pulse Locations Non-zero pulse count Excitation LSb's Excitation Signs Order of the symbols in an individual SILK frame.

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 a 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. To summarize, these weights are coded if and only if This is a stereo Opus frame (), and The current SILK frame corresponds to the mid channel. The prediction weights are coded in three separate pieces, which are decoded by silk_stereo_decode_pred() (decode_stereo_pred.c). The first piece jointly codes the high-order part of a table index for both weights. The second piece codes the low-order part of each table index. The third piece codes an offset used to linearly interpolate between table indices. The details are as follows. Let n be an index decoded with the 25-element stage-1 PDF in . Then let i0 and i1 be indices decoded with the stage-2 and stage-3 PDFs in , respectively, and let i2 and i3 be two more indices decoded with the stage-2 and stage-3 PDFs, all in that order. Stage PDF Stage 1 {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 Stage 2 {85, 86, 85}/256 Stage 3 {51, 51, 52, 51, 51}/256 Then use n, i0, and i2 to form two table indices, wi0 and wi1, according to

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 . Then the two prediction weights, w0_Q13 and w1_Q13, are

> 16)*(2*i3 + 1) w0_Q13 = w_Q13[wi0] + ((w_Q13[wi0+1] - w_Q13[wi0])*6554) >> 16)*(2*i1 + 1) - w1_Q13 ]]> N.b., w1_Q13 is computed first here, because w0_Q13 depends on it. Index Weight (Q13) 0 -13732 1 -10050 2 -8266 3 -7526 4 -6500 5 -5000 6 -2950 7 -820 8 820 9 2950 10 5000 11 6500 12 7526 13 8266 14 10050 15 13732

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 This is a stereo Opus frame (see ), The current SILK frame corresponds to the mid channel, and Either This is a regular SILK frame, or This is an LBRR frame where the corresponding LBRR flags (see and ) indicate the side channel is not coded. It is omitted when there are no stereo weights, and it is also omitted for an LBRR frame when the corresponding LBRR flags indicate the side channel is coded. When the flag is present, the decoder reads a single value using the PDF in , 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 used during the stereo unmixing process. 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 ) SHOULD be invoked to recover a side channel signal. PDF {192, 64}/256

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

A separate quantization gain is coded for each 5 ms subframe. These gains control the step size between quantization levels of the excitation signal and, therefore, the quality of the reconstruction. They are independent of the pitch gains coded for voiced frames. The quantization gains are themselves uniformly quantized to 6 bits on a log scale, giving them a resolution of approximately 1.369 dB and a range of approximately 1.94 dB to 88.21 dB. The subframe gains are either coded independently, or relative to the gain from the most recent coded subframe in the same channel. Independent coding is used if and only if This is the first subframe in the current SILK frame, and Either This is the first LBRR frame for this channel in the current Opus frame, This is an LBRR frame where the LBRR flags (see and ) indicate the previous LBRR frame in the same channel is not coded, or This is the first regular SILK frame for this channel in the current Opus frame. There are a few subtle points here that may benefit from some clarification. The rules for uncoded LBRR frames are very different from the rules for regular SILK frames for the side channel of a stereo Opus frame. Both allow gaps in the sequence of coded frames for a channel, the former based on the LBRR flags, and the latter on the mid-only flag (from ). LBRR frames do not use relative coding to predict across these gaps, while regular SILK frames in the side channel do. In particular, in a 60 ms stereo Opus frame, if the first and third regular SILK frames in the side channel are coded, but the second is not, the first subframe of the third frame is still coded relative to the last subframe in the first frame. In contrast, in a similar situation with LBRR frames, the first subframe of the third frame would use independent coding, even if the mid-only flag for the second frame was 0. In an independently coded subframe gain, the 3 most significant bits of the quantization gain are decoded using a PDF selected from based on the decoded signal type (see ). Signal Type PDF Inactive {32, 112, 68, 29, 12, 1, 1, 1}/256 Unvoiced {2, 17, 45, 60, 62, 47, 19, 4}/256 Voiced {1, 3, 26, 71, 94, 50, 9, 2}/256 The 3 least significant bits are decoded using a uniform PDF: PDF {32, 32, 32, 32, 32, 32, 32, 32}/256 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 yields a delta gain index between 0 and 40, inclusive. PDF {6, 5, 11, 31, 132, 21, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256 The following formula translates this index into a quantization gain for the current subframe using the gain from the previous subframe:

The value here is not clamped at 0, and may decrease as far as -16 over the course of consecutive subframes within a single Opus frame. silk_gains_dequant() (gain_quant.c) dequantizes log_gain for the k'th subframe and converts it into a linear Q16 scale factor via

>16) + 2090) ]]> The function silk_log2lin() (log2lin.c) computes an approximation of 2**(inLog_Q7/128.0), where inLog_Q7 is its Q7 input. Let i = inLog_Q7>>7 be the integer part of inLogQ7 and f = inLog_Q7&127 be the fractional part. Then, if i < 16, then

>16)+f)>>7)*(1< yields the approximate exponential. Otherwise, silk_log2lin uses

>16)+f)*((1<>7) . ]]> The final Q16 gain values lies between 4096 and 1686110208, inclusive (representing scale factors of 0.0625 to 25728, respectively).

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 ). Because of non-linear effects in the decoding process, an implementation SHOULD match the fixed-point arithmetic described in this section exactly. An encoder SHOULD also use the same process. The normalized LSFs are coded using a two-stage vector quantizer (VQ) ( and ). 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 (), the decoder runs them through a stabilization process (), interpolates them between frames (), converts them back into LPC coefficients (), and then runs them through further processes to limit the range of the coefficients () and the gain of the filter (). All of this is necessary to ensure the reconstruction process is stable.

The first VQ stage uses a 32-element codebook, coded with one of the PDFs in , 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 and , but they are not needed until the last stages of reconstructing the LSF coefficients. Audio Bandwidth Signal Type PDF NB or MB Inactive or unvoiced {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 NB or MB Voiced {1, 10, 1, 8, 3, 8, 8, 14, 13, 14, 1, 14, 12, 13, 11, 11, 12, 11, 10, 10, 11, 8, 9, 8, 7, 8, 1, 1, 6, 1, 6, 5}/256 WB Inactive or unvoiced {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 WB Voiced {1, 4, 16, 5, 18, 11, 5, 14, 15, 1, 3, 12, 13, 14, 14, 6, 14, 12, 2, 6, 1, 12, 12, 11, 10, 3, 10, 5, 1, 1, 1, 3}/256

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 , and the 8 (i...p) for WB frames given in . Which PDF is used for which coefficient is driven by the index, I1, decoded in the first stage. lists the letter of the corresponding PDF for each normalized LSF coefficient for NB and MB, and lists the same information for WB. Codebook PDF a {1, 1, 1, 15, 224, 11, 1, 1, 1}/256 b {1, 1, 2, 34, 183, 32, 1, 1, 1}/256 c {1, 1, 4, 42, 149, 55, 2, 1, 1}/256 d {1, 1, 8, 52, 123, 61, 8, 1, 1}/256 e {1, 3, 16, 53, 101, 74, 6, 1, 1}/256 f {1, 3, 17, 55, 90, 73, 15, 1, 1}/256 g {1, 7, 24, 53, 74, 67, 26, 3, 1}/256 h {1, 1, 18, 63, 78, 58, 30, 6, 1}/256 Codebook PDF i {1, 1, 1, 9, 232, 9, 1, 1, 1}/256 j {1, 1, 2, 28, 186, 35, 1, 1, 1}/256 k {1, 1, 3, 42, 152, 53, 2, 1, 1}/256 l {1, 1, 10, 49, 126, 65, 2, 1, 1}/256 m {1, 4, 19, 48, 100, 77, 5, 1, 1}/256 n {1, 1, 14, 54, 100, 72, 12, 1, 1}/256 o {1, 1, 15, 61, 87, 61, 25, 4, 1}/256 p {1, 7, 21, 50, 77, 81, 17, 1, 1}/256 I1 Coefficient 0 1 2 3 4 5 6 7 8 9 0 a a a a a a a a a a 1 b d b c c b c b b b 2 c b b b b b b b b b 3 b c c c c b c b b b 4 c d d d d c c c c c 5 a f d d c c c c b b g a c c c c c c c c b 7 c d g e e e f e f f 8 c e f f e f e g e e 9 c e e h e f e f f e 10 e d d d c d c c c c 11 b f f g e f e f f f 12 c h e g f f f f f f 13 c h f f f f f g f e 14 d d f e e f e f e e 15 c d d f f e e e e e 16 c e e g e f e f f f 17 c f e g f f f e f e 18 c h e f e f e f f f 19 c f e g h g f g f e 20 d g h e g f f g e f 21 c h g e e e f e f f 22 e f f e g g f g f e 23 c f f g f g e g e e 24 e f f f d h e f f e 25 c d e f f g e f f e 26 c d c d d e c d d d 27 b b c c c c c d c c 28 e f f g g g f g e f 29 d f f e e e e d d c 30 c f d h f f e e f e 31 e e f e f g f g f e I1 Coefficient 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 i i i i i i i i i i i i i i i i 1 k l l l l l k k k k k j j j i l 2 k n n l p m m n k n m n n m l l 3 i k j k k j j j j j i i i i i j 4 i o n m o m p n m m m n n m m l 5 i l n n m l l n l l l l l l k m 6 i i i i i i i i i i i i i i i i 7 i k o l p k n l m n n m l l k l 8 i o k o o m n m o n m m n l l l 9 k j i i i i i i i i i i i i i i j0 i j i i i i i i i i i i i i i j 11 k k l m n l l l l l l l k k j l 12 k k l l m l l l l l l l l k j l 13 l m m m o m m n l n m m n m l m 14 i o m n m p n k o n p m m l n l 15 i j i j j j j j j j i i i i j i 16 j o n p n m n l m n m m m l l m 17 j l l m m l l n k l l n n n l m 18 k l l k k k l k j k j k j j j m 19 i k l n l l k k k j j i i i i i 20 l m l n l l k k j j j j j k k m 21 k o l p p m n m n l n l l k l l 22 k l n o o l n l m m l l l l k m 23 j l l m m m m l n n n l j j j j 24 k n l o o m p m m n l m m l l l 25 i o j j i i i i i i i i i i i i 26 i o o l n k n n l m m p p m m m 27 l l p l n m l l l k k l l l k l 28 i i j i i i k j k j j k k k j j 29 i l k n l l k l k j i i j i i j 30 l n n m p n l l k l k k j i j i 31 k l n l m l l l k j k o m i i i Decoding the second stage residual proceeds as follows. For each coefficient, the decoder reads a symbol using the PDF corresponding to I1 from either or , 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 , and adds the value of this second symbol to the index, using the same sign. This gives the index, I2[k], a total range of -10 to 10, inclusive. PDF {156, 60, 24, 9, 4, 2, 1}/256 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. contains lists of prediction weights for each coefficient. There are two lists for NB and MB, and another two lists for WB, giving two possible prediction weights for each coefficient. Coefficient A B C D 0 179 116 175 68 1 138 67 148 62 2 140 82 160 66 3 148 59 176 60 4 151 92 178 72 5 149 72 173 117 6 153 100 174 85 7 151 89 164 90 8 163 92 177 118 9 174 136 10 196 151 11 182 142 12 198 160 13 192 142 14 182 155 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. gives the selections for each coefficient for NB and MB, and gives the selections for WB. Let d_LPC be the order of the codebook, i.e., 10 for NB and MB, and 16 for WB, and let pred_Q8[k] be the weight for the k'th coefficient selected by this process for 0 <= k < d_LPC-1. Then, the stage-2 residual for each coefficient is computed via

>8 : 0) + ((((I2[k]<<10) + sign(I2[k])*102)*qstep)>>16) , ]]> where qstep is the Q16 quantization step size, which is 11796 for NB and MB and 9830 for WB (representing step sizes of approximately 0.18 and 0.15, respectively). I1 Coefficient 0 1 2 3 4 5 6 7 8 0 A B A A A A A A A 1 B A A A A A A A A 2 A A A A A A A A A 3 B B B A A A A B A 4 A B A A A A A A A 5 A B A A A A A A A 6 B A B B A A A B A 7 A B B A A B B A A 8 A A B B A B A B B 9 A A B B A A B B B 10 A A A A A A A A A 11 A B A B B B B B A 12 A B A B B B B B A 13 A B B B B B B B A 14 B A B B A B B B B 15 A B B B B B A B A 16 A A B B A B A B A 17 A A B B B A B B B 18 A B B A A B B B A 19 A A A B B B A B A 20 A B B A A B A B A 21 A B B A A A B B A 22 A A A A A B B B B 23 A A B B A A A B B 24 A A A B A B B B B 25 A B B B B B B B A 26 A A A A A A A A A 27 A A A A A A A A A 28 A A B A B B A B A 29 A A A B A A A A A 30 A A A B B A B A B 31 B A B B A B B B B I1 Coefficient 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 C C C C C C C C C C C C C C D 1 C C C C C C C C C C C C C C C 2 C C D C C D D D C D D D D C C 3 C C C C C C C C C C C C D C C 4 C D D C D C D D C D D D D D C 5 C D C C C C C C C C C C C C C 6 D C C C C C C C C C C D C D C 7 C D D C C C D C D D D C D C D 8 C D C D D C D C D C D D D D D 9 C C C C C C C C C C C C C C D 10 C D C C C C C C C C C C C C C 11 C C D C D D D D D D D C D C C 12 C C D C C D C D C D C C D C C 13 C C C C D D C D C D D D D C C 14 C D C C C D D C D D D C D D D 15 C C D D C C C C C C C C D D C 16 C D D C D C D D D D D C D C C 17 C C D C C C C D C C D D D C C 18 C C C C C C C C C C C C C C D 19 C C C C C C C C C C C C D C C 20 C C C C C C C C C C C C C C C 21 C D C D C D D C D C D C D D C 22 C C D D D D C D D C C D D C C 23 C D D C D C D C D C C C C D C 24 C C C D D C D C D D D D D D D 25 C C C C C C C C C C C C C C D 26 C D D C C C D D C C D D D D D 27 C C C C C D C D D D D C D D D 28 C C C C C C C C C C C C C C D 29 C C C C C C C C C C C C C C D 30 D C C C C C C C C C C D C C C 31 C C D C C D D D C C D C C D C

Once the stage-1 index I1 and the stage-2 residual res_Q10[] have been decoded, the final normalized LSF coefficients can be reconstructed. The spectral distortion introduced by the quantization of each LSF coefficient varies, so the stage-2 residual is weighted accordingly, using the low-complexity Inverse Harmonic Mean Weighting (IHMW) function proposed in . 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 or . Then for 0 <= k < d_LPC the following expression computes the square of the weight as a Q18 value:

where cb1_Q8[-1] = 0 and cb1_Q8[d_LPC] = 256, and the division is exact integer division. This is reduced to an unsquared, Q9 value using the following square-root approximation:

>(i-8)) & 127 y = ((i&1) ? 32768 : 46214) >> ((32-i)>>1) w_Q9[k] = y + ((213*f*y)>>16) ]]> The cb1_Q8[] vector completely determines these weights, and they may be tabulated and stored as 13-bit unsigned values (with a range of 1819 to 5227, inclusive) to avoid computing them when decoding. The reference implementation already requires code to compute these weights on unquantized coefficients in the encoder, in silk_NLSF_VQ_weights_laroia() (NLSF_VQ_weights_laroia.c) and its callers, so it reuses that code in the decoder instead of using a pre-computed table to reduce the amount of ROM required. I1 Codebook (Q8) 0 1 2 3 4 5 6 7 8 9 0 12 35 60 83 108 132 157 180 206 228 1 15 32 55 77 101 125 151 175 201 225 2 19 42 66 89 114 137 162 184 209 230 3 12 25 50 72 97 120 147 172 200 223 4 26 44 69 90 114 135 159 180 205 225 5 13 22 53 80 106 130 156 180 205 228 6 15 25 44 64 90 115 142 168 196 222 7 19 24 62 82 100 120 145 168 190 214 8 22 31 50 79 103 120 151 170 203 227 9 21 29 45 65 106 124 150 171 196 224 10 30 49 75 97 121 142 165 186 209 229 11 19 25 52 70 93 116 143 166 192 219 12 26 34 62 75 97 118 145 167 194 217 13 25 33 56 70 91 113 143 165 196 223 14 21 34 51 72 97 117 145 171 196 222 15 20 29 50 67 90 117 144 168 197 221 16 22 31 48 66 95 117 146 168 196 222 17 24 33 51 77 116 134 158 180 200 224 18 21 28 70 87 106 124 149 170 194 217 19 26 33 53 64 83 117 152 173 204 225 20 27 34 65 95 108 129 155 174 210 225 21 20 26 72 99 113 131 154 176 200 219 22 34 43 61 78 93 114 155 177 205 229 23 23 29 54 97 124 138 163 179 209 229 24 30 38 56 89 118 129 158 178 200 231 25 21 29 49 63 85 111 142 163 193 222 26 27 48 77 103 133 158 179 196 215 232 27 29 47 74 99 124 151 176 198 220 237 28 33 42 61 76 93 121 155 174 207 225 29 29 53 87 112 136 154 170 188 208 227 30 24 30 52 84 131 150 166 186 203 229 31 37 48 64 84 104 118 156 177 201 230 I1 Codebook (Q8) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 7 23 38 54 69 85 100 116 131 147 162 178 193 208 223 239 1 13 25 41 55 69 83 98 112 127 142 157 171 187 203 220 236 2 15 21 34 51 61 78 92 106 126 136 152 167 185 205 225 240 3 10 21 36 50 63 79 95 110 126 141 157 173 189 205 221 237 4 17 20 37 51 59 78 89 107 123 134 150 164 184 205 224 240 5 10 15 32 51 67 81 96 112 129 142 158 173 189 204 220 236 6 8 21 37 51 65 79 98 113 126 138 155 168 179 192 209 218 7 12 15 34 55 63 78 87 108 118 131 148 167 185 203 219 236 8 16 19 32 36 56 79 91 108 118 136 154 171 186 204 220 237 9 11 28 43 58 74 89 105 120 135 150 165 180 196 211 226 241 10 6 16 33 46 60 75 92 107 123 137 156 169 185 199 214 225 11 11 19 30 44 57 74 89 105 121 135 152 169 186 202 218 234 12 12 19 29 46 57 71 88 100 120 132 148 165 182 199 216 233 13 17 23 35 46 56 77 92 106 123 134 152 167 185 204 222 237 14 14 17 45 53 63 75 89 107 115 132 151 171 188 206 221 240 15 9 16 29 40 56 71 88 103 119 137 154 171 189 205 222 237 16 16 19 36 48 57 76 87 105 118 132 150 167 185 202 218 236 17 12 17 29 54 71 81 94 104 126 136 149 164 182 201 221 237 18 15 28 47 62 79 97 115 129 142 155 168 180 194 208 223 238 19 8 14 30 45 62 78 94 111 127 143 159 175 192 207 223 239 20 17 30 49 62 79 92 107 119 132 145 160 174 190 204 220 235 21 14 19 36 45 61 76 91 108 121 138 154 172 189 205 222 238 22 12 18 31 45 60 76 91 107 123 138 154 171 187 204 221 236 23 13 17 31 43 53 70 83 103 114 131 149 167 185 203 220 237 24 17 22 35 42 58 78 93 110 125 139 155 170 188 206 224 240 25 8 15 34 50 67 83 99 115 131 146 162 178 193 209 224 239 26 13 16 41 66 73 86 95 111 128 137 150 163 183 206 225 241 27 17 25 37 52 63 75 92 102 119 132 144 160 175 191 212 231 28 19 31 49 65 83 100 117 133 147 161 174 187 200 213 227 242 29 18 31 52 68 88 103 117 126 138 149 163 177 192 207 223 239 30 16 29 47 61 76 90 106 119 133 147 161 176 193 209 224 240 31 15 21 35 50 61 73 86 97 110 119 129 141 175 198 218 237 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

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.

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). 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. Coefficient NB and MB WB 0 250 100 1 3 3 2 6 40 3 3 3 4 3 3 5 3 3 6 4 5 7 3 14 8 3 14 9 3 10 10 461 11 11 3 12 8 13 9 14 7 15 3 16 347 The procedure starts off by trying to make small adjustments which attempt to minimize the amount of distortion introduced. After 20 such adjustments, it falls back to a more direct method which guarantees the constraints are enforced but may require large adjustments. Let NDeltaMin_Q15[k] be the minimum required spacing for the current audio bandwidth from . First, the procedure finds the index i where NLSF_Q15[i] - NLSF_Q15[i-1] - NDeltaMin_Q15[i] is the smallest, breaking ties by using the lower value of i. If this value is non-negative, then the stabilization stops; the coefficients satisfy all the constraints. Otherwise, if i == 0, it sets NLSF_Q15[0] to NDeltaMin_Q15[0], and if i == d_LPC, it sets NLSF_Q15[d_LPC-1] to (32768 - NDeltaMin_Q15[d_LPC]). For all other values of i, both NLSF_Q15[i-1] and NLSF_Q15[i] are updated as follows:

>1) + \ NDeltaMin[k] /_ k=0 d_LPC __ max_center_Q15 = 32768 - (NDeltaMin[i]>>1) - \ NDeltaMin[k] /_ k=i+1 center_freq_Q15 = clamp(min_center_Q15[i], (NLSF_Q15[i-1] + NLSF_Q15[i] + 1)>>1, max_center_Q15[i]) NLSF_Q15[i-1] = center_freq_Q15 - (NDeltaMin_Q15[i]>>1) NLSF_Q15[i] = NLSF_Q15[i-1] + NDeltaMin_Q15[i] . ]]> Then the procedure repeats again, until it has either executed 20 times or has stopped because the coefficients satisfy all the constraints. After the 20th repetition of the above procedure, the following fallback procedure executes once. First, the values of NLSF_Q15[k] for 0 <= k < d_LPC are sorted in ascending order. Then for each value of k from 0 to d_LPC-1, NLSF_Q15[k] is set to

Next, for each value of k from d_LPC-1 down to 0, NLSF_Q15[k] is set to

For 20 ms SILK frames, the first half of the frame (i.e., the first two subframes) may use normalized LSF coefficients that are interpolated between the decoded LSFs for the most recent coded frame (in the same channel) and the current frame. A Q2 interpolation factor follows the LSF coefficient indices in the bitstream, which is decoded using the PDF in . This happens in silk_decode_indices() (decode_indices.c). For the first frame after a decoder reset, when no prior LSF coefficients are available, the decoder still decodes this factor, but ignores its value and always uses 4 instead. For 10 ms SILK frames, this factor is not stored at all. PDF {13, 22, 29, 11, 181}/256 Let n2_Q15[k] be the normalized LSF coefficients decoded by the procedure in , n0_Q15[k] be the LSF coefficients decoded for the prior frame, and w_Q2 be the interpolation factor. Then the normalized LSF coefficients used for the first half of a 20 ms frame, n1_Q15[k], are

> 2) . ]]> This interpolation is performed in silk_decode_parameters() (decode_parameters.c).

Any LPC filter A(z) can be split into a symmetric part P(z) and an anti-symmetric part Q(z) such that

with

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

However, SILK performs this reconstruction using a fixed-point approximation so that all decoders can reproduce it in a bit-exact manner to avoid prediction drift. The function silk_NLSF2A() (NLSF2A.c) implements this procedure. To start, it approximates cos(pi*n[k]) using a table lookup with linear interpolation. The encoder SHOULD use the inverse of this piecewise linear approximation, rather than the true inverse of the cosine function, when deriving the normalized LSF coefficients. The top 7 bits of each normalized LSF coefficient index a value in the table, and the next 8 bits interpolate between it and the next value. Let i = n[k]>>8 be the integer index and f = n[k]&255 be the fractional part of a given coefficient. Then the approximated cosine, c_Q17[k], is

> 4 , ]]> where cos_Q13[i] is the corresponding entry of . 0 1 2 3 0 8192 8190 8182 8170 4 8152 8130 8104 8072 8 8034 7994 7946 7896 12 7840 7778 7714 7644 16 7568 7490 7406 7318 20 7226 7128 7026 6922 24 6812 6698 6580 6458 28 6332 6204 6070 5934 32 5792 5648 5502 5352 36 5198 5040 4880 4718 40 4552 4382 4212 4038 44 3862 3684 3502 3320 48 3136 2948 2760 2570 52 2378 2186 1990 1794 56 1598 1400 1202 1002 60 802 602 402 202 64 0 -202 -402 -602 68 -802-1002-1202-1400 72 -1598-1794-1990-2186 76 -2378-2570-2760-2948 80 -3136-3320-3502-3684 84 -3862-4038-4212-4382 88 -4552-4718-4880-5040 92 -5198-5352-5502-5648 96 -5792-5934-6070-6204 100 -6332-6458-6580-6698 104 -6812-6922-7026-7128 108 -7226-7318-7406-7490 112 -7568-7644-7714-7778 116 -7840-7896-7946-7994 120 -8034-8072-8104-8130 124 -8152-8170-8182-8190 128 -8192 Given the list of cosine values, silk_NLSF2A_find_poly() (NLSF2A.c) computes the coefficients of P and Q, described here via a simple recurrence. Let p_Q16[k][j] and q_Q16[k][j] be the coefficients of the products of the first (k+1) root pairs for P and Q, with j indexing the coefficient number. Only the first (k+2) coefficients are needed, as the products are symmetric. Let p_Q16[0][0] = q_Q16[0][0] = 1<<16, p_Q16[0][1] = -c_Q17[0], q_Q16[0][1] = -c_Q17[1], and d2 = d_LPC/2. As boundary conditions, assume p_Q16[k][j] = q_Q16[k][j] = 0 for all j < 0. Also, assume p_Q16[k][k+2] = p_Q16[k][k] and q_Q16[k][k+2] = q_Q16[k][k] (because of the symmetry). Then, for 0 < k < d2 and 0 <= j <= k+1,

>16) , q_Q16[k][j] = q_Q16[k-1][j] + q_Q16[k-1][j-2] - ((c_Q17[2*k+1]*q_Q16[k-1][j-1] + 32768)>>16) . ]]> The use of Q17 values for the cosine terms in an otherwise Q16 expression implicitly scales them by a factor of 2. The multiplications in this recurrence may require up to 48 bits of precision in the result to avoid overflow. In practice, each row of the recurrence only depends on the previous row, so an implementation does not need to store all of them. silk_NLSF2A() uses the values from the last row of this recurrence to reconstruct a 32-bit version of the LPC filter (without the leading 1.0 coefficient), a32_Q17[k], 0 <= k < d2:

The sum and difference of two terms from each of the p_Q16 and q_Q16 coefficient lists reflect the (1 + z**-1) and (1 - z**-1) factors of P and Q, respectively. The promotion of the expression from Q16 to Q17 implicitly scales the result by 1/2.

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

> 5, 163838) . ]]> If this is larger than 32767, the procedure derives the chirp factor, sc_Q16[0], to use in the bandwidth expansion as

> 2 ]]> where the division here is exact integer division. This is an approximation of the chirp factor needed to reduce the target coefficient to 32767, though it is both less than 0.999 and, for k > 0 when maxabs_Q12 is much greater than 32767, still slightly too large. silk_bwexpander_32() (bwexpander_32.c) performs the bandwidth expansion (again, only when maxabs_Q12 is greater than 32767) using the following recurrence:

> 16 sc_Q16[k+1] = (sc_Q16[0]*sc_Q16[k] + 32768) >> 16 ]]> The first multiply may require up to 48 bits of precision in the result to avoid overflow. The second multiply must be unsigned to avoid overflow with only 32 bits of precision. The reference implementation uses a slightly more complex formulation that avoids the 32-bit overflow using signed multiplication, but is otherwise equivalent. After 10 rounds of bandwidth expansion are performed, they are simply saturated to 16 bits:

> 5, 32767) << 5 . ]]> Because this performs the actual saturation in the Q12 domain, but converts the coefficients back to the Q17 domain for the purposes of prediction gain limiting, this step must be performed after the 10th round of bandwidth expansion, regardless of whether or not the Q12 version of any coefficient still overflows a 16-bit integer. This saturation is not performed if maxabs_Q12 drops to 32767 or less prior to the 10th round.

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

However, silk_LPC_inverse_pred_gain_QA() approximates this using fixed-point arithmetic to guarantee reproducible results across platforms and implementations. It is important to run on the real Q12 coefficients that will be used during reconstruction, because small changes in the coefficients can make a stable filter unstable, but increasing the precision to Q24 allows more accurate computation of the reflection coefficients. Thus, let

> 5) << 12 ]]> be the Q24 representation of the Q12 version of the LPC coefficients that will eventually be used. Then for each k from d_LPC-1 down to 0, if abs(a32_Q24[k][k]) > 16773022, the filter is unstable and the recurrence stops. Otherwise, row k-1 of a32_Q24 is computed from row k as

> 32) , b1[k] = ilog(div_Q30[k]) , b2[k] = b1[k] - 16 , (1<<29) - 1 inv_Qb2[k] = ----------------------- , div_Q30[k] >> (b2[k]+1) err_Q29[k] = (1<<29) - ((div_Q30[k]<<(15-b2[k]))*inv_Qb2[k] >> 16) , gain_Qb1[k] = ((inv_Qb2[k] << 16) + (err_Q29[k]*inv_Qb2[k] >> 13)) , num_Q24[k-1][n] = a32_Q24[k][n] - ((a32_Q24[k][k-n-1]*rc_Q31[k] + (1<<30)) >> 31) , a32_Q24[k-1][n] = (num_Q24[k-1][n]*gain_Qb1[k] + (1<<(b1[k]-1))) >> b1[k] , ]]> where 0 <= n < k-1. Here, rc_Q30[k] are the reflection coefficients. div_Q30[k] is the denominator for each iteration, and gain_Qb1[k] is its multiplicative inverse (with b1[k] fractional bits, where b1[k] ranges from 20 to 31). inv_Qb2[k], which ranges from 16384 to 32767, is a low-precision version of that inverse (with b2[k] fractional bits). err_Q29[k] is the residual error, ranging from -32763 to 32392, which is used to improve the accuracy. The values t_Q24[k-1][n] for each n are the numerators for the next row of coefficients in the recursion, and a32_Q24[k-1][n] is the final version of that row. Every multiply in this procedure except the one used to compute gain_Qb1[k] requires more than 32 bits of precision, but otherwise all intermediate results fit in 32 bits or less. In practice, because each row only depends on the next one, an implementation does not need to store them all. If abs(a32_Q24[k][k]) <= 16773022 for 0 <= k < d_LPC, then the filter is considered stable. However, the problem of determining stability is ill-conditioned when the filter contains several reflection coefficients whose magnitude is very close to one. This fixed-point algorithm is not mathematically guaranteed to correctly classify filters as stable or unstable in this case, though it does very well in practice. On round i, 1 <= i <= 18, if the filter passes this stability check, then this procedure stops, and the final LPC coefficients to use for reconstruction in are

> 5 . ]]> Otherwise, a round of bandwidth expansion is applied using the same procedure as in , with

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

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

The primary lag index is coded either relative to the primary lag of the prior frame or as an absolute index. Like the quantization gains, the primary pitch lag is coded either as an absolute index, or relative to the most recent coded frame in the same channel. Absolute coding is used if and only if This is the first LBRR frame for this channel in the current Opus frame, This is an LBRR frame where the LBRR flags (see and ) indicate the previous LBRR frame in the same channel is not coded, This is the first regular SILK frame for this channel in the current Opus frame, or The most recently coded frame in the current channel was not voiced (see ). In particular, unlike an LBRR frame where the previous frame is not coded, in a 60 ms stereo Opus frame, if the first and third regular SILK frames in the side channel are coded, voiced frames, but the second is not coded, the third still uses relative coding. With absolute coding, the primary pitch lag may range from 2 ms (inclusive) up to 18 ms (exclusive), corresponding to pitches from 500 Hz down to 55.6 Hz, respectively. It is comprised of a high part and a low part, where the decoder reads the high part using the 32-entry codebook in and the low part using the codebook corresponding to the current audio bandwidth from . The final primary pitch lag is then

where lag_high is the high part, lag_low is the low part, and lag_scale and lag_min are the values from the "Scale" and "Minimum Lag" columns of , respectively. PDF {3, 3, 6, 11, 21, 30, 32, 19, 11, 10, 12, 13, 13, 12, 11, 9, 8, 7, 6, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256 Audio Bandwidth PDF Scale Minimum Lag Maximum Lag NB {64, 64, 64, 64}/256 4 16 144 MB {43, 42, 43, 43, 42, 43}/256 6 24 216 WB {32, 32, 32, 32, 32, 32, 32, 32}/256 8 32 288 All frames that do not use absolute coding for the primary lag index use relative coding instead. The decoder reads a single delta value using the 21-entry PDF in . If the resulting value is zero, it falls back to the absolute coding procedure from the prior paragraph. Otherwise, the final primary pitch lag is then

where lag_prev is the primary pitch lag from the most recent frame in the same channel and delta_lag_index is the value just decoded. This allows a per-frame change in the pitch lag of -8 to +11 samples. The decoder does no clamping at this point, so this value can fall outside the range of 2 ms to 18 ms, and the decoder must use this unclamped value when using relative coding in the next SILK frame (if any). However, because an Opus frame can use relative coding for at most two consecutive SILK frames, integer overflow should not be an issue. PDF {46, 2, 2, 3, 4, 6, 10, 15, 26, 38, 30, 22, 15, 10, 7, 6, 4, 4, 2, 2, 2}/256 After the primary pitch lag, a "pitch contour", stored as a single entry from one of four small VQ codebooks, gives lag offsets for each subframe in the current SILK frame. The codebook index is decoded using one of the PDFs in depending on the current frame size and audio bandwidth. Tables through give the corresponding offsets to apply to the primary pitch lag for each subframe given the decoded codebook index. Audio Bandwidth SILK Frame Size Codebook Size PDF NB 10 ms 3 {143, 50, 63}/256 NB 20 ms 11 {68, 12, 21, 17, 19, 22, 30, 24, 17, 16, 10}/256 MB or WB 10 ms 12 {91, 46, 39, 19, 14, 12, 8, 7, 6, 5, 5, 4}/256 MB or WB 20 ms 34 {33, 22, 18, 16, 15, 14, 14, 13, 13, 10, 9, 9, 8, 6, 6, 6, 5, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1}/256 Index Subframe Offsets 0 0 0 1 1 0 2 0 1 Index Subframe Offsets 0 0 0 0 0 1 2 1 0 -1 2 -1 0 1 2 3 -1 0 0 1 4 -1 0 0 0 5 0 0 0 1 6 0 0 1 1 7 1 1 0 0 8 1 0 0 0 9 0 0 0 -1 10 1 0 0 -1 Index Subframe Offsets 0 0 0 1 0 1 2 1 0 3 -1 1 4 1 -1 5 -1 2 6 2 -1 7 -2 2 8 2 -2 9 -2 3 10 3 -2 11 -3 3 Index Subframe Offsets 0 0 0 0 0 1 0 0 1 1 2 1 1 0 0 3 -1 0 0 0 4 0 0 0 1 5 1 0 0 0 6 -1 0 0 1 7 0 0 0 -1 8 -1 0 1 2 9 1 0 0 -1 10 -2 -1 1 2 11 2 1 0 -1 12 -2 0 0 2 13 -2 0 1 3 14 2 1 -1 -2 15 -3 -1 1 3 16 2 0 0 -2 17 3 1 0 -2 18 -3 -1 2 4 19 -4 -1 1 4 20 3 1 -1 -3 21 -4 -1 2 5 22 4 2 -1 -3 23 4 1 -1 -4 24 -5 -1 2 6 25 5 2 -1 -4 26 -6 -2 2 6 27 -5 -2 2 5 28 6 2 -1 -5 29 -7 -2 3 8 30 6 2 -2 -6 31 5 2 -2 -5 32 8 3 -2 -7 33 -9 -3 3 9 The final pitch lag for each subframe is assembled in silk_decode_pitch() (decode_pitch.c). Let lag be the primary pitch lag for the current SILK frame, contour_index be index of the VQ codebook, and lag_cb[contour_index][k] be the corresponding entry of the codebook from the appropriate table given above for the k'th subframe. Then the final pitch lag for that subframe is

where lag_min and lag_max are the values from the "Minimum Lag" and "Maximum Lag" columns of , respectively.

SILK can use a separate 5-tap pitch filter for each subframe. It selects the filter to use from one of three codebooks. The three codebooks each represent different rate-distortion trade-offs, with average rates of 1.61 bits/subframe, 3.68 bits/subframe, and 4.85 bits/subframe, respectively. The importance of the filter coefficients generally depends on two factors: the periodicity of the signal and relative energy between the current subframe and the signal from one period earlier. Greater periodicity and decaying energy both lead to more important filter coefficients, and thus should be coded with lower distortion and higher rate. These properties are relatively stable over the duration of a single SILK frame, hence all of the subframes in a SILK frame must choose their filter from the same codebook. This is signaled with an explicitly-coded "periodicity index". This immediately follows the subframe pitch lags, and is coded using the 3-entry PDF from . PDF {77, 80, 99}/256 The index of the filter to use for each subframe follows. They are all coded using the PDF from corresponding to the periodicity index. Tables through contain the corresponding filter taps as signed Q7 integers. Periodicity Index Codebook Size PDF 0 8 {185, 15, 13, 13, 9, 9, 6, 6}/256 1 16 {57, 34, 21, 20, 15, 13, 12, 13, 10, 10, 9, 10, 9, 8, 7, 8}/256 2 32 {15, 16, 14, 12, 12, 12, 11, 11, 11, 10, 9, 9, 9, 9, 8, 8, 8, 8, 7, 7, 6, 6, 5, 4, 5, 4, 4, 4, 3, 4, 3, 2}/256 Index Filter Taps (Q7) 0 4 6 24 7 5 1 0 0 2 0 0 2 12 28 41 13 -4 3 -9 15 42 25 14 4 1 -2 62 41 -9 5 -10 37 65 -4 3 6 -6 4 66 7 -8 7 16 14 38 -3 33 Index Filter Taps (Q7) 0 13 22 39 23 12 1 -1 36 64 27 -6 2 -7 10 55 43 17 3 1 1 8 1 1 4 6 -11 74 53 -9 5 -12 55 76 -12 8 6 -3 3 93 27 -4 7 26 39 59 3 -8 8 2 0 77 11 9 9 -8 22 44 -6 7 10 40 9 26 3 9 11 -7 20 101 -7 4 12 3 -8 42 26 0 13 -15 33 68 2 23 14 -2 55 46 -2 15 15 3 -1 21 16 41 Index Filter Taps (Q7) 0 -6 27 61 39 5 1 -11 42 88 4 1 2 -2 60 65 6 -4 3 -1 -5 73 56 1 4 -9 19 94 29 -9 5 0 12 99 6 4 6 8 -19 102 46 -13 7 3 2 13 3 2 8 9 -21 84 72 -18 9 -11 46 104 -22 8 10 18 38 48 23 0 11 -16 70 83 -21 11 12 5 -11 117 22 -8 13 -6 23 117 -12 3 14 3 -8 95 28 4 15 -10 15 77 60 -15 16 -1 4 124 2 -4 17 3 38 84 24 -25 18 2 13 42 13 31 19 21 -4 56 46 -1 20 -1 35 79 -13 19 21 -7 65 88 -9 -14 22 20 4 81 49 -29 23 20 0 75 3 -17 24 5 -9 44 92 -8 25 1 -3 22 69 31 26 -6 95 41 -12 5 27 39 67 16 -4 1 28 0 -6 120 55 -36 29 -13 44 122 4 -24 30 81 5 11 3 7 31 2 0 9 10 88

An LTP scaling parameter appears after the LTP filter coefficients if and only if This is a voiced frame (see ), and Either This is the first LBRR frame for this channel in the current Opus frame, This is an LBRR frame where the LBRR flags (see and ) indicate the previous LBRR frame in the same channel is not coded, or This is the first regular SILK frame for this channel in the current Opus frame. This allows the encoder to trade off the prediction gain between packets against the recovery time after packet loss. Unlike absolute-coding for pitch lags, a regular SILK frame other than the first one in a channel will not include this field even if the prior frame was not voiced. If present, the value is coded using the 3-entry PDF in . The three possible values represent Q14 scale factors of 15565, 12288, and 8192, respectively (corresponding to approximately 0.95, 0.75, and 0.5). Frames that do not code the scaling parameter use the default factor of 15565 (approximately 0.95). PDF {128, 64, 64}/256

SILK uses a linear congruential generator (LCG) to inject pseudorandom noise into the quantized excitation. To ensure synchronization of this process between the encoder and decoder, each SILK frame stores a 2-bit seed after the LTP parameters (if any). The encoder may consider the choice of this seed during quantization, meaning the flexibility to choose the LCG seed can reduce distortion. The seed is decoded with the uniform 4-entry PDF in , yielding a value between 0 and 3, inclusive. PDF {64, 64, 64, 64}/256

SILK codes the excitation using a modified version of the Pyramid Vector Quantization (PVQ) codebook . The PVQ codebook is designed for Laplace-distributed values and consists of all sums of K signed, unit pulses in a vector of dimension N, where two pulses at the same position are required to have the same sign. Thus the codebook includes all integer codevectors y of dimension N that satisfy

Unlike regular PVQ, SILK uses a variable-length, rather than fixed-length, encoding. This encoding is better suited to the more Gaussian-like distribution of the coefficient magnitudes and the non-uniform distribution of their signs (caused by the quantization offset described below). SILK also handles large codebooks by coding the least significant bits (LSb's) of each coefficient directly. This adds a small coding efficiency loss, but greatly reduces the computation time and ROM size required for decoding, as implemented in silk_decode_pulses() (decode_pulses.c). SILK fixes the dimension of the codebook to N = 16. The excitation is made up of a number of "shell blocks", each 16 samples in size. lists the number of shell blocks required for a SILK frame for each possible audio bandwidth and frame size. 10 ms MB frames nominally contain 120 samples (10 ms at 12 kHz), which is not a multiple of 16. This is handled by coding 8 shell blocks (128 samples) and discarding the final 8 samples of the last block. The decoder contains no special case that prevents an encoder from placing pulses in these samples, and they must be correctly parsed from the bitstream if present, but they are otherwise ignored. Audio Bandwidth Frame Size Number of Shell Blocks NB 10 ms 5 MB 10 ms 8 WB 10 ms 10 NB 20 ms 10 MB 20 ms 15 WB 20 ms 20

The first symbol in the excitation is a "rate level", which is an index from 0 to 8, inclusive, coded using the PDF in corresponding to the signal type of the current frame (from ). The rate level selects the PDF used to decode the number of pulses in the individual shell blocks. It does not directly convey any information about the bitrate or the number of pulses itself, but merely changes the probability of the symbols in . Level 0 provides a more efficient encoding at low rates generally, and level 8 provides a more efficient encoding at high rates generally, though the most efficient level for a particular SILK frame may depend on the exact distribution of the coded symbols. An encoder should, but is not required to, use the most efficient rate level. Signal Type PDF Inactive or Unvoiced {15, 51, 12, 46, 45, 13, 33, 27, 14}/256 Voiced {33, 30, 36, 17, 34, 49, 18, 21, 18}/256

The total number of pulses in each of the shell blocks follows the rate level. The pulse counts for all of the shell blocks are coded consecutively, before the content of any of the blocks. Each block may have anywhere from 0 to 16 pulses, inclusive, coded using the 18-entry PDF in corresponding to the rate level from . The special value 17 indicates that this block has one or more additional LSb's to decode for each coefficient. If the decoder encounters this value, it decodes another value for the actual pulse count of the block, but uses the PDF corresponding to the special rate level 9 instead of the normal rate level. This process repeats until the decoder reads a value less than 17, and it then sets the number of extra LSb's used to the number of 17's decoded for that block. If it reads the value 17 ten times, then the next iteration uses the special rate level 10 instead of 9. The probability of decoding a 17 when using the PDF for rate level 10 is zero, ensuring that the number of LSb's for a block will not exceed 10. The cumulative distribution for rate level 10 is just a shifted version of that for 9 and thus does not require any additional storage. Rate Level PDF 0 {131, 74, 25, 8, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256 1 {58, 93, 60, 23, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256 2 {43, 51, 46, 33, 24, 16, 11, 8, 6, 3, 3, 3, 2, 1, 1, 2, 1, 2}/256 3 {17, 52, 71, 57, 31, 12, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}/256 4 {6, 21, 41, 53, 49, 35, 21, 11, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1}/256 5 {7, 14, 22, 28, 29, 28, 25, 20, 17, 13, 11, 9, 7, 5, 4, 4, 3, 10}/256 6 {2, 5, 14, 29, 42, 46, 41, 31, 19, 11, 6, 3, 2, 1, 1, 1, 1, 1}/256 7 {1, 2, 4, 10, 19, 29, 35, 37, 34, 28, 20, 14, 8, 5, 4, 2, 2, 2}/256 8 {1, 2, 2, 5, 9, 14, 20, 24, 27, 28, 26, 23, 20, 15, 11, 8, 6, 15}/256 9 {1, 1, 1, 6, 27, 58, 56, 39, 25, 14, 10, 6, 3, 3, 2, 1, 1, 2}/256 10 {2, 1, 6, 27, 58, 56, 39, 25, 14, 10, 6, 3, 3, 2, 1, 1, 2, 0}/256

The locations of the pulses in each shell block follow the pulse counts, as decoded by silk_shell_decoder() (shell_coder.c). As with the pulse counts, these locations are coded for all the shell blocks before any of the remaining information for each block. Unlike many other codecs, SILK places no restriction on the distribution of pulses within a shell block. All of the pulses may be placed in a single location, or each one in a unique location, or anything in between. The location of pulses is coded by recursively partitioning each block into halves, and coding how many pulses fall on the left side of the split. All remaining pulses must fall on the right side of the split. The process then recurses into the left half, and after that returns, the right half (preorder traversal). The PDF to use is chosen by the size of the current partition (16, 8, 4, or 2) and the number of pulses in the partition (1 to 16, inclusive). Tables through list the PDFs used for each partition size and pulse count. This process skips partitions without any pulses, i.e., where the initial pulse count from was zero, or where the split in the prior level indicated that all of the pulses fell on the other side. These partitions have nothing to code, so they require no PDF. Pulse Count PDF 1 {126, 130}/256 2 {56, 142, 58}/256 3 {25, 101, 104, 26}/256 4 {12, 60, 108, 64, 12}/256 5 {7, 35, 84, 87, 37, 6}/256 6 {4, 20, 59, 86, 63, 21, 3}/256 7 {3, 12, 38, 72, 75, 42, 12, 2}/256 8 {2, 8, 25, 54, 73, 59, 27, 7, 1}/256 9 {2, 5, 17, 39, 63, 65, 42, 18, 4, 1}/256 10 {1, 4, 12, 28, 49, 63, 54, 30, 11, 3, 1}/256 11 {1, 4, 8, 20, 37, 55, 57, 41, 22, 8, 2, 1}/256 12 {1, 3, 7, 15, 28, 44, 53, 48, 33, 16, 6, 1, 1}/256 13 {1, 2, 6, 12, 21, 35, 47, 48, 40, 25, 12, 5, 1, 1}/256 14 {1, 1, 4, 10, 17, 27, 37, 47, 43, 33, 21, 9, 4, 1, 1}/256 15 {1, 1, 1, 8, 14, 22, 33, 40, 43, 38, 28, 16, 8, 1, 1, 1}/256 16 {1, 1, 1, 1, 13, 18, 27, 36, 41, 41, 34, 24, 14, 1, 1, 1, 1}/256 Pulse Count PDF 1 {127, 129}/256 2 {53, 149, 54}/256 3 {22, 105, 106, 23}/256 4 {11, 61, 111, 63, 10}/256 5 {6, 35, 86, 88, 36, 5}/256 6 {4, 20, 59, 87, 62, 21, 3}/256 7 {3, 13, 40, 71, 73, 41, 13, 2}/256 8 {3, 9, 27, 53, 70, 56, 28, 9, 1}/256 9 {3, 8, 19, 37, 57, 61, 44, 20, 6, 1}/256 10 {3, 7, 15, 28, 44, 54, 49, 33, 17, 5, 1}/256 11 {1, 7, 13, 22, 34, 46, 48, 38, 28, 14, 4, 1}/256 12 {1, 1, 11, 22, 27, 35, 42, 47, 33, 25, 10, 1, 1}/256 13 {1, 1, 6, 14, 26, 37, 43, 43, 37, 26, 14, 6, 1, 1}/256 14 {1, 1, 4, 10, 20, 31, 40, 42, 40, 31, 20, 10, 4, 1, 1}/256 15 {1, 1, 3, 8, 16, 26, 35, 38, 38, 35, 26, 16, 8, 3, 1, 1}/256 16 {1, 1, 2, 6, 12, 21, 30, 36, 38, 36, 30, 21, 12, 6, 2, 1, 1}/256 Pulse Count PDF 1 {127, 129}/256 2 {49, 157, 50}/256 3 {20, 107, 109, 20}/256 4 {11, 60, 113, 62, 10}/256 5 {7, 36, 84, 87, 36, 6}/256 6 {6, 24, 57, 82, 60, 23, 4}/256 7 {5, 18, 39, 64, 68, 42, 16, 4}/256 8 {6, 14, 29, 47, 61, 52, 30, 14, 3}/256 9 {1, 15, 23, 35, 51, 50, 40, 30, 10, 1}/256 10 {1, 1, 21, 32, 42, 52, 46, 41, 18, 1, 1}/256 11 {1, 6, 16, 27, 36, 42, 42, 36, 27, 16, 6, 1}/256 12 {1, 5, 12, 21, 31, 38, 40, 38, 31, 21, 12, 5, 1}/256 13 {1, 3, 9, 17, 26, 34, 38, 38, 34, 26, 17, 9, 3, 1}/256 14 {1, 3, 7, 14, 22, 29, 34, 36, 34, 29, 22, 14, 7, 3, 1}/256 15 {1, 2, 5, 11, 18, 25, 31, 35, 35, 31, 25, 18, 11, 5, 2, 1}/256 16 {1, 1, 4, 9, 15, 21, 28, 32, 34, 32, 28, 21, 15, 9, 4, 1, 1}/256 Pulse Count PDF 1 {128, 128}/256 2 {42, 172, 42}/256 3 {21, 107, 107, 21}/256 4 {12, 60, 112, 61, 11}/256 5 {8, 34, 86, 86, 35, 7}/256 6 {8, 23, 55, 90, 55, 20, 5}/256 7 {5, 15, 38, 72, 72, 36, 15, 3}/256 8 {6, 12, 27, 52, 77, 47, 20, 10, 5}/256 9 {6, 19, 28, 35, 40, 40, 35, 28, 19, 6}/256 10 {4, 14, 22, 31, 37, 40, 37, 31, 22, 14, 4}/256 11 {3, 10, 18, 26, 33, 38, 38, 33, 26, 18, 10, 3}/256 12 {2, 8, 13, 21, 29, 36, 38, 36, 29, 21, 13, 8, 2}/256 13 {1, 5, 10, 17, 25, 32, 38, 38, 32, 25, 17, 10, 5, 1}/256 14 {1, 4, 7, 13, 21, 29, 35, 36, 35, 29, 21, 13, 7, 4, 1}/256 15 {1, 2, 5, 10, 17, 25, 32, 36, 36, 32, 25, 17, 10, 5, 2, 1}/256 16 {1, 2, 4, 7, 13, 21, 28, 34, 36, 34, 28, 21, 13, 7, 4, 2, 1}/256

After the decoder reads the pulse locations for all blocks, it reads the LSb's (if any) for each block in turn. Inside each block, it reads all the LSb's for each coefficient in turn, even those where no pulses were allocated, before proceeding to the next one. They are coded from most significant to least significant, and they all use the PDF in . PDF {136, 120}/256 The number of LSb's read for each coefficient in a block is determined in . The magnitude of the coefficient is initially equal to the number of pulses placed at that location in . As each LSb is decoded, the magnitude is doubled, and then the value of the LSb added to it, to obtain an updated magnitude.

After decoding the pulse locations and the LSb's, the decoder knows the magnitude of each coefficient in the excitation. It then decodes a sign for all coefficients with a non-zero magnitude, using one of the PDFs from . If the value decoded is 0, then the coefficient magnitude is negated. Otherwise, it remains positive. The decoder chooses the PDF for the sign based on the signal type and quantization offset type (from ) and the number of pulses in the block (from ). The number of pulses in the block does not take into account any LSb's. Most PDFs are skewed towards negative signs because of the quantizaton offset, but the PDFs for zero pulses are highly skewed towards positive signs. If a block contains many positive coefficients, it is sometimes beneficial to code it solely using LSb's (i.e., with zero pulses), since the encoder may be able to save enough bits on the signs to justify the less efficient coefficient magnitude encoding. Signal Type Quantization Offset Type Pulse Count PDF Inactive Low 0 {2, 254}/256 Inactive Low 1 {207, 49}/256 Inactive Low 2 {189, 67}/256 Inactive Low 3 {179, 77}/256 Inactive Low 4 {174, 82}/256 Inactive Low 5 {163, 93}/256 Inactive Low 6 or more {157, 99}/256 Inactive High 0 {58, 198}/256 Inactive High 1 {245, 11}/256 Inactive High 2 {238, 18}/256 Inactive High 3 {232, 24}/256 Inactive High 4 {225, 31}/256 Inactive High 5 {220, 36}/256 Inactive High 6 or more {211, 45}/256 Unvoiced Low 0 {1, 255}/256 Unvoiced Low 1 {210, 46}/256 Unvoiced Low 2 {190, 66}/256 Unvoiced Low 3 {178, 78}/256 Unvoiced Low 4 {169, 87}/256 Unvoiced Low 5 {162, 94}/256 Unvoiced Low 6 or more {152, 104}/256 Unvoiced High 0 {48, 208}/256 Unvoiced High 1 {242, 14}/256 Unvoiced High 2 {235, 21}/256 Unvoiced High 3 {224, 32}/256 Unvoiced High 4 {214, 42}/256 Unvoiced High 5 {205, 51}/256 Unvoiced High 6 or more {190, 66}/256 Voiced Low 0 {1, 255}/256 Voiced Low 1 {162, 94}/256 Voiced Low 2 {152, 104}/256 Voiced Low 3 {147, 109}/256 Voiced Low 4 {144, 112}/256 Voiced Low 5 {141, 115}/256 Voiced Low 6 or more {138, 118}/256 Voiced High 0 {8, 248}/256 Voiced High 1 {203, 53}/256 Voiced High 2 {187, 69}/256 Voiced High 3 {176, 80}/256 Voiced High 4 {168, 88}/256 Voiced High 5 {161, 95}/256 Voiced High 6 or more {154, 102}/256

After the signs have been read, there is enough information to reconstruct the complete excitation signal. This requires adding a constant quantization offset to each non-zero sample, and then pseudorandomly inverting and offsetting every sample. The constant quantization offset varies depending on the signal type and quantization offset type (see ). Signal Type Quantization Offset Type Quantization Offset (Q10) Inactive Low 100 Inactive High 240 Unvoiced Low 100 Unvoiced High 240 Voiced Low 32 Voiced High 100 Let e_raw[i] be the raw excitation value at position i, with a magnitude composed of the pulses at that location (see ) combined with any additional LSb's (see ), and with the corresponding sign decoded in . Additionally, let seed be the current pseudorandom seed, which is initialized to the value decoded from for the first sample in the current SILK frame, and updated for each subsequent sample according to the procedure below. Finally, let offset_Q10 be the quantization offset from . Then the following procedure produces the final reconstructed excitation value, e_Q10[i]:

When e_raw[i] is zero, sign() returns 0 by the definition in , implying that no quantization offset gets added. The final e_Q10[i] value may require more than 16 bits per sample, but will not require more than 32.

The remainder of the reconstruction process for the frame does not need to be bit-exact, as small errors should only introduce proportionally small distortions. Although the reference implementation only includes a fixed-point version of the remaining steps, this section describes them in terms of a floating-point version for simplicity. This produces a signal with a nominal range of -1.0 to 1.0. silk_decode_core() (decode_core.c) contains the code for the main reconstruction process. It proceeds subframe-by-subframe, since quantization gains, LTP parameters, and (in 20 ms SILK frames) LPC coefficients can vary from one to the next. Let a_Q12[k] be the LPC coefficients for the current subframe. If this is the first or second subframe of a 20 ms SILK frame and the LSF interpolation factor, w_Q2 (see ), is less than 4, then these correspond to the final LPC coefficients produced by from the interpolated LSF coefficients, n1_Q15[k] (computed in ). Otherwise, they correspond to the final LPC coefficients produced from the uninterpolated LSF coefficients for the current frame, n2_Q15[k]. Also, let n be the number of samples in a subframe (40 for NB, 60 for MB, and 80 for WB), s be the index of the current subframe in this SILK frame (0 or 1 for 10 ms frames, or 0 to 3 for 20 ms frames), and j be the index of the first sample in the residual corresponding to the current subframe.

Voiced SILK frames (see ) pass the excitation through an LTP filter using the parameters decoded in to produce an LPC residual. Let e_Q10[i] be the excitation, res[i] be the LPC residual, and out[i] be the fully reconstructed output signal (from ). The LTP filter requires LPC residual values from before the current subframe as input. However, since the LPCs may have changed, it obtains them by "rewhitening" the corresponding output signal using the LPCs from the current subframe. Let LTP_scale_Q14 be the LTP scaling parameter from for the first two subframes in any SILK frame, as well as the last two subframes in a 20 ms SILK frame where w_Q2 == 4. Otherwise let LTP_scale_Q14 be 16384 (corresponding to 1.0). Then, for i such that (j - pitch_lags[s] - d_LPC - 2) <= i < j, where pitch_lags[s] is the pitch lag for the current subframe from , out[i] is rewhitened into res[i] with

This requires storage to buffer up to 306 values of out[i] from previous subframes. This corresponds to WB with a maximum of 18&mbsp;ms * 16 kHz samples of pitch lag, plus 2 samples for the width of the LTP filter, plus 16 samples for d_LPC. Let b_Q7[k] be the coefficients of the LTP filter taken from the codebook entry in one of Tables through corresponding to the index decoded for the current subframe in . Then for i such that j <= i < (j + n), the LPC residual is

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

LPC synthesis uses the short-term LPC filter to predict the next output coefficient. For i such that (j - d_LPC) <= i < j, let lpc[i] be the result of LPC synthesis from the previous subframe, or zeros in the first subframe after a decoder reset. Then for i such that j <= i (j + n), the result of LPC synthesis for the current subframe is

The decoder saves the final d_LPC values, i.e., lpc[i] such that (j + n - d_LPC) <= i < (j + n), to feed into the LPC synthesis of the next subframe. This requires storage for up to 16 values of lpc[i] (for WB frames). Then, the signal is clamped into the final nominal range:

This clamping occurs entirely after the LPC synthesis filter has run. The decoder saves the unclamped values, lpc[i], to feed into the LPC filter for the next subframe, but saves the clamped values, out[i], for rewhitening in voiced frames.

An overview of the decoder is given in .

| decoder |----+ | +---------+ | | | | +---------+ v | | Fine | +---+ +->| decoder |->| + | | +---------+ +---+ | ^ | +---------+ | | | | Range | | +----------+ v | Decoder |-+ | Bit | +-----+ +---------+ | |Allocation| | 2^x | | +----------+ +-----+ | | | | v v +--------+ | +---------+ +---+ +-------+ | pitch | +->| PVQ |->| * |->| IMDCT |->| post- |---> | | decoder | +---+ +-------+ | filter | | +---------+ +--------+ | ^ +--------------------------------------+ ]]> The decoder is based on the following symbols and sets of symbols: Symbol(s) PDF Condition silence {32767, 1}/32768 post-filter {1, 1}/2 octave uniform (6)post-filter period raw bits (4+octave)post-filter gain raw bits (3)post-filter tapset {2, 1, 1}/4post-filter transient {7, 1}/8 intra {7, 1}/8 coarse energy tf_change tf_select {1, 1}/2 spread {7, 2, 21, 2}/32 dyn. alloc. alloc. trim {2, 2, 5, 10, 22, 46, 22, 10, 5, 2, 2}/128 skip {1, 1}/2 intensity uniform dual {1, 1}/2 fine energy residual anti-collapse{1, 1}/2 finalize Order of the symbols in the CELT section of the bitstream. The decoder extracts information from the range-coded bitstream in the order described in the figure above. In some circumstances, it is possible for a decoded value to be out of range due to a very small amount of redundancy in the encoding of large integers by the range coder. In that case, the decoder should assume there has been an error in the coding, decoding, or transmission and SHOULD take measures to conceal the error and/or report to the application that a problem has occurred.

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

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

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

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

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

Many codecs transmit significant amounts of side information for the purpose of controlling bit allocation within a frame. Often this side information controls bit usage indirectly and must be carefully selected to achieve the desired rate constraints. The band-energy normalized structure of Opus MDCT mode ensures that a constant bit allocation for the shape content of a band will result in a roughly constant tone to noise ratio, which provides for fairly consistent perceptual performance. The effectiveness of this approach is the result of two factors: that the band energy, which is understood to be perceptually important on its own, is always preserved regardless of the shape precision, and because the constant tone-to-noise ratio implies a constant intra-band noise to masking ratio. Intra-band masking is the strongest of the perceptual masking effects. This structure means that the ideal allocation is more consistent from frame to frame than it is for other codecs without an equivalent structure. Because the bit allocation is used to drive the decoding of the range-coder stream, it MUST be recovered exactly so that identical coding decisions are made in the encoder and decoder. Any deviation from the reference's resulting bit allocation will result in corrupted output, though implementers are free to implement the procedure in any way which produces identical results. Because all of the information required to decode a frame must be derived from that frame alone in order to retain robustness to packet loss, the overhead of explicitly signaling the allocation would be considerable, especially for low-latency (small frame size) applications, even though the allocation is relatively static. For this reason, in the MDCT mode Opus uses a primarily implicit bit allocation. The available bitstream capacity is known in advance to both the encoder and decoder without additional signaling, ultimately from the packet sizes expressed by a higher-level protocol. Using this information the codec interpolates an allocation from a hard-coded table. While the band-energy structure effectively models intra-band masking, it ignores the weaker inter-band masking, band-temporal masking, and other less significant perceptual effects. While these effects can often be ignored, they can become significant for particular samples. One mechanism available to encoders would be to simply increase the overall rate for these frames, but this is not possible in a constant rate mode and can be fairly inefficient. As a result three explicitly signaled mechanisms are provided to alter the implicit allocation: Band boost Allocation trim Band skipping The first of these mechanisms, band boost, allows an encoder to boost the allocation in specific bands. The second, allocation trim, works by biasing the overall allocation towards higher or lower frequency bands. The third, band skipping, selects which low-precision high frequency bands will be allocated no shape bits at all. In stereo mode there are two additional parameters potentially coded as part of the allocation procedure: a parameter to allow the selective elimination of allocation for the 'side' in jointly coded bands, and a flag to deactivate joint coding. These values are not signaled if they would be meaningless in the overall context of the allocation. Because every signaled adjustment increases overhead and implementation complexity, none were included speculatively: the reference encoder makes use of all of these mechanisms. While the decision logic in the reference was found to be effective enough to justify the overhead and complexity, further analysis techniques may be discovered which increase the effectiveness of these parameters. As with other signaled parameters, an encoder is free to choose the values in any manner, but unless a technique is known to deliver superior perceptual results the methods used by the reference implementation should be used. The allocation process consists of the following steps: determining the per-band maximum allocation vector, decoding the boosts, decoding the tilt, determining the remaining capacity of the frame, searching the mode table for the entry nearest but not exceeding the available space (subject to the tilt, boosts, band maximums, and band minimums), linear interpolation, reallocation of unused bits with concurrent skip decoding, determination of the fine-energy vs. shape split, and final reallocation. This process results in a per-band shape allocation (in 1/8th bit units), a per-band fine-energy allocation (in 1 bit per channel units), a set of band priorities for controlling the use of remaining bits at the end of the frame, and a remaining balance of unallocated space, which is usually zero except at very high rates. The maximum allocation vector is an approximation of the maximum space that can be used by each band for a given mode. The value is approximate because the shape encoding is variable rate (due to entropy coding of splitting parameters). Setting the maximum too low reduces the maximum achievable quality in a band while setting it too high may result in waste: bitstream capacity available at the end of the frame which can not be put to any use. The maximums specified by the codec reflect the average maximum. In the reference the maximums are provided in partially computed form, in order to fit in less memory as a static table (XXX cache.caps). Implementations are expected to simply use the same table data, but the procedure for generating this table is included in rate.c as part of compute_pulse_cache(). To convert the values in cache.caps into the actual maximums: first set nbBands to the maximum number of bands for this mode, and stereo to zero if stereo is not in use and one otherwise. For each band set N to the number of MDCT bins covered by the band (for one channel), set LM to the shift value for the frame size (e.g. 0 for 120, 1 for 240, 3 for 480), then set i to nbBands*(2*LM+stereo). Then set the maximum for the band to the i-th index of cache.caps + 64 and multiply by the number of channels in the current frame (one or two) and by N, then divide the result by 4 using truncating integer division. The resulting vector will be called cap[]. The elements fit in signed 16-bit integers but do not fit in 8 bits. This procedure is implemented in the reference in the function init_caps() in celt.c. The band boosts are represented by a series of binary symbols which are coded with very low probability. Each band can potentially be boosted multiple times, subject to the frame actually having enough room to obey the boost and having enough room to code the boost symbol. The default coding cost for a boost starts out at six bits, but subsequent boosts in a band cost only a single bit and every time a band is boosted the initial cost is reduced (down to a minimum of two). Since the initial cost of coding a boost is 6 bits, the coding cost of the boost symbols when completely unused is 0.48 bits/frame for a 21 band mode (21*-log2(1-1/2**6)). To decode the band boosts: First set 'dynalloc_logp' to 6, the initial amount of storage required to signal a boost in bits, 'total_bits' to the size of the frame in 8th bits, 'total_boost' to zero, and 'tell' to the total number of 8th bits decoded so far. For each band from the coding start (0 normally, but 17 in hybrid mode) to the coding end (which changes depending on the signaled bandwidth): set 'width' to the number of MDCT bins in this band for all channels. Take the larger of width and 64, then the minimum of that value and the width times eight and set 'quanta' to the result. This represents a boost step size of six bits subject to limits of 1/bit/sample and 1/8th bit/sample. Set 'boost' to zero and 'dynalloc_loop_logp' to dynalloc_logp. While dynalloc_loop_log (the current worst case symbol cost) in 8th bits plus tell is less than total_bits plus total_boost and boost is less than cap[] for this band: Decode a bit from the bitstream with a with dynalloc_loop_logp as the cost of a one, update tell to reflect the current used capacity, if the decoded value is zero break the loop otherwise add quanta to boost and total_boost, subtract quanta from total_bits, and set dynalloc_loop_log to 1. When the while loop finishes boost contains the boost for this band. If boost is non-zero and dynalloc_logp is greater than 2, decrease dynalloc_logp. Once this process has been executed on all bands, the band boosts have been decoded. This procedure is implemented around line 2352 of celt.c. At very low rates it is possible that there won't be enough available space to execute the inner loop even once. In these cases band boost is not possible but its overhead is completely eliminated. Because of the high cost of band boost when activated, a reasonable encoder should not be using it at very low rates. The reference implements its dynalloc decision logic around line 1269 of celt.c. The allocation trim is a integer value from 0-10. The default value of 5 indicates no trim. The trim parameter is entropy coded in order to lower the coding cost of less extreme adjustments. Values lower than 5 bias the allocation towards lower frequencies and values above 5 bias it towards higher frequencies. Like other signaled parameters, signaling of the trim is gated so that it is not included if there is insufficient space available in the bitstream. To decode the trim, first set the trim value to 5, then iff the count of decoded 8th bits so far (ec_tell_frac) plus 48 (6 bits) is less than or equal to the total frame size in 8th bits minus total_boost (a product of the above band boost procedure), decode the trim value using the inverse CDF {127, 126, 124, 119, 109, 87, 41, 19, 9, 4, 2, 0}. For 10 ms and 20 ms frames using short blocks and that have at least LM+2 bits left prior to the allocation process, then one anti-collapse bit is reserved in the allocation process so it can be decoded later. Following the the anti-collapse reservation, one bit is reserved for skip if available. For stereo frames, bits are reserved for intensity stereo and for dual stereo. Intensity stereo requires ilog2(end-start) bits. Those bits are reserved if there is enough bits left. Following this, one bit is reserved for dual stereo if available. The allocation computation begins by setting up some initial conditions. 'total' is set to the remaining available 8th bits, computed by taking the size of the coded frame times 8 and subtracting ec_tell_frac(). From this value, one (8th bit) is subtracted to ensure that the resulting allocation will be conservative. 'anti_collapse_rsv' is set to 8 (8th bits) iff the frame is a transient, LM is greater than 1, and total is greater than or equal to (LM+2) * 8. Total is then decremented by anti_collapse_rsv and clamped to be equal to or greater than zero. 'skip_rsv' is set to 8 (8th bits) if total is greater than 8, otherwise it is zero. Total is then decremented by skip_rsv. This reserves space for the final skipping flag. If the current frame is stereo, intensity_rsv is set to the conservative log2 in 8th bits of the number of coded bands for this frame (given by the table LOG2_FRAC_TABLE). If intensity_rsv is greater than total then intensity_rsv is set to zero. Otherwise total is decremented by intensity_rsv, and if total is still greater than 8, dual_stereo_rsv is set to 8 and total is decremented by dual_stereo_rsv. The allocation process then computes a vector representing the hard minimum amounts allocation any band will receive for shape. This minimum is higher than the technical limit of the PVQ process, but very low rate allocations produce an excessively sparse spectrum and these bands are better served by having no allocation at all. For each coded band, set thresh[band] to twenty-four times the number of MDCT bins in the band and divide by 16. If 8 times the number of channels is greater, use that instead. This sets the minimum allocation to one bit per channel or 48 128th bits per MDCT bin, whichever is greater. The band-size dependent part of this value is not scaled by the channel count, because at the very low rates where this limit is applicable there will usually be no bits allocated to the side. The previously decoded allocation trim is used to derive a vector of per-band adjustments, 'trim_offsets[]'. For each coded band take the alloc_trim and subtract 5 and LM. Then multiply the result by the number of channels, the number of MDCT bins in the shortest frame size for this mode, the number of remaining bands, 2**LM, and 8. Then divide this value by 64. Finally, if the number of MDCT bins in the band per channel is only one, 8 times the number of channels is subtracted in order to diminish the allocation by one bit, because width 1 bands receive greater benefit from the coarse energy coding.

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

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

Decoding of PVQ vectors is implemented in decode_pulses() (cwrs.c). The uique codeword index is decoded as a uniformly-distributed integer value between 0 and V(N,K)-1, where V(N,K) is the number of possible combinations of K pulses in N samples. The index is then converted to a vector in the same way specified in . The indexing is based on the calculation of V(N,K) (denoted N(L,K) in ). The number of combinations can be computed recursively as V(N,K) = V(N-1,K) + V(N,K-1) + V(N-1,K-1), with V(N,0) = 1 and V(0,K) = 0, K != 0. There are many different ways to compute V(N,K), including precomputed tables and direct use of the recursive formulation. The reference implementation applies the recursive formulation one line (or column) at a time to save on memory use, along with an alternate, univariate recurrence to initialize an arbitrary line, and direct polynomial solutions for small N. All of these methods are equivalent, and have different trade-offs in speed, memory usage, and code size. Implementations MAY use any methods they like, as long as they are equivalent to the mathematical definition. The decoded vector is normalised such that its L2-norm equals one.

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

where N is the number of dimensions, K is the number of pulses, and f_r depends on the value of the "spread" parameter in the bit-stream. Spread value f_r 0 infinite (no rotation) 1 15 2 10 3 5 The rotation angle is then calculated as

A 2-D rotation R(i,j) between points x_i and x_j is defined as:

An N-D rotation is then achieved by applying a series of 2-D rotations back and forth, in the following order: R(x_1, x_2), R(x_2, x_3), ..., R(x_N-2, X_N-1), R(x_N-1, X_N), R(x_N-2, X_N-1), ..., R(x_1, x_2). If the decoded vector represents more than one time block, then the following process is applied separately on each time block. Also, if each block represents 8 samples or more, then another N-D rotation, by (pi/2-theta), is applied before the rotation described above. This extra rotation is applied in an interleaved manner with a stride equal to round(sqrt(N/nb_blocks))

To avoid the need for multi-precision calculations when decoding PVQ codevectors, the maximum size allowed for codebooks is 32 bits. When larger codebooks are needed, the vector is instead split in two sub-vectors of size N/2. A quantized gain parameter with precision derived from the current allocation is entropy coded to represent the relative gains of each side of the split, and the entire decoding process is recursively applied. Multiple levels of splitting may be applied up to a frame size dependent limit. The same recursive mechanism is applied for the joint coding of stereo audio.

The time-frequency (TF) parameters are used to control the time-frequency resolution tradeoff in each coded band. For each band, there are two possible TF choices. For the first band coded, the PDF is {3, 1}/4 for frames marked as transient and {15, 1}/16 for the other frames. For subsequent bands, the TF choice is coded relative to the previous TF choice with probability {15, 1}/15 for transient frames and {31, 1}/32 otherwise. The mapping between the decoded TF choices and the adjustment in TF resolution is shown in the tables below. Frame size (ms) 0 1 2.5 0 -1 5 0 -1 10 0 -2 20 0 -2 TF adjustments for non-transient frames and tf_select=0 Frame size (ms) 0 1 2.5 0 -1 5 0 -2 10 0 -3 20 0 -3 TF adjustments for non-transient frames and tf_select=1 Frame size (ms) 0 1 2.5 0 -1 5 1 0 10 2 0 20 3 0 TF adjustments for transient frames and tf_select=0 Frame size (ms) 0 1 2.5 0 -1 5 1 -1 10 1 -1 20 1 -1 TF adjustments for transient frames and tf_select=1 A negative TF adjustment means that the temporal resolution is increased, while a positive TF adjustment means that the frequency resolution is increased. Changes in TF resolution are implemented using the Hadamard transform. To increase the time resolution by N, N "levels" of the Hadamard transform are applied to the decoded vector for each interleaved MDCT vector. To increase the frequency resolution (assumes a transient frame), then N levels of the Hadamard transform are applied across the interleaved MDCT vector. In the case of increased time resolution the decoder uses the "sequency order" because the input vector is sorted in time.

When the frame has the transient bit set, an anti-collapse bit is decoded. When anti-collapse is set, the energy in each small MDCT is prevented from collapsing to zero. For each band of each MDCT where a collapse is detected, a pseudo-random signal is inserted with an energy corresponding to the min energy over the two previous frames. A renormalization step is then required to ensure that the anti-collapse step did not alter the energy preservation property.

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

The inverse MDCT implementation has no special characteristics. The input is N frequency-domain samples and the output is 2*N time-domain samples, while scaling by 1/2. A "low-overlap" window is used to reduce the algorithmic delay. It is derived from a basic (full overlap) 240-sample version of the window used by the Vorbis codec:

The low-overlap window is created by zero-padding the basic window and inserting ones in the middle, such that the resulting window still satisfies power complementarity. The IMDCT and windowing are performed by mdct_backward (mdct.c).

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

During a transition between different gains, a smooth transition is calculated using the square of the MDCT window. It is important that values of y(n) be interpolated one at a time such that the past value of y(n) used is interpolated.

After the post-filter, the signal is de-emphasized using the inverse of the pre-emphasis filter used in the encoder:

where alpha_p=0.8500061035.

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

Clock drift refers to the gradual desynchronization of two endpoints whose sample clocks run at different frequencies while they are streaming live audio. Differences in clock frequencies are generally attributable to manufacturing variation in the endpoints' clock hardware. For long-lived streams, the time difference between sender and receiver can grow without bound. When the sender's clock runs slower than the receiver's, the effect is similar to packet loss: too few packets are received. The receiver can distinguish between drift and loss if the transport provides packet timestamps. A receiver for live streams SHOULD conceal the effects of drift, and MAY do so by invoking the PLC. When the sender's clock runs faster than the receiver's, too many packets will be received. The receiver MAY respond by skipping any packet (i.e. not submitting the packet for decoding). This is likely to produce a less severe artifact than if the frame were dropped after decoding. A decoder MAY employ a more sophisticated drift compensation method. For example, the NetEQ component of the WebRTC.org codebase compensates for drift by adding or removing one period when the signal is highly periodic. The reference implementation of Opus allows a caller to learn whether the current frame's signal is highly periodic, and if so what the period is, using the OPUS_GET_PITCH() request.

Switching between the Opus coding modes requires careful consideration. More specifically, the transitions that cannot be easily handled are the ones where the lower frequencies have to switch between the SILK LP-based model and the CELT transform model. If nothing is done, a glitch will occur for these transitions. On the other hand, switching between the SILK-only modes and the hybrid mode does not require any special treatment. There are two ways to avoid or reduce glitches during the problematic mode transitions: with side information or without it. Only transitions with side information are normatively specified. For transitions with no side information, it is RECOMMENDED for the decoder to use a concealment technique (e.g. make use of the PLC algorithm) to "fill in" the gap or discontinuity caused by the mode transition. Note that this concealment MUST NOT be applied when switching between the SILK mode and the hybrid mode or vice versa. Similarly, it MUST NOT be applied when merely changing the bandwidth within the same mode.

Switching with side information involves transmitting in-band a 5-ms "redundant" CELT frame within the Opus frame. This frame is designed to fill in the gap or discontinuity without requiring the decoder to conceal it. For transitions from a CELT-only frame to a SILK-only or hybrid frame, the redundant frame is inserted in the frame following the transition (i.e. the SILK-only/hybrid frame). For transitions from a SILK-only/hybrid frame to a CELT-only frame, the redundant frame is inserted in the first frame. For all SILK-only and hybrid frames (not only those involved in a mode transition), a binary symbol of probability 2^-12 needs to be decoded just after the SILK part of the bitstream. When the symbol value is 1, the frame then includes an embedded redundant frame. The redundant frame always starts and ends on a byte boundary. For SILK-only frames, the number of bytes is simply the number of whole remaining bytes. For hybrid frames, the number of bytes is equal to 2, plus a decoded unsigned integer (ec_dec_uint()) between 0 and 255. For hybrid frames, the redundant frame is placed at the end of the frame, after the CELT layer of the hybrid frame. The redundant frame is decoded like any other CELT-only frame, with the exception that it does not contain a TOC byte. The bandwidth is instead set to the same bandwidth of the current frame (for MB frames, the redundant frame is set to WB). For CELT-only to SILK-only/hybrid transitions, the first 2.5 ms of the redundant frame is used as-is for the reconstructed output. The remaining 2.5 ms is overlapped and added (cross-faded using the square of the MDCT power-complementary window) to the decoded SILK/hybrid signal, ensuring a smooth transition. For SILK-only/hyrid to CELT-only transitions, only the second half of the 5-ms decoded redundant frame is used. In that case, only a 2.5-ms cross-fade is applied, still using the power-complementary window.

Opus encoder block diagram.

| rate |--->|encoder|--+ | |conversion| | | | audio | +----------+ +-------+ | +-------+ ------+ +--->| Range | | +------------+ +-------+ |encoder|----> | | Delay | | CELT | +--->| | bitstream +->|Compensation|->|encoder|--+ +-------+ | | | | +------------+ +-------+ ]]>

The range coder also acts as the bit-packer for Opus. It is used in three different ways, to encode: entropy-coded symbols with a fixed probability model using ec_encode(), (entenc.c) integers from 0 to 2**M-1 using ec_enc_uint() or ec_enc_bits(), (entenc.c) integers from 0 to N-1 (where N is not a power of two) using ec_enc_uint(). (entenc.c) The range encoder maintains an internal state vector composed of the four-tuple (low,rng,rem,ext) representing the low end of the current range, the size of the current range, a single buffered output octet, and a count of additional carry-propagating output octets. Both rng and low are 32-bit unsigned integer values, rem is an octet value or the special value -1, and ext is an integer with at least 16 bits. This state vector is initialized at the start of each each frame to the value (0,2**31,-1,0). The reference implementation re-uses the 'val' field of the entropy coder structure to hold low, in order to allow the same structure to be used for encoding and decoding, but we maintain the distinction here for clarity.

The main encoding function is ec_encode() (entenc.c), which takes as an argument a three-tuple (fl,fh,ft) describing the range of the symbol to be encoded in the current context, with 0 <= fl < fh <= ft <= 65535. The values of this tuple are derived from the probability model for the symbol. Let f(i) be the frequency of the i'th symbol in the current context. Then the three-tuple corresponding to the k'th symbol is given by ec_encode() updates the state of the encoder as follows. If fl is greater than zero, then low = low + rng - (rng/ft)*(ft-fl) and rng = (rng/ft)*(fh-fl). Otherwise, low is unchanged and rng = rng - (rng/ft)*(fh-fl). The divisions here are exact integer division. After this update, the range is normalized. To normalize the range, the following process is repeated until rng > 2**23. First, the top 9 bits of low, (low>>23), are placed into a carry buffer. Then, low is set to . This process is carried out by ec_enc_normalize() (entenc.c). The 9 bits produced in each iteration of the normalization loop consist of 8 data bits and a carry flag. The final value of the output bits is not determined until carry propagation is accounted for. Therefore the reference implementation buffers a single (non-propagating) output octet and keeps a count of additional propagating (0xFF) output octets. An implementation MAY choose to use any mathematically equivalent scheme to perform carry propagation. The function ec_enc_carry_out() (entenc.c) performs this buffering. It takes a 9-bit input value, c, from the normalization: 8 bits of output and a carry bit. If c is 0xFF, then ext is incremented and no octets are output. Otherwise, if rem is not the special value -1, then the octet (rem+(c>>8)) is output. Then ext octets are output with the value 0 if the carry bit is set, or 0xFF if it is not, and rem is set to the lower 8 bits of c. After this, ext is set to zero. In the reference implementation, a special version of ec_encode() called ec_encode_bin() (entenc.c) is defined to take a two-tuple (fl,ftb), where , but avoids using division.

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

The function ec_enc_uint() is based on ec_encode() and encodes one of N equiprobable symbols, each with a frequency of 1, where N may be as large as 2**32-1. Because ec_encode() is limited to a total frequency of 2**16-1, this is done by encoding a series of symbols in smaller contexts. ec_enc_uint() (entenc.c) takes a two-tuple (fl,ft), where ft is not necessarily a power of two. Let ftb be the location of the highest 1 bit in the two's-complement representation of (ft-1), or -1 if no bits are set. If ftb>8, then the top 8 bits of fl are encoded using ec_encode() with the three-tuple (fl>>ftb-8,(fl>>ftb-8)+1,(ft-1>>ftb-8)+1), and the remaining bits are encoded as raw bits. Otherwise, fl is encoded with ec_encode() directly using the three-tuple (fl,fl+1,ft).

After all symbols are encoded, the stream must be finalized by outputting a value inside the current range. Let end be the integer in the interval [low,low+rng) with the largest number of trailing zero bits, b, such that end+(1<<b)-1 is also in the interval [low,low+rng). Then while end is not zero, the top 9 bits of end, e.g., >23), are sent to the carry buffer, and end is replaced by (end<<8&0x7FFFFFFF). Finally, if the value in carry buffer, rem, is]]> neither zero nor the special value -1, or the carry count, ext, is greater than zero, then 9 zero bits are sent to the carry buffer. After the carry buffer is finished outputting octets, the rest of the output buffer (if any) is padded with zero bits, until it reaches the raw bits. Finally, rem is set to the special value -1. This process is implemented by ec_enc_done() (entenc.c).

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

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

| | +---------+ | +---------+ | | |Voice | | |LTP | | | +----->|Activity |-----+ +---->|Scaling |---------+--->| | | |Detector | 3 | | |Control |<+ 12 | | | | +---------+ | | +---------+ | | | | | | | +---------+ | | | | | | | |Gains | | 11 | | | | | | +->|Processor|-|---+---|--->| R | | | | | | | | | | | a | | \/ | | +---------+ | | | | n | | +---------+ | | +---------+ | | | | g | | |Pitch | | | |LSF | | | | | e | | +->|Analysis |-+ | |Quantizer|-|---|---|--->| | | | | |4| | | | | 8 | | | E |-> | | +---------+ | | +---------+ | | | | n |14 | | | | 9/\ 10| | | | | c | | | | | | \/ | | | | o | | | +---------+ | | +----------+| | | | d | | | |Noise | +--|->|Prediction|+---|---|--->| e | | +->|Shaping |-|--+ |Analysis || 7 | | | r | | | |Analysis |5| | | || | | | | | | +---------+ | | +----------+| | | | | | | | | /\ | | | | | | | +---------|--|-------+ | | | | | | | | \/ \/ \/ \/ \/ | | | +---------+ | | +---------+ +------------+ | | | |High-Pass| | | | | |Noise | | | -+->|Filter |-+----+----->|Prefilter|------>|Shaping |->| | 1 | | 2 | | 6 |Quantization|13| | +---------+ +---------+ +------------+ +---+ 1: Input speech signal 2: High passed input signal 3: Voice activity estimate 4: Pitch lags (per 5 ms) and voicing decision (per 20 ms) 5: Noise shaping quantization coefficients - Short term synthesis and analysis noise shaping coefficients (per 5 ms) - Long term synthesis and analysis noise shaping coefficients (per 5 ms and for voiced speech only) - Noise shaping tilt (per 5 ms) - Quantizer gain/step size (per 5 ms) 6: Input signal filtered with analysis noise shaping filters 7: Short and long term prediction coefficients LTP (per 5 ms) and LPC (per 20 ms) 8: LSF quantization indices 9: LSF coefficients 10: Quantized LSF coefficients 11: Processed gains, and synthesis noise shape coefficients 12: LTP state scaling coefficient. Controlling error propagation / prediction gain trade-off 13: Quantized signal 14: Range encoded bitstream ]]> Encoder block diagram.

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

The input signal is filtered by a high-pass filter to remove the lowest part of the spectrum that contains little speech energy and may contain background noise. This is a second order Auto Regressive Moving Average (ARMA) filter with a cut-off frequency around 70 Hz. In the future, a music detector may also be used to lower the cut-off frequency when the input signal is detected to be music rather than speech.

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

|sampling|->|Correlator| | | | | | | |4 | +--------+ +----------+ \/ | | 2 +-------+ | | +-->|Speech |5 +---------+ +--------+ | \/ | |Type |-> |LPC | |Down | | +----------+ | | +->|Analysis | +->|sample |-+------------->|Time- | +-------+ | | | | |to 8 kHz| |Correlator|-----------> | +---------+ | +--------+ |__________| 6 | | | |3 | \/ | \/ | +---------+ | +----------+ | |Whitening| | |Time- | -+->|Filter |-+--------------------------->|Correlator|-----------> 1 | | | | 7 +---------+ +----------+ 1: Input signal 2: Lag candidates from stage 1 3: Lag candidates from stage 2 4: Correlation threshold 5: Voiced/unvoiced flag 6: Pitch correlation 7: Pitch lags ]]> Block diagram of the pitch estimator. The pitch analysis finds a binary voiced/unvoiced classification, and, for frames classified as voiced, four pitch lags per frame - one for each 5 ms subframe - and a pitch correlation indicating the periodicity of the signal. The input is first whitened using a Linear Prediction (LP) whitening filter, where the coefficients are computed through standard Linear Prediction Coding (LPC) analysis. The order of the whitening filter is 16 for best results, but is reduced to 12 for medium complexity and 8 for low complexity modes. The whitened signal is analyzed to find pitch lags for which the time correlation is high. The analysis consists of three stages for reducing the complexity: In the first stage, the whitened signal is downsampled to 4 kHz (from 8 kHz) and the current frame is correlated to a signal delayed by a range of lags, starting from a shortest lag corresponding to 500 Hz, to a longest lag corresponding to 56 Hz. The second stage operates on an 8 kHz signal (downsampled from 12, 16, or 24 kHz) and measures time correlations only near the lags corresponding to those that had sufficiently high correlations in the first stage. The resulting correlations are adjusted for a small bias towards short lags to avoid ending up with a multiple of the true pitch lag. The highest adjusted correlation is compared to a threshold depending on: Whether the previous frame was classified as voiced The speech activity level The spectral tilt. If the threshold is exceeded, the current frame is classified as voiced and the lag with the highest adjusted correlation is stored for a final pitch analysis of the highest precision in the third stage. The last stage operates directly on the whitened input signal to compute time correlations for each of the four subframes independently in a narrow range around the lag with highest correlation from the second stage.

The noise shaping analysis finds gains and filter coefficients used in the prefilter and noise shaping quantizer. These parameters are chosen such that they will fulfill several requirements: Balancing quantization noise and bitrate. The quantization gains determine the step size between reconstruction levels of the excitation signal. Therefore, increasing the quantization gain amplifies quantization noise, but also reduces the bitrate by lowering the entropy of the quantization indices. Spectral shaping of the quantization noise; the noise shaping quantizer is capable of reducing quantization noise in some parts of the spectrum at the cost of increased noise in other parts without substantially changing the bitrate. By shaping the noise such that it follows the signal spectrum, it becomes less audible. In practice, best results are obtained by making the shape of the noise spectrum slightly flatter than the signal spectrum. De-emphasizing spectral valleys; by using different coefficients in the analysis and synthesis part of the prefilter and noise shaping quantizer, the levels of the spectral valleys can be decreased relative to the levels of the spectral peaks such as speech formants and harmonics. This reduces the entropy of the signal, which is the difference between the coded signal and the quantization noise, thus lowering the bitrate. Matching the levels of the decoded speech formants to the levels of the original speech formants; an adjustment gain and a first order tilt coefficient are computed to compensate for the effect of the noise shaping quantization on the level and spectral tilt.

Frequency 1: Input signal spectrum 2: De-emphasized and level matched spectrum 3: Quantization noise spectrum ]]> Noise shaping and spectral de-emphasis illustration. shows an example of an input signal spectrum (1). After de-emphasis and level matching, the spectrum has deeper valleys (2). The quantization noise spectrum (3) more or less follows the input signal spectrum, while having slightly less pronounced peaks. The entropy, which provides a lower bound on the bitrate for encoding the excitation signal, is proportional to the area between the de-emphasized spectrum (2) and the quantization noise spectrum (3). Without de-emphasis, the entropy is proportional to the area between input spectrum (1) and quantization noise (3) - clearly higher. The transformation from input signal to de-emphasized signal can be described as a filtering operation with a filter

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

is the analysis part of the de-emphasis filter, consisting of the short-term shaping filter with coefficients a_ana(k), and the long-term shaping filter with coefficients b_ana(k) and pitch lag L. The parameter d determines the number of long-term shaping filter taps. Similarly, but without the tilt adjustment, the synthesis part can be written as

All noise shaping parameters are computed and applied per subframe of 5 ms. First, an LPC analysis is performed on a windowed signal block of 15 ms. The signal block has a look-ahead of 5 ms relative to the current subframe, and the window is an asymmetric sine window. The LPC analysis is done with the autocorrelation method, with an order of 16 for best quality or 12 in low complexity operation. The quantization gain is found by taking the square root of the residual energy from the LPC analysis and multiplying it by a value inversely proportional to the coding quality control parameter and the pitch correlation. Next we find the two sets of short-term noise shaping coefficients a_ana(k) and a_syn(k), by applying different amounts of bandwidth expansion to the coefficients found in the LPC analysis. This bandwidth expansion moves the roots of the LPC polynomial towards the origin, using the formulas

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

where C is the coding quality control parameter between 0 and 1. Applying more bandwidth expansion to the analysis part than to the synthesis part gives the desired de-emphasis of spectral valleys in between formants. The long-term shaping is applied only during voiced frames. It uses three filter taps, described by

For unvoiced frames these coefficients are set to 0. The multiplication factors F_ana and F_syn are chosen between 0 and 1, depending on the coding quality control parameter, as well as the calculated pitch correlation and smoothed subband SNR of the lowest subband. By having F_ana less than F_syn, the pitch harmonics are emphasized relative to the valleys in between the harmonics. The tilt coefficient c_tilt is for unvoiced frames chosen as

for voiced frames, where C again is the coding quality control parameter and is between 0 and 1. The adjustment gain G serves to correct any level mismatch between the original and decoded signals that might arise from the noise shaping and de-emphasis. This gain is computed as the ratio of the prediction gain of the short-term analysis and synthesis filter coefficients. The prediction gain of an LPC synthesis filter is the square root of the output energy when the filter is excited by a unit-energy impulse on the input. An efficient way to compute the prediction gain is by first computing the reflection coefficients from the LPC coefficients through the step-down algorithm, and extracting the prediction gain from the reflection coefficients as

where r_k is the k'th reflection coefficient. Initial values for the quantization gains are computed as the square-root of the residual energy of the LPC analysis, adjusted by the coding quality control parameter. These quantization gains are later adjusted based on the results of the prediction analysis.

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

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

For a frame of voiced speech the pitch pulses will remain dominant in the pre-whitened input signal. Further whitening is desirable as it leads to higher quality at the same available bitrate. To achieve this, a Long-Term Prediction (LTP) analysis is carried out to estimate the coefficients of a fifth-order LTP filter for each of four subframes. The LTP coefficients are used to find an LTP residual signal with the simulated output signal as input to obtain better modeling of the output signal. This LTP residual signal is the input to an LPC analysis where the LPCs are estimated using Burg's method, such that the residual energy is minimized. The estimated LPCs are converted to a Line Spectral Frequency (LSF) vector and quantized as described in . After quantization, the quantized LSF vector is converted back to LPC coefficients using the full procedure in . By using LPC coefficients derived from the quantized LSF coefficients, the encoder remains fully synchronized with the decoder. The LTP coefficients are quantized using a method described in . The quantized LPC and LTP coefficients are then used to filter the high-pass filtered input signal and measure residual energy for each of the four subframes.

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

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

Instead, we minimize an objective function that consists of a weighted sum of rate and distortion, and use a codebook with an associated non-uniform rate table. Thus, we take into account that the probability mass function for selecting the codebook entries is by no means guaranteed to be uniform in our scenario. This approach has several advantages. It ensures that rarely used codebook vector centroids, which are modeling statistical outliers in the training set, are quantized with low error at the expense of a high rate. At the same time, it allows modeling frequently used centroids with low error and a relatively low rate. This approach leads to equal or lower distortion than the fixed-rate codebook at any given average rate, provided that the data is similar to that used for training the codebook.

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

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

This number of possible combinations is far too high to carry out a full search for each frame, so for all stages but the last (i.e., s smaller than S), only the best min(L, Ms) centroids are carried over to stage s+1. In each stage, the objective function (i.e., the weighted sum of accumulated bitrate and distortion) is evaluated for each codebook vector entry and the results are sorted. Only the best paths and their corresponding quantization errors are considered in the next stage. In the last stage, S, the single best path through the multistage codebook is determined. By varying the maximum number of survivors from each stage to the next, L, the complexity can be adjusted in real time, at the cost of a potential increase when evaluating the objective function for the resulting quantized vector. This approach scales all the way between the two extremes, L=1 being a greedy search, and the desirable but infeasible full search, L=T/MS. Performance almost as good as that of the infeasible full search can be obtained at substantially lower complexity by using this approach (see, e.g., ).

If the input is stable, finding the best candidate usually results in a quantized vector that is also stable. Due to the multi-stage approach, however, it is theoretically possible that the best quantization candidate is unstable. Because of this, it is necessary to explicitly ensure that the quantized vectors are stable. Therefore we apply an LSF stabilization method which ensures that the LSF parameters are within valid range, increasingly sorted, and have minimum distances between each other and the border values that have been predetermined as the 0.01 percentile distance values from a large training set.

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

For voiced frames, the prediction analysis described in resulted in four sets (one set per subframe) of five LTP coefficients, plus four weighting matrices. The LTP coefficients for each subframe are quantized using entropy constrained vector quantization. A total of three vector codebooks are available for quantization, with different rate-distortion trade-offs. The three codebooks have 10, 20, and 40 vectors and average rates of about 3, 4, and 5 bits per vector, respectively. Consequently, the first codebook has larger average quantization distortion at a lower rate, whereas the last codebook has smaller average quantization distortion at a higher rate. Given the weighting matrix W_ltp and LTP vector b, the weighted rate-distortion measure for a codebook vector cb_i with rate r_i is give by

where u is a fixed, heuristically-determined parameter balancing the distortion and rate. Which codebook gives the best performance for a given LTP vector depends on the weighting matrix for that LTP vector. For example, for a low valued W_ltp, it is advantageous to use the codebook with 10 vectors as it has a lower average rate. For a large W_ltp, on the other hand, it is often better to use the codebook with 40 vectors, as it is more likely to contain the best codebook vector. The weighting matrix W_ltp depends mostly on two aspects of the input signal. The first is the periodicity of the signal; the more periodic, the larger W_ltp. The second is the change in signal energy in the current subframe, relative to the signal one pitch lag earlier. A decaying energy leads to a larger W_ltp than an increasing energy. Both aspects fluctuate relatively slowly, which causes the W_ltp matrices for different subframes of one frame often to be similar. Because of this, one of the three codebooks typically gives good performance for all subframes, and therefore the codebook search for the subframe LTP vectors is constrained to only allow codebook vectors to be chosen from the same codebook, resulting in a rate reduction. To find the best codebook, each of the three vector codebooks is used to quantize all subframe LTP vectors and produce a combined weighted rate-distortion measure for each vector codebook. The vector codebook with the lowest combined rate-distortion over all subframes is chosen. The quantized LTP vectors are used in the noise shaping quantizer, and the index of the codebook plus the four indices for the four subframe codebook vectors are passed on to the range encoder.

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

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

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

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

Energy quantization (both coarse and fine) can be easily understood from the decoding process. The quantizer simply minimizes the log energy error for each band, with the exception that at very low rate, larger errors are allowed in the coarse energy to minimize the bit-rate. When the avaialble CPU requirements allow it, it is best to try encoding the coarse energy both with and without inter-frame prediction such that the best prediction mode can be selected. The optimal mode depends on the coding rate, the available bit-rate, and the current rate of packet loss.

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

The search for the best codevector y is performed by alg_quant() (vq.c). There are several possible approaches to the search, with a trade-off between quality and complexity. The method used in the reference implementation computes an initial codeword y1 by projecting the normalized spectrum X onto the codebook pyramid of K-1 pulses: y0 = round_towards_zero( (K-1) * X / sum(abs(X))) Depending on N, K and the input data, the initial codeword y0 may contain from 0 to K-1 non-zero values. All the remaining pulses, with the exception of the last one, are found iteratively with a greedy search that minimizes the normalized correlation between y and X:

The search described above is considered to be a good trade-off between quality and computational cost. However, there are other possible ways to search the PVQ codebook and the implementers MAY use any other search methods.