# Difference between revisions of "AMR-NB-WIP"

This text aims to be a simpler and more explicit document of the AMR narrow band decoding processes to aid in development of a decoder. Reference to sections of the specification will be made in the following format: (c.f. §5.2.5). Happy reading.

## Nomenclature weirdness

Throughout the specification, a number of references are made to the same (or very similar) items with fairly confusing variation. They are listed below to aid understanding of the following text but efforts will be made to consistently use one item name throughout or to use both with the lesser used name in parenthesis.

• Fixed / Innovative (also algebraic when referring to the codebook)
• Quantified means not quantised

## Summary

• Mode dependent bitstream parsing
• Indices parsed from bitstream
• Indices decoded to give LSF vectors, fractional pitch lags, innovative code vectors and the pitch and innovative gains
• LSF vectors converted to LP filter coefficients at each subframe
• Subframe decoding
• Excitation vector = adaptive code vector * adaptive (pitch) gain + innovative code vector * innovative gain
• Excitation vector filtered through an LP synthesis filter to reconstruct speech
• Speech signal filtered with adaptive postfilter

## Bitstream parsing

Documented on http://wiki.multimedia.cx/index.php?title=AMR-NB and in 26.101

For implementation, see http://svn.mplayerhq.hu/soc/amr/amrnbdec.c?view=markup

## Decoding of LP filter parameters

The received indices of LSP quantization are used to reconstruct the quantified LSP vectors. (c.f. §5.2.5)

### 12.2kbps mode summary

• indices into code books are parsed from the bit stream
• indices give elements of split matrix quantised (SMQ) residual LSF vectors from the relevant code books
• prediction from the previous frame is added to obtain the mean-removed LSF vectors
• the LSF vectors are converted to cosine domain LSP vectors

#### Decoding SMQ residual LSF vectors

The elements of the SMQ vectors are stored at an index into a code book that varies according to the mode. There are 5 code books for the 12.2kbps mode corresponding to the 5 indices. These tables will be referred to as:

lsf_m_n

the number of indices parsed according to the mode
the index 'position' i.e. 1 for the first index, etc

The 5 indices are stored using 7, 8, 8 + sign bit, 8, 6 bits respectively. The four elements of a 'split quantized sub-matrix' are stored at the index position in the appropriate code book are:

1st index in 1st code book
r1_1, r1_2, r2_1, r2_2
2nd index in 2nd code book
r1_3, r1_4, r2_3, r2_4
3rd index in 3rd code book
r1_5, r1_6, r2_5, r2_6
4th index in 4th code book
r1_7, r1_8, r2_7, r2_8
5th index in 5th code book
r1_9, r1_10, r2_9, r2_10

With rj_i :

the first or second residual lsf vector
the coefficient of a residual lsf vector ( i = 1, ..., 10 )
rj_i
residual line spectral frequencies (LSFs) in Hz

#### Mean-removed LSF vector prediction

z_j(n)
the mean-removed LSF vector at the jth subframe
r_j(n)
prediction residual vector of frame n at the jth subframe
^r_2(n-1)
the quantified residual vector from the previous frame at the 2nd subframe

lsf_mean_m
a table of the means of the LSF coefficients
the number of indices parsed according to the mode
fj
the LSF vectors

#### LSF to LSP vector conversion

q_k[i]
the ith coefficient of the kth line spectral pair (LSP) in the cosine domain
the two lsf vectors give the LSP vectors q2, q4 at the 2nd and 4th subframes; k = 2*j
f_j[i]
ith coefficient of the jth LSF vector; [0,4000] Hz
f_s
sampling frequency in Hz (8kHz)

### Other active modes summary

The process for the other modes is similar to that for the 12.2kbps mode.

• indices into code books are parsed from the bit stream
• indices give elements of a split matrix quantised (SMQ) residual LSF vector from the relevant code books
• prediction from the previous frame is added to obtain the mean-removed LSF vector
• the LSF vector is converted to a cosine domain LSP vector

#### Decoding the SMQ residual LSF vector

The 3 indices are stored with the following numbers of bits:

Mode (kbps) 1st index (bits) 2nd index (bits) 3rd index (bits)
10.2 8 9 9
7.95 9 9 9
7.40 8 9 9
6.70 8 9 9
5.90 8 9 9
5.15 8 8 7
4.75 8 8 7

The four elements of a 'split quantized sub-matrix' are stored at the index position in the appropriate code book are:

1st index in 1st code book
r_1, r_2, r_3
2nd index in 2nd code book
r_4, r_5, r_6
3rd index in 3rd code book
r_7, r_8, r_9, r_10
r_i
residual LSF vector (Hz)
the coefficient of vector ( i = 1, ..., 10 )

#### Mean-removed LSF vector prediction

z_j(n)[i]
the ith coefficient of the mean-removed LSF vector at the jth subframe
r_j(n)[i]
the ith coefficient of the prediction residual vector of frame n at the jth subframe
pred_fac[i]
the ith coefficient of the prediction factor
^r_j(n-1)[i]
the ith coefficient of the the quantified residual vector of the previous frame at the jth subframe

These processes give the LSP vector at the 4th subframe (q4)

### LSP vector to LP filter coefficient conversion (c.f. §5.2.4)

 for i=1..5
f1[i]  = 2*f1[i-2] - 2*q[2i-1]*f1[i-1]
for j=i-1..1
f1[j] +=   f1[j-2] - 2*q[2i-1]*f1[j-1]
end
end


f1[-1] = 0; f1[0] = 0;

Same for f2[i] with q[2i] instead of q[2i-1]

 for i=1..5
f'1[i] = f1[i] + f1[i-1]
f'2[i] = f2[i] - f2[i-1]
end

 for i=1..5
a_i = 0.5*f'1[i]    + 0.5*f'2[i]
end
for i=6..10
a_i = 0.5*f'1[11-i] - 0.5*f'2[11-i]
end

a_i
the LP filter coefficients

## Decoding of the pitch (adaptive codebook) vector

• indices parsed from bitstream
• indices give integer and fractional parts of the pitch lag
• pitch vector v(n) is found by interpolating the past excitation u(n) at the pitch lag using an FIR filter. (c.f. §5.6)

### Decode pitch lag

Note: division in this section is integer division!

#### 12.2kbps mode - 1/6 resolution pitch lag

##### First and third subframes

In the first and third subframes, a fractional pitch lag is used with resolutions:

• 1/6 in the range [17 3/6, 94 3/6]
• 1 in the range [95, 143]

...encoded using 9 bits.

For [17 3/6, 94 3/6] the pitch index is encoded as:

 pitch_index = (pitch_lag_int - 17)*6 + pitch_lag_frac - 3;

pitch_lag_int
integer part of the pitch lag in the range [17, 94]
pitch_lag_frac
fractional part of the pitch lag in 1/6 units in the range [-2, 3]

so...

 if(pitch_index < (94 4/6 - 17 3/6)*6)
// fractional part is encoded in range [17 3/6, 94 3/6]
pitch_lag_int = (pitch_index + 5)/6 + 17;
pitch_lag_frac = pitch_index - pitch_lag_int*6 + (17 3/6)*6;


And for [95, 143] the pitch index is encoded as:

 pitch_index = (pitch_lag_int - 95) + (94 4/6 - 17 3/6)*6;

pitch_lag_int
integer pitch lag in the range [95, 143]

so...

 else
// only integer part encoded in range [95, 143], no fractional part
pitch_lag_int  = pitch_index - (94 4/6 - 17 3/6)*6 + 95;
pitch_lag_frac = 0;

##### Second and fourth subframes

In the second and fourth subframes, a pitch lag resolution of 1/6 is always used in the range [T1 - 5 3/6, T1 + 4 3/6], where T1 is nearest integer to the fractional pitch lag of the previous (1st or 3rd) subframe. The search range is bounded by [18, 143]. In this case the pitch delay is encoded using 6 bits and is therefore in the range [0,63].

So the search range for the pitch lag is:

 search_range_min = max(pitch_lag_int_prev - 5, 18);
search_range_max = search_range_min + 9;
if(search_range_max > 143) {
search_range_max = 143;
search_range_min = search_range_max - 9;
}

pitch_lag_int_prev
the integer part of the pitch lag from the previous sub frame

The pitch index is encoded as:

 pitch_index = (pitch_lag_int - (search_range_min - 1))*6 + pitch_lag_frac - 3;

pitch_lag_int
the integer part of the pitch lag in the range [search_range_min - 1, search_range_max]
pitch_lag_frac
the fractional part of the pitch lag in the range [-2, 3]

The formula for the pitch_index has been chosen to map pitch_lag_int [search_range_min - 1, search_range_max] and pitch_lag_frac [-2, 3] to [0,60]. (pitch_index = [0, 10]*6 + [-2, 3] - 3 = [0, 6, ..., 60] + [-5, 0] = [0,60])

So the pitch lag is calculated through:

 // integer part of pitch lag = position in range [search_range_min - 1, search_range_max] + lower bound of range
pitch_lag_int  = (pitch_index + 5)/6 + search_range_min - 1;
// fractional part of pitch lag = pitch index - (integer part without offset)*6 - 1 3/6 offset to bring the values to the correct range
pitch_lag_frac = pitch_index - ((pitch_index + 5)/6)*6 - 9;


Note that when using integers and integer division to conduct (pitch_index + 5)/6 the result is similar to taking the ceiling of pitch_index/6.0.

#### Others modes - 1/3 resolution pitch lag

##### First and third subframes

In the first and third subframes, a fractional pitch lag is used with resolutions:

• 1/3 in the range [19 1/3, 84 2/3]
• 1 in the range [85, 143]

...encoded using 8 bits.

For [19 1/3, 84 2/3] the pitch lag is encoded as:

 pitch_index = pitch_lag_int*3 + pitch_lag_frac - (19 1/3)*3;

pitch_lag_int
integer part of the pitch lag in the range [19, 84]
pitch_lag_frac
fractional part of the pitch lag in 1/3 units in the range [0, 2]

so...

 if(pitch_index < (85 - 19 1/3)*3)
// fractional part is encoded in range [19 1/3, 84 2/3]
pitch_lag_int = (pitch_index + 2)/3 + 19;
pitch_lag_frac = pitch_index - pitch_lag_int*3 + (19 1/3)*3;


And for [85, 143] the pitch index is encoded as:

 pitch_index = pitch_lag_int - 85 + (85 - 19 1/3)*3;

pitch_lag_int
integer pitch lag in the range [85, 143]

so...

 else
// only integer part encoded in range [85, 143], no fractional part
pitch_lag_int  = pitch_index - (85 - 19 1/3)*3 + 85;
pitch_lag_frac = 0;

##### Second and fourth subframes

In the second and fourth subframes, the pitch lag resolution varies depending on the mode as follows:

• 7.95 kbps mode
• resolution of 1/3 is always used in the range [T1 - 10 2/3, T1 + 9 2/3]
• encoded using 6 bits => pitch_index is in the range [0, 63]
• 10.2 and 7.40 kbps modes
• resolution of 1/3 is always used in the range [T1 - 5 2/3, T1 + 4 2/3]
• encoded using 5 bits => pitch_index is in the range [0, 31]
• 6.70, 5.90, 5.15 and 4.75 kbps modes
• resolution of 1 is used in the range [T1 - 5, T1 + 4]
• resolution of 1/3 is always used in the range [T1 - 1 2/3, T1 + 2/3]
• encoded using 4 bits => pitch_index is in the range [0, 15]

Where T1 is nearest integer to the fractional pitch lag of the previous (1st or 3rd) subframe. The search range is bounded by [20, 143].

So the search range for the pitch lag is:

 lower_bound = 5;
range = 9;
if(mode == 7.95) {
lower_bound = 10;
range = 19;
}
search_range_min = max(pitch_lag_int_prev - lower_bound, 20);
search_range_max = search_range_min + range;
if(search_range_max > 143) {
search_range_max = 143;
search_range_min = search_range_max - range;
}

pitch_lag_int_prev
the integer part of the pitch lag from the previous sub frame

For modes 7.40, 7.95 and 10.2 the pitch index is encoded as:

 pitch_index = (pitch_lag_int - search_range_min)*3 + pitch_lag_frac + 2;

pitch_lag_int
the integer part of the pitch lag in the range [search_range_min, search_range_max]
pitch_lag_frac
the fractional part of the pitch lag in the range [-1, 1]

So the pitch lag is calculated through:

 // integer part of pitch lag = position of pitch lag in range [search_range_min, search_range_max] + lower bound of the range
pitch_lag_int = (pitch_index + 2)/3 - 1 + search_range_min;
// fractional part of pitch lag = pitch index - (integer part without offset)*3 - 2/3 to bring the values to the correct range
pitch_lag_frac = pitch_index - ((pitch_index + 2)/3 - 1)*3 - 2;


For modes 4.75, 5.15, 5.90 and 6.70:

 t1_temp = max( min(pitch_lag_int_prev, search_range_min + 5), search_range_max - 4 );

t1_temp
predicted pitch lag from the previous frame adjusted to fit into the 0 position of the search range

The pitch index is encoded as:

 // if pitch lag is below T1 - 1 2/3
if( pitch_lag_int*3 + pitch_lag_frac <= (t1_temp - 2)*3 ) {
// encode with resolution 1
index = (pitch_lag_int - t1_temp) + 5;
// else if pitch lag is below T1 + 1
}else if( pitch_lag_int*3 + pitch_lag_frac < (t1_temp + 1)*3 ) {
// encode with resolution 1/3
index = ( pitch_lag_int*3 + pitch_lag_frac - (t1_temp - 2)*3 ) + 3;
// else pitch lag is above T1 + 2/3
}else {
// encode with resolution 1
index = (pitch_lag_int - t1_temp) + 11;
}

pitch_lag_int
the integer part of the pitch lag in the range [search_range_min, search_range_max]
pitch_lag_frac
the fractional part of the pitch lag in the range [-1, 1]

The possible pitch indices and values are:

 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -5 -4 -3 -2 -1 2/3 -1 1/3 -1 -2/3 -1/3 0 1/3 2/3 1 2 3 4

So the pitch lag is calculated through:

 if(pitch_index < 4) {
// integer part of pitch lag = pitch lag position in range [t1_temp - 5, t1_temp - 2] + lower bound of range
pitch_lag_int = pitch_index + (t1_temp - 5);
// this range is coded with resolution 1 so no fractional part
pitch_lag_frac = 0;
}else if(pitch_index < 12) {
pitch_lag_int = (pitch_index - 2)/3 + (t1_temp - 2);
pitch_lag_frac = (pitch_index - 4) - ((pitch_index - 2)/3)*3 - 11;
}else {
// integer part of pitch lag = pitch lag position in range [t1_temp + 1, t1_temp + 4] + lower bound of range
pitch_lag_int = pitch_index - 12 + t1_temp + 1;
// this range is coded with resolution 1 so no fractional part
pitch_lag_frac = 0;
}


### Calculate pitch vector

integer pitch lag
sample position in the vectors 0, ..., 39
0, ..., 5 corresponding to fractions 0, 1/6, 2/6, 3/6, -2/6, -1/6 respectively

This equation can be used for both 1/3 and 1/6 resolution simply by multiplying t by 2 in the 1/3 case.

(Note: the coefficients b60 are in the reference source in an array called inter6)

## Decoding of the fixed (innovative or algebraic) vector

• the excitation pulse positions and signs are parsed from the bit stream
• the pulse positions and signs are encoded differently depending on the mode
• the fixed code book vector, c(n), is then constructed from the pulse positions and signs
• if pitch_lag_int is less than the subframe size (40), the pitch sharpening procedure is applied

### Decoding the pulse positions

#### 12.2 kbps mode

• 10 pulse positions each coded using 3 bit Gray codes
• signs coded using 1 bit each for 5 pulse pairs

Pulse Positions
i0,i5 0, 5, 10, 15, 20, 25, 30, 35
i1,i6 1, 6, 11, 16, 21, 26, 31, 36
i2,i7 2, 7, 12, 17, 22, 27, 32, 37
i3,i8 3, 8, 13, 18, 23, 28, 33, 38
i4,i9 4, 9, 14, 19, 24, 29, 34, 39

#### 10.2 kbps mode

• 8 pulse positions, 4 pairs, coded as 3 values using 10, 10 and 7 bits
• signs coded using 1 bit each for 4 pulse pairs

Pulse Positions
i0,i4 0, 4, 8, 12, 16, 20, 24, 28, 32, 36
i1,i5 1, 5, 9, 13, 17, 21, 25, 29, 33, 37
i2,i6 2, 6, 10, 14, 18, 22, 26, 30, 34, 38
i3,i7 3, 7, 11, 15, 19, 23, 27, 31, 35, 39

#### 7.95 and 7.40 kbps modes

• 4 pulse positions coded using 3, 3, 3 and 4 bits
• signs coded using 1 bit for each pulse

Pulse Positions
i0 0, 5, 10, 15, 20, 25, 30, 35
i1 1, 6, 11, 16, 21, 26, 31, 36
i2 2, 7, 12, 17, 22, 27, 32, 37
i3 3, 8, 13, 18, 23, 28, 33, 38

4, 9, 14, 19, 24, 29, 34, 39

#### 6.70 kbps mode

• 3 pulse positions coded using 3, 4 and 4 bits
• signs coded using 1 bit for each pulse

Pulse Positions
i0 0, 5, 10, 15, 20, 25, 30, 35
i1 1, 6, 11, 16, 21, 26, 31, 36

3, 8, 13, 18, 23, 28, 33, 38

i2 2, 7, 12, 17, 22, 27, 32, 37

4, 9, 14, 19, 24, 29, 34, 39

#### 5.90 kbps mode

• 2 pulse positions coded using 4 and 5 bits
• signs coded using 1 bit for each pulse

Pulse Positions
i0 1, 6, 11, 16, 21, 26, 31, 36

3, 8, 13, 18, 23, 28, 33, 38

i1 0, 5, 10, 15, 20, 25, 30, 35

1, 6, 11, 16, 21, 26, 31, 36
2, 7, 12, 17, 22, 27, 32, 37
4, 9, 14, 19, 24, 29, 34, 39

#### 5.15 and 4.75 kbps modes

• 2 pulse positions coded using 1 bit for the position subset and 3 bits per pulse
• signs coded using 1 bit for each pulse

Subframe Subset Pulse Positions
1 1 i0 0, 5, 10, 15, 20, 25, 30, 35
i1 2, 7, 12, 17, 22, 27, 32, 37
2 i0 1, 6, 11, 16, 21, 26, 31, 36
i1 3, 8, 13, 18, 23, 28, 33, 38
2 1 i0 0, 5, 10, 15, 20, 25, 30, 35
i1 3, 8, 13, 18, 23, 28, 33, 38
2 i0 2, 7, 12, 17, 22, 27, 32, 37
i1 4, 9, 14, 19, 24, 29, 34, 39
3 1 i0 0, 5, 10, 15, 20, 25, 30, 35
i1 2, 7, 12, 17, 22, 27, 32, 37
2 i0 1, 6, 11, 16, 21, 26, 31, 36
i1 4, 9, 14, 19, 24, 29, 34, 39
4 1 i0 0, 5, 10, 15, 20, 25, 30, 35
i1 3, 8, 13, 18, 23, 28, 33, 38
2 i0 1, 6, 11, 16, 21, 26, 31, 36
i1 4, 9, 14, 19, 24, 29, 34, 39

### Fixed codebook vector construction

All c(n) are zero if there is no pulse at position n. If there is a pulse at position n then it has the corresponding sign as parsed above.

### Pitch sharpening

β
the decoded pitch gain, ^g_p, bounded by [0.0,1.0] for 12.2. kbps or [0.0,0.8] for other modes

## Decoding of the pitch and fixed codebook gains

### Fixed gain prediction

A moving average prediction of the innovation (fixed) energy is conducted.

g_c'
fixed gain prediction
\tilde{E}
predicted energy [dB]
\bar{E}
desired mean innovation (fixed) energy [dB]
E_I
calculated mean innovation (fixed) energy [dB]

4-tap MA prediction coefficients [0.68, 0.58, 0.34, 0.19]
^R(k)
quantified prediction errors at subframe k (tabulated in ref source)

Desired mean innovation (fixed) energy:

Mode (kbps) Mean energy (dB)
12.2 36
10.2 33
7.95 36
7.40 30
6.70 28.75
5.90 33
5.15 33
4.75 33

E_I
calculated mean innovation (fixed) energy [dB]
subframe size 40
c(n)
fixed codebook vector

### Dequantisation of the gains

#### 12.2kbps and 7.95kbps - scalar quantised gains

The received indices are used to find the quantified pitch gain, ^g_p, and the quantified fixed gain correction factor, ^γ_gc.

##### Pitch gain

The parsed gain index is used to obtain the quantified pitch gain, ^g_p, from the corresponding codebook. (qua_gain_pitch in the reference source.) In the 12.2 kbps mode, the two least significant bits are cleared.

##### Fixed gain correction factor

The parsed gain index is used to obtain the quantified fixed gain, ^g_c, from the corresponding codebook. (qua_gain_code in the reference source.) The table stores ^γ_gc, and the quantised energy error in two forms (these are needed for the moving average calculation of the predicted fixed gain):

qua_ener_MR122 = log2(^γ_gc) qua_ener = 20*log10(^γ_gc)

^γ_gc is stored at Q11 (i.e. it's multiplied by 2^11.) qua_ener_MR122 and qua_ener are stored at Q10.

#### Other modes - vector quantised gains

The received index gives both the quantified adaptive codebook gain, ^g_p, and the quantified algebraic codebook gain correction factor, ^γ_gc.

The tables contains the following data:

^g_p           (Q14),
^γ_gc          (Q12), (^g_c = g_c'*^γ_gc),
qua_ener_MR122 (Q10), (log2(^γ_gc))
qua_ener       (Q10)  (20*log10(^γ_gc))

The log2() and log10() values are calculated on the fixed point value
(g_fac Q12) and not on the original floating point value of g_fac
to make the quantizer/MA predictdor use corresponding values.


Q: Should we bother using all the same tables at the same Q#? Should we calculate qua_ener* on-the-fly from ^γ_gc? A: log2 is fast on ints but generally log is quite slow. 20log10() standard expression for energy. If speed doesn't matter, calculate on the fly. My conclusion: Use tables, probably the ones from the reference source.

The codebook used depends on the mode:

6.70, 7.40, 10.2 kbps modes - table_gain_highrates in the reference source Four consecutive entries give ^g_p, ^γ_gc, qua_ener_MR122 and qua_ener. Apparently the values for qua_ener are the original ones from IS641 to ensure bit-exactness but are not exactly the rounded value of 20log10(^γ_gc).

5.15, 5.90 kbps modes - table_gain_lowrates Similar to table_gain_highrates, four consecutive entries give ^g_p, ^γ_gc, qua_ener_MR122 and qua_ener. There are no special notes for this table.

4.75 kbps mode - table_gain_MR475 Unlike the above mentioned tables, four consecutive values give ^g_p, ^γ_gc for subframes 0,2, and then ^g_p, ^γ_gc for subframes 1,3.

At the very least I think these tables could be redesigned a little.

## Smoothing of the fixed codebook gain

10.2, 6.70, 5.90, 5.15, 4.75 kbit/s modes only

### Calculate averaged LSP vector

\bar{q}(n)
averaged LSP vector at frame n
^q_4(n)
quantified LSP vector for the 4th subframe at frame n

### Calculate fixed gain smoothing factor

diff_m
difference measure at subframe m
loops over LSPs
loops over subframes
^q_m
quantified LSP vector at subframe m

k_m
fixed gain smoothing factor
K_1
0.4
K_2
0.25
diff_m
difference measure at subframe m

If diff_m has been greater than 0.65 for 10 subframes, k_m is set to 1.0 (i.e. no smoothing) for 40 subframes.

### Calculate mean fixed gain

\bar{g_c}(m)
mean fixed gain at subframe m
^g_c(k)
quantified fixed gain at subframe k

### Calculate smoothed fixed gain

^g_c
quantified fixed gain
\bar{g_c}
averaged fixed gain

## Anti-sparseness processing

7.95, 6.70, 5.90, 5.15, 4.75 kbit/s modes only

### Evaluate impulse response filter strength

The fixed vector, c(n), has only a few pulses per subframe. In certain conditions, to reduce perceptual artifacts arising from this, the vector is circularly convolved with a predefined impulse response. The selection of the strength of filter is made based on the decoded gains.

 if ^g_p < 0.6
impNr = 0
else if ^g_p < 0.9
impNr = 1
else
impNr = 2

impNr = 0
strong impulse response filter
impNr = 1
medium impulse response filter
impNr = 2
no filtering
 if ^g_c(k) > 2 * ^g_c(k-1)
impNr = min( impNr + 1, 2 )
else if impNr = 0 AND median of last five ^g_p >= 0.6
impNr = min( impNr(k), min(impNr(k-1) + 1, 2) )


### Circular convolution of fixed vector and impulse response filter

(c * h)
convolution of vectors c and h, in this case circular convolution
h[n]
nth coefficient of the impulse response used for filtering
c[n]
nth coefficient of the fixed vector

n = 0, ..., 39

To make the convolution circular, make the impulse response circular by taking h[-m] = h[39-m]

## Computing the reconstructed speech

### Construct excitation

u(n)
excitation vector
^g_p
quantified pitch gain
v(n)
pitch vector
^g_c
quantified fixed gain
c(n)
fixed vector

### Emphasise pitch vector contribution

This is apparently a post-processing technique.

^u(n)
excitation vector with emphasised pitch vector contribution
β
^g_p bounded by [0.0, 0.8] or [0.0, 1.0] depending on mode

### Apply adaptive gain control (AGC) through gain scaling

η
gain scaling factor for emphasised excitation

^u'(n)
gain-scaled emphasised excitation

### Calculate reconstructed speech samples

^s(n)
reconstructed speech samples
^a_i
LP filter coefficients

Note: for n-i < 0, ^s(n-i) should be take from previous speech samples if existing, else we will consider them 0 as this behaviour is undefined the specification.

If an overflow occurs during synthesis, the pitch vector, v(n), is scaled down by a factor of 4 and synthesis is conducted again bypassing emphasising the pitch vector contribution and adaptive gain control.

Q: What classifies an overflow? A: s(n) < -32768 or s(n) > 32767 (i.e. 16-bit signed int)

## Post-processing

(c.f. §6.2.1)

#### IIR filtering

The speech samples, ^s(n), are filtered through a formant filter and a tilt compensation filter.

γ_n, γ_d
control the amount of formant post-filtering

The speech samples, ^s(n), are filtered through ^A(z/γ_n) to produce the residual signal ^r(n).

^r(n) is filtered through 1/^A(z/γ_d).

The output is filtered through H_t(z) (the tilt compensation filter) resulting in the post-filtered speech signal, ^s_f(n).

L_h
22 - the truncation of the impulse response
h_f
impulse response of the formant filter H_f

So, in summation notation:

##### 12.2 and 10.2 kbps modes

γ_n = 0.7

γ_d = 0.75

γ_t = 0.8 if k_1' > 0; 0 otherwise

γ_n = 0.55

γ_d = 0.7

γ_t = 0.8