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b97b4b58a6
ffmpeg/libavcodec/mpegaudioenc.c
Michael Niedermayer b97b4b58a6 Merge remote branch 'qatar/master' 
* qatar/master:
  mpegaudiodec: group #includes more sanely
  mpegaudio: remove #if 0 blocks
  ffmpeg.c: reset avoptions after each input/output file.
  ffmpeg.c: store per-output stream sws flags.
  mpegaudio: remove CONFIG_MPEGAUDIO_HP option
  mpegtsenc: Clear st->priv_data when freeing it
  udp: Fix receiving RTP data over multicast
  rtpproto: Remove an unused variable
  regtest: fix wma tests
NOT pulled:  mpegaudio: remove CONFIG_AUDIO_NONSHORT
  regtest: separate flags for encoding and decoding

Conflicts:
	ffmpeg.c

Merged-by: Michael Niedermayer <michaelni@gmx.at>
2011-05-10 04:03:17 +02:00

787 lines
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/*
* The simplest mpeg audio layer 2 encoder
* Copyright (c) 2000, 2001 Fabrice Bellard
*
* This file is part of FFmpeg.
*
* FFmpeg is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* FFmpeg is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with FFmpeg; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
/**
* @file
* The simplest mpeg audio layer 2 encoder.
*/
#include "avcodec.h"
#include "put_bits.h"
#define FRAC_BITS 15 /* fractional bits for sb_samples and dct */
#define WFRAC_BITS 14 /* fractional bits for window */
#include "mpegaudio.h"
/* currently, cannot change these constants (need to modify
quantization stage) */
#define MUL(a,b) (((int64_t)(a) * (int64_t)(b)) >> FRAC_BITS)
#define SAMPLES_BUF_SIZE 4096
typedef struct MpegAudioContext {
PutBitContext pb;
int nb_channels;
int lsf; /* 1 if mpeg2 low bitrate selected */
int bitrate_index; /* bit rate */
int freq_index;
int frame_size; /* frame size, in bits, without padding */
/* padding computation */
int frame_frac, frame_frac_incr, do_padding;
short samples_buf[MPA_MAX_CHANNELS][SAMPLES_BUF_SIZE]; /* buffer for filter */
int samples_offset[MPA_MAX_CHANNELS]; /* offset in samples_buf */
int sb_samples[MPA_MAX_CHANNELS][3][12][SBLIMIT];
unsigned char scale_factors[MPA_MAX_CHANNELS][SBLIMIT][3]; /* scale factors */
/* code to group 3 scale factors */
unsigned char scale_code[MPA_MAX_CHANNELS][SBLIMIT];
int sblimit; /* number of used subbands */
const unsigned char *alloc_table;
} MpegAudioContext;
/* define it to use floats in quantization (I don't like floats !) */
#define USE_FLOATS
#include "mpegaudiodata.h"
#include "mpegaudiotab.h"
static av_cold int MPA_encode_init(AVCodecContext *avctx)
{
MpegAudioContext *s = avctx->priv_data;
int freq = avctx->sample_rate;
int bitrate = avctx->bit_rate;
int channels = avctx->channels;
int i, v, table;
float a;
if (channels <= 0 || channels > 2){
av_log(avctx, AV_LOG_ERROR, "encoding %d channel(s) is not allowed in mp2\n", channels);
return -1;
}
bitrate = bitrate / 1000;
s->nb_channels = channels;
avctx->frame_size = MPA_FRAME_SIZE;
/* encoding freq */
s->lsf = 0;
for(i=0;i<3;i++) {
if (ff_mpa_freq_tab[i] == freq)
break;
if ((ff_mpa_freq_tab[i] / 2) == freq) {
s->lsf = 1;
break;
}
}
if (i == 3){
av_log(avctx, AV_LOG_ERROR, "Sampling rate %d is not allowed in mp2\n", freq);
return -1;
}
s->freq_index = i;
/* encoding bitrate & frequency */
for(i=0;i<15;i++) {
if (ff_mpa_bitrate_tab[s->lsf][1][i] == bitrate)
break;
}
if (i == 15){
av_log(avctx, AV_LOG_ERROR, "bitrate %d is not allowed in mp2\n", bitrate);
return -1;
}
s->bitrate_index = i;
/* compute total header size & pad bit */
a = (float)(bitrate * 1000 * MPA_FRAME_SIZE) / (freq * 8.0);
s->frame_size = ((int)a) * 8;
/* frame fractional size to compute padding */
s->frame_frac = 0;
s->frame_frac_incr = (int)((a - floor(a)) * 65536.0);
/* select the right allocation table */
table = ff_mpa_l2_select_table(bitrate, s->nb_channels, freq, s->lsf);
/* number of used subbands */
s->sblimit = ff_mpa_sblimit_table[table];
s->alloc_table = ff_mpa_alloc_tables[table];
av_dlog(avctx, "%d kb/s, %d Hz, frame_size=%d bits, table=%d, padincr=%x\n",
bitrate, freq, s->frame_size, table, s->frame_frac_incr);
for(i=0;i<s->nb_channels;i++)
s->samples_offset[i] = 0;
for(i=0;i<257;i++) {
int v;
v = ff_mpa_enwindow[i];
#if WFRAC_BITS != 16
v = (v + (1 << (16 - WFRAC_BITS - 1))) >> (16 - WFRAC_BITS);
#endif
filter_bank[i] = v;
if ((i & 63) != 0)
v = -v;
if (i != 0)
filter_bank[512 - i] = v;
}
for(i=0;i<64;i++) {
v = (int)(pow(2.0, (3 - i) / 3.0) * (1 << 20));
if (v <= 0)
v = 1;
scale_factor_table[i] = v;
#ifdef USE_FLOATS
scale_factor_inv_table[i] = pow(2.0, -(3 - i) / 3.0) / (float)(1 << 20);
#else
#define P 15
scale_factor_shift[i] = 21 - P - (i / 3);
scale_factor_mult[i] = (1 << P) * pow(2.0, (i % 3) / 3.0);
#endif
}
for(i=0;i<128;i++) {
v = i - 64;
if (v <= -3)
v = 0;
else if (v < 0)
v = 1;
else if (v == 0)
v = 2;
else if (v < 3)
v = 3;
else
v = 4;
scale_diff_table[i] = v;
}
for(i=0;i<17;i++) {
v = ff_mpa_quant_bits[i];
if (v < 0)
v = -v;
else
v = v * 3;
total_quant_bits[i] = 12 * v;
}
avctx->coded_frame= avcodec_alloc_frame();
avctx->coded_frame->key_frame= 1;
return 0;
}
/* 32 point floating point IDCT without 1/sqrt(2) coef zero scaling */
static void idct32(int *out, int *tab)
{
int i, j;
int *t, *t1, xr;
const int *xp = costab32;
for(j=31;j>=3;j-=2) tab[j] += tab[j - 2];
t = tab + 30;
t1 = tab + 2;
do {
t[0] += t[-4];
t[1] += t[1 - 4];
t -= 4;
} while (t != t1);
t = tab + 28;
t1 = tab + 4;
do {
t[0] += t[-8];
t[1] += t[1-8];
t[2] += t[2-8];
t[3] += t[3-8];
t -= 8;
} while (t != t1);
t = tab;
t1 = tab + 32;
do {
t[ 3] = -t[ 3];
t[ 6] = -t[ 6];
t[11] = -t[11];
t[12] = -t[12];
t[13] = -t[13];
t[15] = -t[15];
t += 16;
} while (t != t1);
t = tab;
t1 = tab + 8;
do {
int x1, x2, x3, x4;
x3 = MUL(t[16], FIX(SQRT2*0.5));
x4 = t[0] - x3;
x3 = t[0] + x3;
x2 = MUL(-(t[24] + t[8]), FIX(SQRT2*0.5));
x1 = MUL((t[8] - x2), xp[0]);
x2 = MUL((t[8] + x2), xp[1]);
t[ 0] = x3 + x1;
t[ 8] = x4 - x2;
t[16] = x4 + x2;
t[24] = x3 - x1;
t++;
} while (t != t1);
xp += 2;
t = tab;
t1 = tab + 4;
do {
xr = MUL(t[28],xp[0]);
t[28] = (t[0] - xr);
t[0] = (t[0] + xr);
xr = MUL(t[4],xp[1]);
t[ 4] = (t[24] - xr);
t[24] = (t[24] + xr);
xr = MUL(t[20],xp[2]);
t[20] = (t[8] - xr);
t[ 8] = (t[8] + xr);
xr = MUL(t[12],xp[3]);
t[12] = (t[16] - xr);
t[16] = (t[16] + xr);
t++;
} while (t != t1);
xp += 4;
for (i = 0; i < 4; i++) {
xr = MUL(tab[30-i*4],xp[0]);
tab[30-i*4] = (tab[i*4] - xr);
tab[ i*4] = (tab[i*4] + xr);
xr = MUL(tab[ 2+i*4],xp[1]);
tab[ 2+i*4] = (tab[28-i*4] - xr);
tab[28-i*4] = (tab[28-i*4] + xr);
xr = MUL(tab[31-i*4],xp[0]);
tab[31-i*4] = (tab[1+i*4] - xr);
tab[ 1+i*4] = (tab[1+i*4] + xr);
xr = MUL(tab[ 3+i*4],xp[1]);
tab[ 3+i*4] = (tab[29-i*4] - xr);
tab[29-i*4] = (tab[29-i*4] + xr);
xp += 2;
}
t = tab + 30;
t1 = tab + 1;
do {
xr = MUL(t1[0], *xp);
t1[0] = (t[0] - xr);
t[0] = (t[0] + xr);
t -= 2;
t1 += 2;
xp++;
} while (t >= tab);
for(i=0;i<32;i++) {
out[i] = tab[bitinv32[i]];
}
}
#define WSHIFT (WFRAC_BITS + 15 - FRAC_BITS)
static void filter(MpegAudioContext *s, int ch, const short *samples, int incr)
{
short *p, *q;
int sum, offset, i, j;
int tmp[64];
int tmp1[32];
int *out;
// print_pow1(samples, 1152);
offset = s->samples_offset[ch];
out = &s->sb_samples[ch][0][0][0];
for(j=0;j<36;j++) {
/* 32 samples at once */
for(i=0;i<32;i++) {
s->samples_buf[ch][offset + (31 - i)] = samples[0];
samples += incr;
}
/* filter */
p = s->samples_buf[ch] + offset;
q = filter_bank;
/* maxsum = 23169 */
for(i=0;i<64;i++) {
sum = p[0*64] * q[0*64];
sum += p[1*64] * q[1*64];
sum += p[2*64] * q[2*64];
sum += p[3*64] * q[3*64];
sum += p[4*64] * q[4*64];
sum += p[5*64] * q[5*64];
sum += p[6*64] * q[6*64];
sum += p[7*64] * q[7*64];
tmp[i] = sum;
p++;
q++;
}
tmp1[0] = tmp[16] >> WSHIFT;
for( i=1; i<=16; i++ ) tmp1[i] = (tmp[i+16]+tmp[16-i]) >> WSHIFT;
for( i=17; i<=31; i++ ) tmp1[i] = (tmp[i+16]-tmp[80-i]) >> WSHIFT;
idct32(out, tmp1);
/* advance of 32 samples */
offset -= 32;
out += 32;
/* handle the wrap around */
if (offset < 0) {
memmove(s->samples_buf[ch] + SAMPLES_BUF_SIZE - (512 - 32),
s->samples_buf[ch], (512 - 32) * 2);
offset = SAMPLES_BUF_SIZE - 512;
}
}
s->samples_offset[ch] = offset;
// print_pow(s->sb_samples, 1152);
}
static void compute_scale_factors(unsigned char scale_code[SBLIMIT],
unsigned char scale_factors[SBLIMIT][3],
int sb_samples[3][12][SBLIMIT],
int sblimit)
{
int *p, vmax, v, n, i, j, k, code;
int index, d1, d2;
unsigned char *sf = &scale_factors[0][0];
for(j=0;j<sblimit;j++) {
for(i=0;i<3;i++) {
/* find the max absolute value */
p = &sb_samples[i][0][j];
vmax = abs(*p);
for(k=1;k<12;k++) {
p += SBLIMIT;
v = abs(*p);
if (v > vmax)
vmax = v;
}
/* compute the scale factor index using log 2 computations */
if (vmax > 1) {
n = av_log2(vmax);
/* n is the position of the MSB of vmax. now
use at most 2 compares to find the index */
index = (21 - n) * 3 - 3;
if (index >= 0) {
while (vmax <= scale_factor_table[index+1])
index++;
} else {
index = 0; /* very unlikely case of overflow */
}
} else {
index = 62; /* value 63 is not allowed */
}
#if 0
printf("%2d:%d in=%x %x %d\n",
j, i, vmax, scale_factor_table[index], index);
#endif
/* store the scale factor */
assert(index >=0 && index <= 63);
sf[i] = index;
}
/* compute the transmission factor : look if the scale factors
are close enough to each other */
d1 = scale_diff_table[sf[0] - sf[1] + 64];
d2 = scale_diff_table[sf[1] - sf[2] + 64];
/* handle the 25 cases */
switch(d1 * 5 + d2) {
case 0*5+0:
case 0*5+4:
case 3*5+4:
case 4*5+0:
case 4*5+4:
code = 0;
break;
case 0*5+1:
case 0*5+2:
case 4*5+1:
case 4*5+2:
code = 3;
sf[2] = sf[1];
break;
case 0*5+3:
case 4*5+3:
code = 3;
sf[1] = sf[2];
break;
case 1*5+0:
case 1*5+4:
case 2*5+4:
code = 1;
sf[1] = sf[0];
break;
case 1*5+1:
case 1*5+2:
case 2*5+0:
case 2*5+1:
case 2*5+2:
code = 2;
sf[1] = sf[2] = sf[0];
break;
case 2*5+3:
case 3*5+3:
code = 2;
sf[0] = sf[1] = sf[2];
break;
case 3*5+0:
case 3*5+1:
case 3*5+2:
code = 2;
sf[0] = sf[2] = sf[1];
break;
case 1*5+3:
code = 2;
if (sf[0] > sf[2])
sf[0] = sf[2];
sf[1] = sf[2] = sf[0];
break;
default:
assert(0); //cannot happen
code = 0; /* kill warning */
}
#if 0
printf("%d: %2d %2d %2d %d %d -> %d\n", j,
sf[0], sf[1], sf[2], d1, d2, code);
#endif
scale_code[j] = code;
sf += 3;
}
}
/* The most important function : psycho acoustic module. In this
encoder there is basically none, so this is the worst you can do,
but also this is the simpler. */
static void psycho_acoustic_model(MpegAudioContext *s, short smr[SBLIMIT])
{
int i;
for(i=0;i<s->sblimit;i++) {
smr[i] = (int)(fixed_smr[i] * 10);
}
}
#define SB_NOTALLOCATED 0
#define SB_ALLOCATED 1
#define SB_NOMORE 2
/* Try to maximize the smr while using a number of bits inferior to
the frame size. I tried to make the code simpler, faster and
smaller than other encoders :-) */
static void compute_bit_allocation(MpegAudioContext *s,
short smr1[MPA_MAX_CHANNELS][SBLIMIT],
unsigned char bit_alloc[MPA_MAX_CHANNELS][SBLIMIT],
int *padding)
{
int i, ch, b, max_smr, max_ch, max_sb, current_frame_size, max_frame_size;
int incr;
short smr[MPA_MAX_CHANNELS][SBLIMIT];
unsigned char subband_status[MPA_MAX_CHANNELS][SBLIMIT];
const unsigned char *alloc;
memcpy(smr, smr1, s->nb_channels * sizeof(short) * SBLIMIT);
memset(subband_status, SB_NOTALLOCATED, s->nb_channels * SBLIMIT);
memset(bit_alloc, 0, s->nb_channels * SBLIMIT);
/* compute frame size and padding */
max_frame_size = s->frame_size;
s->frame_frac += s->frame_frac_incr;
if (s->frame_frac >= 65536) {
s->frame_frac -= 65536;
s->do_padding = 1;
max_frame_size += 8;
} else {
s->do_padding = 0;
}
/* compute the header + bit alloc size */
current_frame_size = 32;
alloc = s->alloc_table;
for(i=0;i<s->sblimit;i++) {
incr = alloc[0];
current_frame_size += incr * s->nb_channels;
alloc += 1 << incr;
}
for(;;) {
/* look for the subband with the largest signal to mask ratio */
max_sb = -1;
max_ch = -1;
max_smr = INT_MIN;
for(ch=0;ch<s->nb_channels;ch++) {
for(i=0;i<s->sblimit;i++) {
if (smr[ch][i] > max_smr && subband_status[ch][i] != SB_NOMORE) {
max_smr = smr[ch][i];
max_sb = i;
max_ch = ch;
}
}
}
#if 0
printf("current=%d max=%d max_sb=%d alloc=%d\n",
current_frame_size, max_frame_size, max_sb,
bit_alloc[max_sb]);
#endif
if (max_sb < 0)
break;
/* find alloc table entry (XXX: not optimal, should use
pointer table) */
alloc = s->alloc_table;
for(i=0;i<max_sb;i++) {
alloc += 1 << alloc[0];
}
if (subband_status[max_ch][max_sb] == SB_NOTALLOCATED) {
/* nothing was coded for this band: add the necessary bits */
incr = 2 + nb_scale_factors[s->scale_code[max_ch][max_sb]] * 6;
incr += total_quant_bits[alloc[1]];
} else {
/* increments bit allocation */
b = bit_alloc[max_ch][max_sb];
incr = total_quant_bits[alloc[b + 1]] -
total_quant_bits[alloc[b]];
}
if (current_frame_size + incr <= max_frame_size) {
/* can increase size */
b = ++bit_alloc[max_ch][max_sb];
current_frame_size += incr;
/* decrease smr by the resolution we added */
smr[max_ch][max_sb] = smr1[max_ch][max_sb] - quant_snr[alloc[b]];
/* max allocation size reached ? */
if (b == ((1 << alloc[0]) - 1))
subband_status[max_ch][max_sb] = SB_NOMORE;
else
subband_status[max_ch][max_sb] = SB_ALLOCATED;
} else {
/* cannot increase the size of this subband */
subband_status[max_ch][max_sb] = SB_NOMORE;
}
}
*padding = max_frame_size - current_frame_size;
assert(*padding >= 0);
}
/*
* Output the mpeg audio layer 2 frame. Note how the code is small
* compared to other encoders :-)
*/
static void encode_frame(MpegAudioContext *s,
unsigned char bit_alloc[MPA_MAX_CHANNELS][SBLIMIT],
int padding)
{
int i, j, k, l, bit_alloc_bits, b, ch;
unsigned char *sf;
int q[3];
PutBitContext *p = &s->pb;
/* header */
put_bits(p, 12, 0xfff);
put_bits(p, 1, 1 - s->lsf); /* 1 = mpeg1 ID, 0 = mpeg2 lsf ID */
put_bits(p, 2, 4-2); /* layer 2 */
put_bits(p, 1, 1); /* no error protection */
put_bits(p, 4, s->bitrate_index);
put_bits(p, 2, s->freq_index);
put_bits(p, 1, s->do_padding); /* use padding */
put_bits(p, 1, 0); /* private_bit */
put_bits(p, 2, s->nb_channels == 2 ? MPA_STEREO : MPA_MONO);
put_bits(p, 2, 0); /* mode_ext */
put_bits(p, 1, 0); /* no copyright */
put_bits(p, 1, 1); /* original */
put_bits(p, 2, 0); /* no emphasis */
/* bit allocation */
j = 0;
for(i=0;i<s->sblimit;i++) {
bit_alloc_bits = s->alloc_table[j];
for(ch=0;ch<s->nb_channels;ch++) {
put_bits(p, bit_alloc_bits, bit_alloc[ch][i]);
}
j += 1 << bit_alloc_bits;
}
/* scale codes */
for(i=0;i<s->sblimit;i++) {
for(ch=0;ch<s->nb_channels;ch++) {
if (bit_alloc[ch][i])
put_bits(p, 2, s->scale_code[ch][i]);
}
}
/* scale factors */
for(i=0;i<s->sblimit;i++) {
for(ch=0;ch<s->nb_channels;ch++) {
if (bit_alloc[ch][i]) {
sf = &s->scale_factors[ch][i][0];
switch(s->scale_code[ch][i]) {
case 0:
put_bits(p, 6, sf[0]);
put_bits(p, 6, sf[1]);
put_bits(p, 6, sf[2]);
break;
case 3:
case 1:
put_bits(p, 6, sf[0]);
put_bits(p, 6, sf[2]);
break;
case 2:
put_bits(p, 6, sf[0]);
break;
}
}
}
}
/* quantization & write sub band samples */
for(k=0;k<3;k++) {
for(l=0;l<12;l+=3) {
j = 0;
for(i=0;i<s->sblimit;i++) {
bit_alloc_bits = s->alloc_table[j];
for(ch=0;ch<s->nb_channels;ch++) {
b = bit_alloc[ch][i];
if (b) {
int qindex, steps, m, sample, bits;
/* we encode 3 sub band samples of the same sub band at a time */
qindex = s->alloc_table[j+b];
steps = ff_mpa_quant_steps[qindex];
for(m=0;m<3;m++) {
sample = s->sb_samples[ch][k][l + m][i];
/* divide by scale factor */
#ifdef USE_FLOATS
{
float a;
a = (float)sample * scale_factor_inv_table[s->scale_factors[ch][i][k]];
q[m] = (int)((a + 1.0) * steps * 0.5);
}
#else
{
int q1, e, shift, mult;
e = s->scale_factors[ch][i][k];
shift = scale_factor_shift[e];
mult = scale_factor_mult[e];
/* normalize to P bits */
if (shift < 0)
q1 = sample << (-shift);
else
q1 = sample >> shift;
q1 = (q1 * mult) >> P;
q[m] = ((q1 + (1 << P)) * steps) >> (P + 1);
}
#endif
if (q[m] >= steps)
q[m] = steps - 1;
assert(q[m] >= 0 && q[m] < steps);
}
bits = ff_mpa_quant_bits[qindex];
if (bits < 0) {
/* group the 3 values to save bits */
put_bits(p, -bits,
q[0] + steps * (q[1] + steps * q[2]));
} else {
put_bits(p, bits, q[0]);
put_bits(p, bits, q[1]);
put_bits(p, bits, q[2]);
}
}
}
/* next subband in alloc table */
j += 1 << bit_alloc_bits;
}
}
}
/* padding */
for(i=0;i<padding;i++)
put_bits(p, 1, 0);
/* flush */
flush_put_bits(p);
}
static int MPA_encode_frame(AVCodecContext *avctx,
unsigned char *frame, int buf_size, void *data)
{
MpegAudioContext *s = avctx->priv_data;
const short *samples = data;
short smr[MPA_MAX_CHANNELS][SBLIMIT];
unsigned char bit_alloc[MPA_MAX_CHANNELS][SBLIMIT];
int padding, i;
for(i=0;i<s->nb_channels;i++) {
filter(s, i, samples + i, s->nb_channels);
}
for(i=0;i<s->nb_channels;i++) {
compute_scale_factors(s->scale_code[i], s->scale_factors[i],
s->sb_samples[i], s->sblimit);
}
for(i=0;i<s->nb_channels;i++) {
psycho_acoustic_model(s, smr[i]);
}
compute_bit_allocation(s, smr, bit_alloc, &padding);
init_put_bits(&s->pb, frame, MPA_MAX_CODED_FRAME_SIZE);
encode_frame(s, bit_alloc, padding);
return put_bits_ptr(&s->pb) - s->pb.buf;
}
static av_cold int MPA_encode_close(AVCodecContext *avctx)
{
av_freep(&avctx->coded_frame);
return 0;
}
AVCodec ff_mp2_encoder = {
"mp2",
AVMEDIA_TYPE_AUDIO,
CODEC_ID_MP2,
sizeof(MpegAudioContext),
MPA_encode_init,
MPA_encode_frame,
MPA_encode_close,
NULL,
.sample_fmts = (const enum AVSampleFormat[]){AV_SAMPLE_FMT_S16,AV_SAMPLE_FMT_NONE},
.supported_samplerates= (const int[]){44100, 48000, 32000, 22050, 24000, 16000, 0},
.long_name = NULL_IF_CONFIG_SMALL("MP2 (MPEG audio layer 2)"),
};
#undef FIX
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