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FFmpeg/libavcodec/aacpsy.c

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/*
* AAC encoder psychoacoustic model
* Copyright (C) 2008 Konstantin Shishkov
*
* 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
* AAC encoder psychoacoustic model
*/
#include "avcodec.h"
#include "aactab.h"
#include "psymodel.h"
/***********************************
* TODOs:
* thresholds linearization after their modifications for attaining given bitrate
* try other bitrate controlling mechanism (maybe use ratecontrol.c?)
* control quality for quality-based output
**********************************/
/**
* constants for 3GPP AAC psychoacoustic model
* @{
*/
#define PSY_3GPP_SPREAD_LOW 1.5f // spreading factor for ascending threshold spreading (15 dB/Bark)
#define PSY_3GPP_SPREAD_HI 3.0f // spreading factor for descending threshold spreading (30 dB/Bark)
#define PSY_3GPP_RPEMIN 0.01f
#define PSY_3GPP_RPELEV 2.0f
/* LAME psy model constants */
#define PSY_LAME_FIR_LEN 21 ///< LAME psy model FIR order
#define AAC_BLOCK_SIZE_LONG 1024 ///< long block size
#define AAC_BLOCK_SIZE_SHORT 128 ///< short block size
#define AAC_NUM_BLOCKS_SHORT 8 ///< number of blocks in a short sequence
#define PSY_LAME_NUM_SUBBLOCKS 3 ///< Number of sub-blocks in each short block
/**
* @}
*/
/**
* information for single band used by 3GPP TS26.403-inspired psychoacoustic model
*/
typedef struct AacPsyBand{
float energy; ///< band energy
float ffac; ///< form factor
float thr; ///< energy threshold
float min_snr; ///< minimal SNR
float thr_quiet; ///< threshold in quiet
}AacPsyBand;
/**
* single/pair channel context for psychoacoustic model
*/
typedef struct AacPsyChannel{
AacPsyBand band[128]; ///< bands information
AacPsyBand prev_band[128]; ///< bands information from the previous frame
float win_energy; ///< sliding average of channel energy
float iir_state[2]; ///< hi-pass IIR filter state
uint8_t next_grouping; ///< stored grouping scheme for the next frame (in case of 8 short window sequence)
enum WindowSequence next_window_seq; ///< window sequence to be used in the next frame
/* LAME psy model specific members */
float attack_threshold; ///< attack threshold for this channel
float prev_energy_subshort[AAC_NUM_BLOCKS_SHORT * PSY_LAME_NUM_SUBBLOCKS];
int prev_attack; ///< attack value for the last short block in the previous sequence
}AacPsyChannel;
/**
* psychoacoustic model frame type-dependent coefficients
*/
typedef struct AacPsyCoeffs{
float ath [64]; ///< absolute threshold of hearing per bands
float barks [64]; ///< Bark value for each spectral band in long frame
float spread_low[64]; ///< spreading factor for low-to-high threshold spreading in long frame
float spread_hi [64]; ///< spreading factor for high-to-low threshold spreading in long frame
}AacPsyCoeffs;
/**
* 3GPP TS26.403-inspired psychoacoustic model specific data
*/
typedef struct AacPsyContext{
AacPsyCoeffs psy_coef[2];
AacPsyChannel *ch;
}AacPsyContext;
/**
* LAME psy model preset struct
*/
typedef struct {
int quality; ///< Quality to map the rest of the vaules to.
/* This is overloaded to be both kbps per channel in ABR mode, and
* requested quality in constant quality mode.
*/
float st_lrm; ///< short threshold for L, R, and M channels
} PsyLamePreset;
/**
* LAME psy model preset table for ABR
*/
static const PsyLamePreset psy_abr_map[] = {
/* TODO: Tuning. These were taken from LAME. */
/* kbps/ch st_lrm */
{ 8, 6.60},
{ 16, 6.60},
{ 24, 6.60},
{ 32, 6.60},
{ 40, 6.60},
{ 48, 6.60},
{ 56, 6.60},
{ 64, 6.40},
{ 80, 6.00},
{ 96, 5.60},
{112, 5.20},
{128, 5.20},
{160, 5.20}
};
/**
* LAME psy model preset table for constant quality
*/
static const PsyLamePreset psy_vbr_map[] = {
/* vbr_q st_lrm */
{ 0, 4.20},
{ 1, 4.20},
{ 2, 4.20},
{ 3, 4.20},
{ 4, 4.20},
{ 5, 4.20},
{ 6, 4.20},
{ 7, 4.20},
{ 8, 4.20},
{ 9, 4.20},
{10, 4.20}
};
/**
* LAME psy model FIR coefficient table
*/
static const float psy_fir_coeffs[] = {
-8.65163e-18 * 2, -0.00851586 * 2, -6.74764e-18 * 2, 0.0209036 * 2,
-3.36639e-17 * 2, -0.0438162 * 2, -1.54175e-17 * 2, 0.0931738 * 2,
-5.52212e-17 * 2, -0.313819 * 2
};
/**
* calculates the attack threshold for ABR from the above table for the LAME psy model
*/
static float lame_calc_attack_threshold(int bitrate)
{
/* Assume max bitrate to start with */
int lower_range = 12, upper_range = 12;
int lower_range_kbps = psy_abr_map[12].quality;
int upper_range_kbps = psy_abr_map[12].quality;
int i;
/* Determine which bitrates the value specified falls between.
* If the loop ends without breaking our above assumption of 320kbps was correct.
*/
for (i = 1; i < 13; i++) {
if (FFMAX(bitrate, psy_abr_map[i].quality) != bitrate) {
upper_range = i;
upper_range_kbps = psy_abr_map[i ].quality;
lower_range = i - 1;
lower_range_kbps = psy_abr_map[i - 1].quality;
break; /* Upper range found */
}
}
/* Determine which range the value specified is closer to */
if ((upper_range_kbps - bitrate) > (bitrate - lower_range_kbps))
return psy_abr_map[lower_range].st_lrm;
return psy_abr_map[upper_range].st_lrm;
}
/**
* LAME psy model specific initialization
*/
static void lame_window_init(AacPsyContext *ctx, AVCodecContext *avctx) {
int i;
for (i = 0; i < avctx->channels; i++) {
AacPsyChannel *pch = &ctx->ch[i];
if (avctx->flags & CODEC_FLAG_QSCALE)
pch->attack_threshold = psy_vbr_map[avctx->global_quality / FF_QP2LAMBDA].st_lrm;
else
pch->attack_threshold = lame_calc_attack_threshold(avctx->bit_rate / avctx->channels / 1000);
for (i = 0; i < AAC_NUM_BLOCKS_SHORT * PSY_LAME_NUM_SUBBLOCKS; i++)
pch->prev_energy_subshort[i] = 10.0f;
}
}
/**
* Calculate Bark value for given line.
*/
static av_cold float calc_bark(float f)
{
return 13.3f * atanf(0.00076f * f) + 3.5f * atanf((f / 7500.0f) * (f / 7500.0f));
}
#define ATH_ADD 4
/**
* Calculate ATH value for given frequency.
* Borrowed from Lame.
*/
static av_cold float ath(float f, float add)
{
f /= 1000.0f;
return 3.64 * pow(f, -0.8)
- 6.8 * exp(-0.6 * (f - 3.4) * (f - 3.4))
+ 6.0 * exp(-0.15 * (f - 8.7) * (f - 8.7))
+ (0.6 + 0.04 * add) * 0.001 * f * f * f * f;
}
static av_cold int psy_3gpp_init(FFPsyContext *ctx) {
AacPsyContext *pctx;
float bark;
int i, j, g, start;
float prev, minscale, minath;
ctx->model_priv_data = av_mallocz(sizeof(AacPsyContext));
pctx = (AacPsyContext*) ctx->model_priv_data;
minath = ath(3410, ATH_ADD);
for (j = 0; j < 2; j++) {
AacPsyCoeffs *coeffs = &pctx->psy_coef[j];
float line_to_frequency = ctx->avctx->sample_rate / (j ? 256.f : 2048.0f);
i = 0;
prev = 0.0;
for (g = 0; g < ctx->num_bands[j]; g++) {
i += ctx->bands[j][g];
bark = calc_bark((i-1) * line_to_frequency);
coeffs->barks[g] = (bark + prev) / 2.0;
prev = bark;
}
for (g = 0; g < ctx->num_bands[j] - 1; g++) {
coeffs->spread_low[g] = pow(10.0, -(coeffs->barks[g+1] - coeffs->barks[g]) * PSY_3GPP_SPREAD_LOW);
coeffs->spread_hi [g] = pow(10.0, -(coeffs->barks[g+1] - coeffs->barks[g]) * PSY_3GPP_SPREAD_HI);
}
start = 0;
for (g = 0; g < ctx->num_bands[j]; g++) {
minscale = ath(start * line_to_frequency, ATH_ADD);
for (i = 1; i < ctx->bands[j][g]; i++)
minscale = FFMIN(minscale, ath((start + i) * line_to_frequency, ATH_ADD));
coeffs->ath[g] = minscale - minath;
start += ctx->bands[j][g];
}
}
pctx->ch = av_mallocz(sizeof(AacPsyChannel) * ctx->avctx->channels);
lame_window_init(pctx, ctx->avctx);
return 0;
}
/**
* IIR filter used in block switching decision
*/
static float iir_filter(int in, float state[2])
{
float ret;
ret = 0.7548f * (in - state[0]) + 0.5095f * state[1];
state[0] = in;
state[1] = ret;
return ret;
}
/**
* window grouping information stored as bits (0 - new group, 1 - group continues)
*/
static const uint8_t window_grouping[9] = {
0xB6, 0x6C, 0xD8, 0xB2, 0x66, 0xC6, 0x96, 0x36, 0x36
};
/**
* Tell encoder which window types to use.
* @see 3GPP TS26.403 5.4.1 "Blockswitching"
*/
static FFPsyWindowInfo psy_3gpp_window(FFPsyContext *ctx,
const int16_t *audio, const int16_t *la,
int channel, int prev_type)
{
int i, j;
int br = ctx->avctx->bit_rate / ctx->avctx->channels;
int attack_ratio = br <= 16000 ? 18 : 10;
AacPsyContext *pctx = (AacPsyContext*) ctx->model_priv_data;
AacPsyChannel *pch = &pctx->ch[channel];
uint8_t grouping = 0;
int next_type = pch->next_window_seq;
FFPsyWindowInfo wi;
memset(&wi, 0, sizeof(wi));
if (la) {
float s[8], v;
int switch_to_eight = 0;
float sum = 0.0, sum2 = 0.0;
int attack_n = 0;
int stay_short = 0;
for (i = 0; i < 8; i++) {
for (j = 0; j < 128; j++) {
v = iir_filter(la[(i*128+j)*ctx->avctx->channels], pch->iir_state);
sum += v*v;
}
s[i] = sum;
sum2 += sum;
}
for (i = 0; i < 8; i++) {
if (s[i] > pch->win_energy * attack_ratio) {
attack_n = i + 1;
switch_to_eight = 1;
break;
}
}
pch->win_energy = pch->win_energy*7/8 + sum2/64;
wi.window_type[1] = prev_type;
switch (prev_type) {
case ONLY_LONG_SEQUENCE:
wi.window_type[0] = switch_to_eight ? LONG_START_SEQUENCE : ONLY_LONG_SEQUENCE;
next_type = switch_to_eight ? EIGHT_SHORT_SEQUENCE : ONLY_LONG_SEQUENCE;
break;
case LONG_START_SEQUENCE:
wi.window_type[0] = EIGHT_SHORT_SEQUENCE;
grouping = pch->next_grouping;
next_type = switch_to_eight ? EIGHT_SHORT_SEQUENCE : LONG_STOP_SEQUENCE;
break;
case LONG_STOP_SEQUENCE:
wi.window_type[0] = switch_to_eight ? LONG_START_SEQUENCE : ONLY_LONG_SEQUENCE;
next_type = switch_to_eight ? EIGHT_SHORT_SEQUENCE : ONLY_LONG_SEQUENCE;
break;
case EIGHT_SHORT_SEQUENCE:
stay_short = next_type == EIGHT_SHORT_SEQUENCE || switch_to_eight;
wi.window_type[0] = stay_short ? EIGHT_SHORT_SEQUENCE : LONG_STOP_SEQUENCE;
grouping = next_type == EIGHT_SHORT_SEQUENCE ? pch->next_grouping : 0;
next_type = switch_to_eight ? EIGHT_SHORT_SEQUENCE : LONG_STOP_SEQUENCE;
break;
}
pch->next_grouping = window_grouping[attack_n];
pch->next_window_seq = next_type;
} else {
for (i = 0; i < 3; i++)
wi.window_type[i] = prev_type;
grouping = (prev_type == EIGHT_SHORT_SEQUENCE) ? window_grouping[0] : 0;
}
wi.window_shape = 1;
if (wi.window_type[0] != EIGHT_SHORT_SEQUENCE) {
wi.num_windows = 1;
wi.grouping[0] = 1;
} else {
int lastgrp = 0;
wi.num_windows = 8;
for (i = 0; i < 8; i++) {
if (!((grouping >> i) & 1))
lastgrp = i;
wi.grouping[lastgrp]++;
}
}
return wi;
}
/**
* Calculate band thresholds as suggested in 3GPP TS26.403
*/
static void psy_3gpp_analyze(FFPsyContext *ctx, int channel,
const float *coefs, const FFPsyWindowInfo *wi)
{
AacPsyContext *pctx = (AacPsyContext*) ctx->model_priv_data;
AacPsyChannel *pch = &pctx->ch[channel];
int start = 0;
int i, w, g;
const int num_bands = ctx->num_bands[wi->num_windows == 8];
const uint8_t* band_sizes = ctx->bands[wi->num_windows == 8];
AacPsyCoeffs *coeffs = &pctx->psy_coef[wi->num_windows == 8];
//calculate energies, initial thresholds and related values - 5.4.2 "Threshold Calculation"
for (w = 0; w < wi->num_windows*16; w += 16) {
for (g = 0; g < num_bands; g++) {
AacPsyBand *band = &pch->band[w+g];
band->energy = 0.0f;
for (i = 0; i < band_sizes[g]; i++)
band->energy += coefs[start+i] * coefs[start+i];
band->energy *= 1.0f / (512*512);
band->thr = band->energy * 0.001258925f;
start += band_sizes[g];
ctx->psy_bands[channel*PSY_MAX_BANDS+w+g].energy = band->energy;
}
}
//modify thresholds - spread, threshold in quiet - 5.4.3 "Spreaded Energy Calculation"
for (w = 0; w < wi->num_windows*16; w += 16) {
AacPsyBand *band = &pch->band[w];
for (g = 1; g < num_bands; g++)
band[g].thr = FFMAX(band[g].thr, band[g-1].thr * coeffs->spread_low[g-1]);
for (g = num_bands - 2; g >= 0; g--)
band[g].thr = FFMAX(band[g].thr, band[g+1].thr * coeffs->spread_hi [g]);
for (g = 0; g < num_bands; g++) {
band[g].thr_quiet = FFMAX(band[g].thr, coeffs->ath[g]);
if (wi->num_windows != 8 && wi->window_type[1] != EIGHT_SHORT_SEQUENCE)
band[g].thr_quiet = FFMAX(PSY_3GPP_RPEMIN*band[g].thr_quiet,
FFMIN(band[g].thr_quiet,
PSY_3GPP_RPELEV*pch->prev_band[w+g].thr_quiet));
band[g].thr = FFMAX(band[g].thr, band[g].thr_quiet * 0.25);
ctx->psy_bands[channel*PSY_MAX_BANDS+w+g].threshold = band[g].thr;
}
}
memcpy(pch->prev_band, pch->band, sizeof(pch->band));
}
static av_cold void psy_3gpp_end(FFPsyContext *apc)
{
AacPsyContext *pctx = (AacPsyContext*) apc->model_priv_data;
av_freep(&pctx->ch);
av_freep(&apc->model_priv_data);
}
static void lame_apply_block_type(AacPsyChannel *ctx, FFPsyWindowInfo *wi, int uselongblock)
{
int blocktype = ONLY_LONG_SEQUENCE;
if (uselongblock) {
if (ctx->next_window_seq == EIGHT_SHORT_SEQUENCE)
blocktype = LONG_STOP_SEQUENCE;
} else {
blocktype = EIGHT_SHORT_SEQUENCE;
if (ctx->next_window_seq == ONLY_LONG_SEQUENCE)
ctx->next_window_seq = LONG_START_SEQUENCE;
if (ctx->next_window_seq == LONG_STOP_SEQUENCE)
ctx->next_window_seq = EIGHT_SHORT_SEQUENCE;
}
wi->window_type[0] = ctx->next_window_seq;
ctx->next_window_seq = blocktype;
}
static FFPsyWindowInfo psy_lame_window(FFPsyContext *ctx,
const int16_t *audio, const int16_t *la,
int channel, int prev_type)
{
AacPsyContext *pctx = (AacPsyContext*) ctx->model_priv_data;
AacPsyChannel *pch = &pctx->ch[channel];
int grouping = 0;
int uselongblock = 1;
int attacks[AAC_NUM_BLOCKS_SHORT + 1] = { 0 };
int i;
FFPsyWindowInfo wi;
memset(&wi, 0, sizeof(wi));
if (la) {
float hpfsmpl[AAC_BLOCK_SIZE_LONG];
float const *pf = hpfsmpl;
float attack_intensity[(AAC_NUM_BLOCKS_SHORT + 1) * PSY_LAME_NUM_SUBBLOCKS];
float energy_subshort[(AAC_NUM_BLOCKS_SHORT + 1) * PSY_LAME_NUM_SUBBLOCKS];
float energy_short[AAC_NUM_BLOCKS_SHORT + 1] = { 0 };
int chans = ctx->avctx->channels;
const int16_t *firbuf = la + (AAC_BLOCK_SIZE_SHORT/4 - PSY_LAME_FIR_LEN) * chans;
int j, att_sum = 0;
/* LAME comment: apply high pass filter of fs/4 */
for (i = 0; i < AAC_BLOCK_SIZE_LONG; i++) {
float sum1, sum2;
sum1 = firbuf[(i + ((PSY_LAME_FIR_LEN - 1) / 2)) * chans];
sum2 = 0.0;
for (j = 0; j < ((PSY_LAME_FIR_LEN - 1) / 2) - 1; j += 2) {
sum1 += psy_fir_coeffs[j] * (firbuf[(i + j) * chans] + firbuf[(i + PSY_LAME_FIR_LEN - j) * chans]);
sum2 += psy_fir_coeffs[j + 1] * (firbuf[(i + j + 1) * chans] + firbuf[(i + PSY_LAME_FIR_LEN - j - 1) * chans]);
}
hpfsmpl[i] = sum1 + sum2;
}
/* Calculate the energies of each sub-shortblock */
for (i = 0; i < PSY_LAME_NUM_SUBBLOCKS; i++) {
energy_subshort[i] = pch->prev_energy_subshort[i + ((AAC_NUM_BLOCKS_SHORT - 1) * PSY_LAME_NUM_SUBBLOCKS)];
assert(pch->prev_energy_subshort[i + ((AAC_NUM_BLOCKS_SHORT - 2) * PSY_LAME_NUM_SUBBLOCKS + 1)] > 0);
attack_intensity[i] = energy_subshort[i] / pch->prev_energy_subshort[i + ((AAC_NUM_BLOCKS_SHORT - 2) * PSY_LAME_NUM_SUBBLOCKS + 1)];
energy_short[0] += energy_subshort[i];
}
for (i = 0; i < AAC_NUM_BLOCKS_SHORT * PSY_LAME_NUM_SUBBLOCKS; i++) {
float const *const pfe = pf + AAC_BLOCK_SIZE_LONG / (AAC_NUM_BLOCKS_SHORT * PSY_LAME_NUM_SUBBLOCKS);
float p = 1.0f;
for (; pf < pfe; pf++)
if (p < fabsf(*pf))
p = fabsf(*pf);
pch->prev_energy_subshort[i] = energy_subshort[i + PSY_LAME_NUM_SUBBLOCKS] = p;
energy_short[1 + i / PSY_LAME_NUM_SUBBLOCKS] += p;
/* FIXME: The indexes below are [i + 3 - 2] in the LAME source.
* Obviously the 3 and 2 have some significance, or this would be just [i + 1]
* (which is what we use here). What the 3 stands for is ambigious, as it is both
* number of short blocks, and the number of sub-short blocks.
* It seems that LAME is comparing each sub-block to sub-block + 1 in the
* previous block.
*/
if (p > energy_subshort[i + 1])
p = p / energy_subshort[i + 1];
else if (energy_subshort[i + 1] > p * 10.0f)
p = energy_subshort[i + 1] / (p * 10.0f);
else
p = 0.0;
attack_intensity[i + PSY_LAME_NUM_SUBBLOCKS] = p;
}
/* compare energy between sub-short blocks */
for (i = 0; i < (AAC_NUM_BLOCKS_SHORT + 1) * PSY_LAME_NUM_SUBBLOCKS; i++)
if (!attacks[i / PSY_LAME_NUM_SUBBLOCKS])
if (attack_intensity[i] > pch->attack_threshold)
attacks[i / PSY_LAME_NUM_SUBBLOCKS] = (i % PSY_LAME_NUM_SUBBLOCKS) + 1;
/* should have energy change between short blocks, in order to avoid periodic signals */
/* Good samples to show the effect are Trumpet test songs */
/* GB: tuned (1) to avoid too many short blocks for test sample TRUMPET */
/* RH: tuned (2) to let enough short blocks through for test sample FSOL and SNAPS */
for (i = 1; i < AAC_NUM_BLOCKS_SHORT + 1; i++) {
float const u = energy_short[i - 1];
float const v = energy_short[i];
float const m = FFMAX(u, v);
if (m < 40000) { /* (2) */
if (u < 1.7f * v && v < 1.7f * u) { /* (1) */
if (i == 1 && attacks[0] < attacks[i])
attacks[0] = 0;
attacks[i] = 0;
}
}
att_sum += attacks[i];
}
if (attacks[0] <= pch->prev_attack)
attacks[0] = 0;
att_sum += attacks[0];
/* 3 below indicates the previous attack happened in the last sub-block of the previous sequence */
if (pch->prev_attack == 3 || att_sum) {
uselongblock = 0;
if (attacks[1] && attacks[0])
attacks[1] = 0;
if (attacks[2] && attacks[1])
attacks[2] = 0;
if (attacks[3] && attacks[2])
attacks[3] = 0;
if (attacks[4] && attacks[3])
attacks[4] = 0;
if (attacks[5] && attacks[4])
attacks[5] = 0;
if (attacks[6] && attacks[5])
attacks[6] = 0;
if (attacks[7] && attacks[6])
attacks[7] = 0;
if (attacks[8] && attacks[7])
attacks[8] = 0;
}
} else {
/* We have no lookahead info, so just use same type as the previous sequence. */
uselongblock = !(prev_type == EIGHT_SHORT_SEQUENCE);
}
lame_apply_block_type(pch, &wi, uselongblock);
wi.window_type[1] = prev_type;
if (wi.window_type[0] != EIGHT_SHORT_SEQUENCE) {
wi.num_windows = 1;
wi.grouping[0] = 1;
if (wi.window_type[0] == LONG_START_SEQUENCE)
wi.window_shape = 0;
else
wi.window_shape = 1;
} else {
int lastgrp = 0;
wi.num_windows = 8;
wi.window_shape = 0;
for (i = 0; i < 8; i++) {
if (!((pch->next_grouping >> i) & 1))
lastgrp = i;
wi.grouping[lastgrp]++;
}
}
/* Determine grouping, based on the location of the first attack, and save for
* the next frame.
* FIXME: Move this to analysis.
* TODO: Tune groupings depending on attack location
* TODO: Handle more than one attack in a group
*/
for (i = 0; i < 9; i++) {
if (attacks[i]) {
grouping = i;
break;
}
}
pch->next_grouping = window_grouping[grouping];
pch->prev_attack = attacks[8];
return wi;
}
const FFPsyModel ff_aac_psy_model =
{
.name = "3GPP TS 26.403-inspired model",
.init = psy_3gpp_init,
.window = psy_lame_window,
.analyze = psy_3gpp_analyze,
.end = psy_3gpp_end,
};