# MDCT and IMDCT based on FFTW3¶

• Author or source: moc.liamg@gnahz.auhuhs
• Type: analysis and synthesis filterbank
• Created: 2009-07-31 06:37:37
notes
```MDCT/IMDCT is the most widely used filterbank in digital audio coding, e.g. MP3, AAC, WMA,
OGG Vorbis, ATRAC.

suppose input x and N=size(x,1)/2. the MDCT transform matrix is
C=cos(pi/N*([0:2*N-1]'+.5+.5*N)*([0:N-1]+.5));
then MDCT spectrum for input x is
y=C'*x;
A well known fast algorithm is based on FFT :
(1) fold column-wisely the 2*N rows into N rows
(2) complex arrange the N rows into N/2 rows
(3) pre-twiddle, N/2-point complex fft, post-twiddle
(4) reorder to form the MDCT spectrum
in fact, (2)-(4) is a fast DCT-IV algorithm.

Implementation of the algorithm can be found in faac, but a little bit mess to extract for
standalone use, and I ran into that problem. So I wrote some c codes to implement
MDCT/IMDCT for any length that is a multiple of 4. Hopefully they will be useful to people
here.

I benchmarked the codes using 3 FFT routines, FFT in faac, kiss_fft, and the awful FFTW.
MDCT based on FFTW is the fastest, 2048-point MDCT single precision 10^5 times in 1.54s,
about 50% of FFT in faac on my Petium IV 3G Hz.

An author of the FFTW, Steven G. Johnson, has a hard-coded fixed size MDCT of 256 input
samples(http://jdj.mit.edu/~stevenj/mdct_128nr.c). My code is 13% slower than his.

Using the codes is very simple:
(1) init (declare first "extern void* mdctf_init(int)")
void* m_plan = mdctf_init(N);
(2) run mdct/imdct as many times as you wish
mdctf(freq, time, m_plan);
(3) free
mdctf_free(m_plan);

Of course you need the the fftw library. On Linux, gcc options are "-O2 -lfftw3f -lm".
This is single precision.

Enjoy :)
```
code
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206``` ```/********************************************************* MDCT/IMDCT of 4x length, Single Precision, based on FFTW shuhua dot zhang at gmail dot com Dept. of E.E., Tsinghua University *********************************************************/ #include #include #include #include typedef struct { int N; // Number of time data points float* twiddle; // Twiddle factor fftwf_complex* fft_in; // fft workspace, input fftwf_complex* fft_out; // fft workspace, output fftwf_plan fft_plan; // fft configuration } mdctf_plan; mdctf_plan* mdctf_init(int N); void mdctf_free(mdctf_plan* m_plan); void mdctf(float* mdct_line, float* time_signal, mdctf_plan* m_plan); void imdctf(float* time_signal, float* mdct_line, mdctf_plan* m_plan); mdctf_plan* mdctf_init(int N) { mdctf_plan* m_plan; double alpha, omiga, scale; int n; if( 0x00 != (N & 0x03)) { fprintf(stderr, " Expecting N a multiple of 4\n"); return NULL; } m_plan = (mdctf_plan*) malloc(sizeof(mdctf_plan)); m_plan->N = N; m_plan->twiddle = (float*) malloc(sizeof(float) * N >> 1); alpha = 2.f * M_PI / (8.f * N); omiga = 2.f * M_PI / N; scale = sqrt(sqrt(2.f / N)); for(n = 0; n < (N >> 2); n++) { m_plan->twiddle[2*n+0] = (float) (scale * cos(omiga * n + alpha)); m_plan->twiddle[2*n+1] = (float) (scale * sin(omiga * n + alpha)); } m_plan->fft_in = (fftwf_complex*) fftwf_malloc(sizeof(fftwf_complex) * N >> 2); m_plan->fft_out = (fftwf_complex*) fftwf_malloc(sizeof(fftwf_complex) * N >> 2); m_plan->fft_plan = fftwf_plan_dft_1d(N >> 2, m_plan->fft_in, m_plan->fft_out, FFTW_FORWARD, FFTW_MEASURE); return m_plan; } void mdctf_free(mdctf_plan* m_plan) { fftwf_destroy_plan(m_plan->fft_plan); fftwf_free(m_plan->fft_in); fftwf_free(m_plan->fft_out); free(m_plan->twiddle); free(m_plan); } void mdctf(float* mdct_line, float* time_signal, mdctf_plan* m_plan) { float *xr, *xi, r0, i0; float *cos_tw, *sin_tw, c, s; int N4, N2, N34, N54, n; N4 = (m_plan->N) >> 2; N2 = 2 * N4; N34 = 3 * N4; N54 = 5 * N4; cos_tw = m_plan->twiddle; sin_tw = cos_tw + 1; /* odd/even folding and pre-twiddle */ xr = (float*) m_plan->fft_in; xi = xr + 1; for(n = 0; n < N4; n += 2) { r0 = time_signal[N34-1-n] + time_signal[N34+n]; i0 = time_signal[N4+n] - time_signal[N4-1-n]; c = cos_tw[n]; s = sin_tw[n]; xr[n] = r0 * c + i0 * s; xi[n] = i0 * c - r0 * s; } for(; n < N2; n += 2) { r0 = time_signal[N34-1-n] - time_signal[-N4+n]; i0 = time_signal[N4+n] + time_signal[N54-1-n]; c = cos_tw[n]; s = sin_tw[n]; xr[n] = r0 * c + i0 * s; xi[n] = i0 * c - r0 * s; } /* complex FFT of N/4 long */ fftwf_execute(m_plan->fft_plan); /* post-twiddle */ xr = (float*) m_plan->fft_out; xi = xr + 1; for(n = 0; n < N2; n += 2) { r0 = xr[n]; i0 = xi[n]; c = cos_tw[n]; s = sin_tw[n]; mdct_line[n] = - r0 * c - i0 * s; mdct_line[N2-1-n] = - r0 * s + i0 * c; } } void imdctf(float* time_signal, float* mdct_line, mdctf_plan* m_plan) { float *xr, *xi, r0, i0, r1, i1; float *cos_tw, *sin_tw, c, s; int N4, N2, N34, N54, n; N4 = (m_plan->N) >> 2; N2 = 2 * N4; N34 = 3 * N4; N54 = 5 * N4; cos_tw = m_plan->twiddle; sin_tw = cos_tw + 1; /* pre-twiddle */ xr = (float*) m_plan->fft_in; xi = xr + 1; for(n = 0; n < N2; n += 2) { r0 = mdct_line[n]; i0 = mdct_line[N2-1-n]; c = cos_tw[n]; s = sin_tw[n]; xr[n] = -2.f * (i0 * s + r0 * c); xi[n] = -2.f * (i0 * c - r0 * s); } /* complex FFT of N/4 long */ fftwf_execute(m_plan->fft_plan); /* odd/even expanding and post-twiddle */ xr = (float*) m_plan->fft_out; xi = xr + 1; for(n = 0; n < N4; n += 2) { r0 = xr[n]; i0 = xi[n]; c = cos_tw[n]; s = sin_tw[n]; r1 = r0 * c + i0 * s; i1 = r0 * s - i0 * c; time_signal[N34-1-n] = r1; time_signal[N34+n] = r1; time_signal[N4+n] = i1; time_signal[N4-1-n] = -i1; } for(; n < N2; n += 2) { r0 = xr[n]; i0 = xi[n]; c = cos_tw[n]; s = sin_tw[n]; r1 = r0 * c + i0 * s; i1 = r0 * s - i0 * c; time_signal[N34-1-n] = r1; time_signal[-N4+n] = -r1; time_signal[N4+n] = i1; time_signal[N54-1-n] = i1; } } ```

• Date: 2009-08-05 14:42:44
• By: none
```Hi, your "freq, time" example in your comments feed into the main function as "mdct_line, time_signal" float pointers.
Can you explain what these are?
Thanks
D
```
```Hi,

Here I past a complete test bench for the MDCT/IMDCT routine. Suppose the MDCT/IMDCT routines named "mdctf.c" and the following benchmark routine named "ftestbench.c", the gcc compilation command will be
gcc -o ftestbench -O2 ftestbench.c mdctf.c -lfftw3f -lm

Shuhua Zhang, Aug. 11, 2009

/* benchmark MDCT and IMDCT, floating point */
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>

extern void* mdctf_init(int);

int main(int argc, char* argv[])
{
int N, r, i;
float* time;
float* freq;
void* m_plan;
clock_t t0, t1;

if(3 != argc)
{
fprintf(stderr, " Usage: %s <MDCT_SIZE> <run_times> \n", argv);
return -1;
}

sscanf(argv, "%d", &N);
sscanf(argv, "%d", &r);

time = (float*)malloc(sizeof(float) * N);
freq = (float*)malloc(sizeof(float) * (N >> 1));
for(i = 0; i < N; i++)
time[i] = 2.f * rand() / RAND_MAX - 1.f;

/* MDCT/IMDCT floating point initialization */
m_plan = mdctf_init(N);
if(NULL == m_plan)
{
free(freq);
free(time);
return -1;
}

/* benchmark MDCT floating point*/
t0 = clock();
for(i = 0; i < r; i++)
mdctf(freq, time, m_plan);
t1 = clock();
fprintf(stdout, "MDCT of size %d, float, running %d times, uses %.2e s\n",
N, r, (float) (t1 - t0) / CLOCKS_PER_SEC);

/* benchmark IMDCT floating point*/
t0 = clock();
for(i = 0; i < r; i++)
imdctf(time, freq, m_plan);
t1 = clock();
fprintf(stdout, "IMDCT of size %d, float, running %d times, uses %.2e s\n",
N, r, (float) (t1 - t0) / CLOCKS_PER_SEC);

/* free MDCT/IMDCT workspace */
mdctf_free(m_plan);

free(freq);
free(time);

return 0;

}
```