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This Appendix contains source code listings corresponding to the FFT implementations with precomputed coefficients in Implementation details .
#include <math.h>#include <complex.h>#include <stdio.h>#include <stdlib.h>
typedef complex float data_t;
#define W(N,k) (cexp(-2.0f * M_PI * I * (float)k / (float)N))
data_t *LUT;
void ditfft2(data_t *in, data_t *out, int log2stride, int stride, int N) {if(N == 2) {
out[0] = in[0] + in[stride];
out[N/2] = in[0] - in[stride];
}else{ditfft2(in, out, log2stride+1, stride << 1, N >> 1);
ditfft2(in+stride, out+N/2, log2stride+1, stride << 1, N >> 1);{ /* k=0 -> no multiplication */
data_t Ek = out[0];
data_t Ok = out[N/2];
out[0] = Ek + Ok;
out[N/2] = Ek - Ok;
}
int k;for(k=1;k<N/2;k++) {
data_t Ek = out[k];
data_t Ok = out[(k+N/2)];
data_t w = LUT[k<<log2stride];out[k] = Ek + w * Ok;out[(k+N/2) ] = Ek - w * Ok;}
}}
void fft_init(int N) {
LUT = malloc(N/2 * sizeof(data_t));int i;
for(i=0;i<N/2;i++) LUT[i] = W(N,i);}
Simple radix-2 FFT with precomputed LUT
#include <complex.h>#include <stdio.h>#include <stdlib.h>
typedef complex float data_t;
#define W(N,k) (cexp(-2.0f * M_PI * I * (float)k / (float)N))data_t *LUT1;
data_t *LUT3;
void splitfft(data_t *in, data_t *out, int log2stride, int stride, int N) {if(N == 1) {
out[0] = in[0];}else if(N == 2) {
out[0] = in[0] + in[stride];
out[N/2] = in[0] - in[stride];
}else{splitfft(in, out, log2stride+1, stride << 1, N >> 1);
splitfft(in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
splitfft(in+3*stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);{
data_t Uk = out[0];
data_t Zk = out[0+N/2];
data_t Uk2 = out[0+N/4];
data_t Zdk = out[0+3*N/4];
out[0] = Uk + (Zk + Zdk);
out[0+N/2] = Uk - (Zk + Zdk);
out[0+N/4] = Uk2 - I*(Zk - Zdk);
out[0+3*N/4] = Uk2 + I*(Zk - Zdk);
}int k;
for(k=1;k<N/4;k++) {
data_t Uk = out[k];
data_t Zk = out[k+N/2];
data_t Uk2 = out[k+N/4];
data_t Zdk = out[k+3*N/4];
data_t w1 = LUT1[k<<log2stride];data_t w3 = LUT3[k<<log2stride];out[k] = Uk + (w1*Zk + w3*Zdk);out[k+N/2] = Uk - (w1*Zk + w3*Zdk);out[k+N/4] = Uk2 - I*(w1*Zk - w3*Zdk);out[k+3*N/4] = Uk2 + I*(w1*Zk - w3*Zdk);}
}}
void fft_init(int N) {
LUT1 = malloc(N/4 * sizeof(data_t));LUT3 = malloc(N/4 * sizeof(data_t));
int i;for(i=0;i<N/4;i++) LUT1[i] = W(N,i);for(i=0;i<N/4;i++) LUT3[i] = W(N,3*i);}
Simple split-radix FFT with precomputed LUT
#include <math.h>#include <complex.h>#include <stdio.h>#include <stdlib.h>
typedef complex float data_t;
#define W(N,k) (cexp(-2.0f * M_PI * I * (float)k / (float)N))data_t *LUT;
void conjfft(data_t *base, int TN,
data_t *in, data_t *out, int log2stride, int stride, int N) {if(N == 1) {
if(in < base) in += TN;
out[0] = in[0];}else if(N == 2) {
data_t *i0 = in, *i1 = in + stride;if(i0 < base) i0 += TN;
if(i1 < base) i1 += TN;
out[0] = *i0 + *i1;
out[N/2] = *i0 - *i1;
}else{conjfft(base, TN, in, out, log2stride+1, stride << 1, N >> 1);
conjfft(base, TN, in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
conjfft(base, TN, in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);{
data_t Uk = out[0];
data_t Zk = out[0+N/2];
data_t Uk2 = out[0+N/4];
data_t Zdk = out[0+3*N/4];
out[0] = Uk + (Zk + Zdk);
out[0+N/2] = Uk - (Zk + Zdk);
out[0+N/4] = Uk2 - I*(Zk - Zdk);
out[0+3*N/4] = Uk2 + I*(Zk - Zdk);
}int k;
for(k=1;k<N/4;k++) {
data_t Uk = out[k];
data_t Zk = out[k+N/2];
data_t Uk2 = out[k+N/4];
data_t Zdk = out[k+3*N/4];
data_t w = LUT[k<<log2stride];out[k] = Uk + (w*Zk + conj(w)*Zdk);out[k+N/2] = Uk - (w*Zk + conj(w)*Zdk);out[k+N/4] = Uk2 - I*(w*Zk - conj(w)*Zdk);out[k+3*N/4] = Uk2 + I*(w*Zk - conj(w)*Zdk);}
}}void fft_init(int N) {
LUT = malloc(N/4 * sizeof(data_t));int i;
for(i=0;i<N/4;i++) LUT[i] = W(N,i);}
Simple conjugate-pair FFT with precomputed LUT
#include <math.h>#include <complex.h>#include <stdio.h>#include <stdlib.h>
typedef complex float data_t;
#define W(N,k) (cexp(-2.0f * M_PI * I * (float)(k) / (float)(N)))
floats(int n, int k) {
if (n <= 4) return 1.0f;
int k4 = k % (n/4);
if (k4 <= n/8)
return (s(n/4,k4) * cosf(2.0f * M_PI * (float)k4 / (float)n)); return (s(n/4,k4) * sinf(2.0f * M_PI * (float)k4 / (float)n));
}
data_t *LUT, *LUT0, *LUT1, *LUT2;float *s2, *s4;
void tangentfft8(data_t *base, int TN, data_t *in, data_t *out, int log2stride,
int stride, int N) { if(N == 1) {
if(in < base) in += TN;
out[0] = in[0]; }else if(N == 2) {
data_t *i0 = in, *i1 = in + stride;if(i0 < base) i0 += TN;
if(i1 < base) i1 += TN;
out[0] = *i0 + *i1;
out[N/2] = *i0 - *i1;
}else if(N == 4) { tangentfft8(base, TN, in, out, log2stride+1, stride << 1, N >> 1);
tangentfft8(base, TN, in+stride, out+2, log2stride+1, stride << 1, N >> 1);
data_t temp1 = out[0] + out[2];
data_t temp2 = out[0] - out[2]; out[0] = temp1; out[2] = temp2; temp1 = out[1] - I*out[3];
temp2 = out[1] + I*out[3]; out[1] = temp1; out[3] = temp2;
}else{ tangentfft8(base, TN, in, out, log2stride+2, stride << 2, N >> 2);
tangentfft8(base, TN, in+(stride*2), out+2*N/8, log2stride+3, stride << 3, N >> 3);
tangentfft8(base, TN, in-(stride*2), out+3*N/8, log2stride+3, stride << 3, N >> 3);
tangentfft8(base, TN, in+(stride), out+4*N/8, log2stride+2, stride << 2, N >> 2);
tangentfft8(base, TN, in-(stride), out+6*N/8, log2stride+2, stride << 2, N >> 2);
int k; for(k=0;k<N/8;k++) {
data_t w0 = LUT0[k<<log2stride];data_t w1 = LUT1[k<<log2stride];data_t w2 = LUT2[k<<log2stride];
data_t zk_p = w0 * out[k+4*N/8];
data_t zk_n = conj(w0) * out[k+6*N/8];
data_t zk2_p = w1 * out[k+5*N/8];
data_t zk2_n = conj(w1) * out[k+7*N/8];
data_t uk = out[k] * s4[k<<log2stride]; data_t uk2 = out[k+N/8] * s4[k+N/8 << log2stride]; data_t yk_p = w2 * out[k+2*N/8]; data_t yk_n = conj(w2) * out[k+3*N/8];data_t y0 = (yk_p + yk_n)*s2[k<<log2stride];data_t y1 = (yk_p - yk_n)*I*s2[k+N/8 << log2stride];
out[k] = uk + y0 + (zk_p + zk_n);
out[k+4*N/8] = uk + y0 - (zk_p + zk_n);
out[k+2*N/8] = uk - y0 - I*(zk_p - zk_n);
out[k+6*N/8] = uk - y0 + I*(zk_p - zk_n);
out[k+1*N/8] = uk2 - y1 + (zk2_p + zk2_n);
out[k+3*N/8] = uk2 + y1 - I*(zk2_p - zk2_n);
out[k+5*N/8] = uk2 - y1 - (zk2_p + zk2_n);
out[k+7*N/8] = uk2 + y1 + I*(zk2_p - zk2_n);
} }
}
void tangentfft4(data_t *base, int TN, data_t *in, data_t *out, int log2stride,
int stride, int N) {if(N == 1) {
if(in < base) in += TN;
out[0] = in[0];}else if(N == 2) {
data_t *i0 = in, *i1 = in + stride;if(i0 < base) i0 += TN;
if(i1 < base) i1 += TN;
out[0] = *i0 + *i1;
out[N/2] = *i0 - *i1;
}else{tangentfft4(base, TN, in, out, log2stride+1, stride << 1, N >> 1);
tangentfft8(base, TN, in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
tangentfft8(base, TN, in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);{
data_t Uk = out[0];
data_t Zk = out[0+N/2];
data_t Uk2 = out[0+N/4];
data_t Zdk = out[0+3*N/4];
out[0] = Uk + (Zk + Zdk);
out[0+N/2] = Uk - (Zk + Zdk);
out[0+N/4] = Uk2 - I*(Zk - Zdk);
out[0+3*N/4] = Uk2 + I*(Zk - Zdk);
}int k;
for(k=1;k<N/4;k++) {
data_t Uk = out[k];
data_t Zk = out[k+N/2];
data_t Uk2 = out[k+N/4];
data_t Zdk = out[k+3*N/4];
data_t w = LUT[k<<log2stride];out[k] = Uk + (w*Zk + conj(w)*Zdk);out[k+N/2] = Uk - (w*Zk + conj(w)*Zdk);out[k+N/4] = Uk2 - I*(w*Zk - conj(w)*Zdk);out[k+3*N/4] = Uk2 + I*(w*Zk - conj(w)*Zdk);}
}}
void fft_init(int N) {
LUT0 = malloc(N/8 * sizeof(data_t));LUT1 = malloc(N/8 * sizeof(data_t));
LUT2 = malloc(N/8 * sizeof(data_t)); LUT = malloc(N/4 * sizeof(data_t));
s2 = malloc(N/4 * sizeof(float));
s4 = malloc(N/4 * sizeof(float));
int i;for(i=0;i<N/8;i++) LUT0[i] = W(N,i)*s(N/4,i)/s(N,i);for(i=0;i<N/8;i++) LUT1[i] = W(N,i+N/8)*s(N/4,i+N/8)/s(N,i+N/8);for(i=0;i<N/8;i++) LUT2[i] = W(N,2*i)*s(N/8,i)/s(N/2,i);for(i=0;i<N/4;i++) LUT[i] = W(N,i)*s(N/4,i);for(i=0;i<N/4;i++) s4[i] = s(N/4,i)/s(N,i);for(i=0;i<N/4;i++) s2[i] = s(N/2,i)/s(N,i);}
Simple tangent FFT with precomputed LUT
Read also:
OpenStax, Computing the fast fourier transform on simd microprocessors. OpenStax CNX. Jul 15, 2012 Download for free at http://cnx.org/content/col11438/1.2
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