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Appendix 2 - FFTs with precomputed LUTs

Module by: Anthony Blake. E-mail the author

This Appendix contains source code listings corresponding to the FFT implementations with precomputed coefficients in Implementation details.

Listing 1: Simple radix-2 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 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);
}
Listing 2: Simple split-radix 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);
}
Listing 3: 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))
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);
}
Listing 4: Simple tangent 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)))
 
float
s(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);
}

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