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Appendix 3 - FFTs with vectorized loops

Module by: Anthony Blake. E-mail the author

This Appendix contains source code listings corresponding to the vectorized FFT implementations in Implementation details.

Listing 1: Radix-2 FFT with vectorized loops
#include <math.h>
#include <complex.h>
#include <stdio.h>
#include <stdlib.h>
#include <xmmintrin.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 if(N == 4){
		ditfft2(in, out, log2stride+1, stride << 1, N >> 1);
		ditfft2(in+stride, out+N/2, log2stride+1, stride << 1, N >> 1);
	
    data_t temp0 = out[0] + out[2];
    data_t temp1 = out[0] - out[2];
    data_t temp2 = out[1] - I*out[3];
    data_t temp3 = out[1] + I*out[3];
    if(log2stride) {
			out[0] = creal(temp0) + creal(temp2)*I;
			out[1] = creal(temp1) + creal(temp3)*I;
			out[2] = cimag(temp0) + cimag(temp2)*I;
			out[3] = cimag(temp1) + cimag(temp3)*I;
		}else{
			out[0] = temp0;
			out[2] = temp1;
			out[1] = temp2;
			out[3] = temp3;
		}
	}else if(!log2stride){
		ditfft2(in, out, log2stride+1, stride << 1, N >> 1);
		ditfft2(in+stride, out+N/2, log2stride+1, stride << 1, N >> 1);
 
		int k;
		for(k=0;k<N/2;k+=4) {
			__m128 Ok_re = _mm_load_ps((float *)&out[k+N/2]);
			__m128 Ok_im = _mm_load_ps((float *)&out[k+N/2+2]);
			__m128 w_re = _mm_load_ps((float *)&LUT[log2stride][k]);
			__m128 w_im = _mm_load_ps((float *)&LUT[log2stride][k+2]);
			__m128 Ek_re = _mm_load_ps((float *)&out[k]);
			__m128 Ek_im = _mm_load_ps((float *)&out[k+2]);
			__m128 wOk_re = _mm_sub_ps(_mm_mul_ps(Ok_re,w_re),_mm_mul_ps(Ok_im,w_im));
			__m128 wOk_im = _mm_add_ps(_mm_mul_ps(Ok_re,w_im),_mm_mul_ps(Ok_im,w_re));
			__m128 out0_re = _mm_add_ps(Ek_re, wOk_re);
			__m128 out0_im = _mm_add_ps(Ek_im, wOk_im);
			__m128 out1_re = _mm_sub_ps(Ek_re, wOk_re);
			__m128 out1_im = _mm_sub_ps(Ek_im, wOk_im);
			_mm_store_ps((float *)(out+k), _mm_unpacklo_ps(out0_re, out0_im));
			_mm_store_ps((float *)(out+k+2), _mm_unpackhi_ps(out0_re, out0_im));
			_mm_store_ps((float *)(out+k+N/2), _mm_unpacklo_ps(out1_re, out1_im));
			_mm_store_ps((float *)(out+k+N/2+2), _mm_unpackhi_ps(out1_re, out1_im));
		}
	}else{
		ditfft2(in, out, log2stride+1, stride << 1, N >> 1);
		ditfft2(in+stride, out+N/2, log2stride+1, stride << 1, N >> 1);
		
		int k;
		for(k=0;k<N/2;k+=4) {
			__m128 Ok_re = _mm_load_ps((float *)&out[k+N/2]);
			__m128 Ok_im = _mm_load_ps((float *)&out[k+N/2+2]);
			__m128 w_re = _mm_load_ps((float *)&LUT[log2stride][k]);
			__m128 w_im = _mm_load_ps((float *)&LUT[log2stride][k+2]);
			__m128 Ek_re = _mm_load_ps((float *)&out[k]);
			__m128 Ek_im = _mm_load_ps((float *)&out[k+2]);
			__m128 wOk_re = _mm_sub_ps(_mm_mul_ps(Ok_re,w_re),_mm_mul_ps(Ok_im,w_im));
			__m128 wOk_im = _mm_add_ps(_mm_mul_ps(Ok_re,w_im),_mm_mul_ps(Ok_im,w_re));
			_mm_store_ps((float *)(out+k), _mm_add_ps(Ek_re, wOk_re));
			_mm_store_ps((float *)(out+k+2), _mm_add_ps(Ek_im, wOk_im));
			_mm_store_ps((float *)(out+k+N/2), _mm_sub_ps(Ek_re, wOk_re));
			_mm_store_ps((float *)(out+k+N/2+2), _mm_sub_ps(Ek_im, wOk_im));
		}
	}
}
 
void fft_init(int N) {
	int i;
	
	#define log2(x) ((int)(log(x)/log(2)))
	
	int n_luts = log2(N)-2;
	LUT = malloc(n_luts * sizeof(data_t *));
	for(i=0;i<n_luts;i++) {
		int n = N / pow(2,i);
		LUT[i] = _mm_malloc(n/2 * sizeof(data_t), 16);
		
		
		int j;		
		for(j=0;j<n/2;j+=4) {
			data_t w[4];
			int k;
			for(k=0;k<4;k++) w[k] = W(n,j+k);
 
			LUT[i][j]   = creal(w[0]) + creal(w[1])*I;
			LUT[i][j+1] = creal(w[2]) + creal(w[3])*I;
			LUT[i][j+2] = cimag(w[0]) + cimag(w[1])*I;
			LUT[i][j+3] = cimag(w[2]) + cimag(w[3])*I;
		}
	}
	
}
Listing 2: Vectorized math functions for split-radix implementations
typedef struct _reg_t {
	__m128 re, im;
} reg_t;
 
static inline reg_t MUL(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_sub_ps(_mm_mul_ps(a.re,b.re),_mm_mul_ps(a.im,b.im));
	r.im = _mm_add_ps(_mm_mul_ps(a.re,b.im),_mm_mul_ps(a.im,b.re));
	return r;
}
static inline reg_t MULJ(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_add_ps(_mm_mul_ps(a.re,b.re),_mm_mul_ps(a.im,b.im));
	r.im = _mm_sub_ps(_mm_mul_ps(a.im,b.re),_mm_mul_ps(a.re,b.im));
	return r;
}
 
static inline reg_t ADD(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_add_ps(a.re,b.re);
	r.im = _mm_add_ps(a.im,b.im);
	return r;
}
static inline reg_t SUB(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_sub_ps(a.re,b.re);
	r.im = _mm_sub_ps(a.im,b.im);
	return r;
}
static inline reg_t ADD_I(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_sub_ps(a.re,b.im);
	r.im = _mm_add_ps(a.im,b.re);
	return r;
}
static inline reg_t SUB_I(reg_t a, reg_t b) {
	reg_t r;
	r.re = _mm_add_ps(a.re,b.im);
	r.im = _mm_sub_ps(a.im,b.re);
	return r;
}
 
static inline reg_t LOAD(float *a) {
	reg_t r;
	r.re = _mm_load_ps(a);
	r.im = _mm_load_ps(a+4);
	return r;
}
static inline void STORE(float *a, reg_t r) {
	_mm_store_ps(a, r.re);
	_mm_store_ps(a+4, r.im);
}
static inline void STOREIL(float *a, reg_t r) {
	_mm_store_ps(a, _mm_unpacklo_ps(r.re, r.im));
	_mm_store_ps(a+4, _mm_unpackhi_ps(r.re, r.im));
}
Listing 3: Split-radix FFT with vectorized loops
#include <math.h>
#include <complex.h>
#include <stdio.h>
#include <stdlib.h>
#include <xmmintrin.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;
 
#include "vecmath.h"
 
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[1] = in[0] - in[stride];
	}else if(N == 4) {
		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 temp0 = out[0]  + (out[2] + out[3]);
			data_t temp1 = out[0]  - (out[2] + out[3]);
			data_t temp2 = out[1] - I*(out[2] - out[3]);
			data_t temp3 = out[1] + I*(out[2] - out[3]);
    if(log2stride) {
  		out[0] = creal(temp0) + creal(temp2)*I;
  		out[1] = creal(temp1) + creal(temp3)*I;
  		out[2] = cimag(temp0) + cimag(temp2)*I;
  		out[3] = cimag(temp1) + cimag(temp3)*I;
  	}else{
			out[0] = temp0;
			out[2] = temp1;
			out[1] = temp2;
			out[3] = temp3;
  	}
	
	}else if(N == 8) {
		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 o[8];
		{
			data_t Uk  = creal(out[0]) + creal(out[2])*I;
			data_t Zk  = out[4];
			data_t Uk2  = creal(out[1]) + creal(out[3])*I;
			data_t Zdk = out[6];
			
			o[0] = Uk  + (Zk + Zdk);
			o[4] = Uk  - (Zk + Zdk);
			o[2] = Uk2 - I*(Zk - Zdk);
			o[6] = Uk2 + I*(Zk - Zdk);
		}
		{
			data_t Uk = cimag(out[0]) + cimag(out[2])*I;
			data_t Zk  = out[5];
			data_t Uk2 = cimag(out[1]) + cimag(out[3])*I;
			data_t Zdk = out[7];
			data_t w1 = LUT1[log2stride][1];
			data_t w3 = LUT3[log2stride][1];
			
			o[1] = Uk  + (w1*Zk + w3*Zdk);
			o[5] = Uk  - (w1*Zk + w3*Zdk);
			o[3] = Uk2 - I*(w1*Zk - w3*Zdk);
			o[7] = Uk2 + I*(w1*Zk - w3*Zdk);
		}
		if(log2stride) {
			out[0] = creal(o[0]) + creal(o[1])*I;
			out[1] = creal(o[2]) + creal(o[3])*I;
			out[2] = cimag(o[0]) + cimag(o[1])*I;
			out[3] = cimag(o[2]) + cimag(o[3])*I;
			out[4] = creal(o[4]) + creal(o[5])*I;
			out[5] = creal(o[6]) + creal(o[7])*I;
			out[6] = cimag(o[4]) + cimag(o[5])*I;
			out[7] = cimag(o[6]) + cimag(o[7])*I;
		}else{
			int i;
			for(i=0;i<8;i++) out[i] = o[i];
		}
	}else if(!log2stride){
 
		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);
		int k;	
		for(k=0;k<N/4;k+=4) {
			reg_t Uk  = LOAD((float *)&out[k]);
			reg_t Zk  = LOAD((float *)&out[k+N/2]);
			reg_t Uk2 = LOAD((float *)&out[k+N/4]);
			reg_t Zdk = LOAD((float *)&out[k+3*N/4]);
			reg_t w1 = LOAD((float *)&LUT1[log2stride][k]);
			reg_t w3 = LOAD((float *)&LUT3[log2stride][k]);
			
			reg_t w3Zdk = MUL(w3, Zdk);
			reg_t w1Zk = MUL(w1, Zk);
			reg_t sum = ADD(w1Zk, w3Zdk);
			reg_t dif = SUB(w1Zk, w3Zdk);
			
			STOREIL((float *)&out[k], ADD(Uk, sum));
			STOREIL((float *)&out[k+N/2], SUB(Uk, sum));
			STOREIL((float *)&out[k+N/4], SUB_I(Uk2, dif));
			STOREIL((float *)&out[k+3*N/4], ADD_I(Uk2, dif));
		}
 
	}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);
		
		int k;
		for(k=0;k<N/4;k+=4) {
			reg_t Uk  = LOAD((float *)&out[k]);
			reg_t Zk  = LOAD((float *)&out[k+N/2]);
			reg_t Uk2 = LOAD((float *)&out[k+N/4]);
			reg_t Zdk = LOAD((float *)&out[k+3*N/4]);
			reg_t w1 = LOAD((float *)&LUT1[log2stride][k]);
			reg_t w3 = LOAD((float *)&LUT3[log2stride][k]);
			
			reg_t w3Zdk = MUL(w3, Zdk);
			reg_t w1Zk = MUL(w1, Zk);
			reg_t sum = ADD(w1Zk, w3Zdk);
			reg_t dif = SUB(w1Zk, w3Zdk);
			
			STORE((float *)&out[k], ADD(Uk, sum));
			STORE((float *)&out[k+N/2], SUB(Uk, sum));
			STORE((float *)&out[k+N/4], SUB_I(Uk2, dif));
			STORE((float *)&out[k+3*N/4], ADD_I(Uk2, dif));
		}
	}
}
 
void fft_init(int N) {
	#define log2(x) ((int)(log(x)/log(2)))
	
	int n_luts = log2(N)-1;
	LUT1 = malloc(n_luts * sizeof(data_t *));
	LUT3 = malloc(n_luts * sizeof(data_t *));
	int i;
	for(i=0;i<n_luts;i++) {
		int n = N / pow(2,i);
		LUT1[i] = _mm_malloc(n/4 * sizeof(data_t),16);
		LUT3[i] = _mm_malloc(n/4 * sizeof(data_t),16);
		
		if(n == 8) {
			int j;
			for(j=0;j<n/4;j++) {
				LUT1[i][j] = W(n,j);
				LUT3[i][j] = W(n,3*j);
			}
		}else{
			int j;		
			for(j=0;j<n/4;j+=4) {
				data_t w1[4], w3[4];
				int k;
				for(k=0;k<4;k++) w1[k] = W(n,j+k);
				for(k=0;k<4;k++) w3[k] = W(n,3*(j+k));
 
				LUT1[i][j]   = creal(w1[0]) + creal(w1[1])*I;
				LUT1[i][j+1] = creal(w1[2]) + creal(w1[3])*I;
				LUT1[i][j+2] = cimag(w1[0]) + cimag(w1[1])*I;
				LUT1[i][j+3] = cimag(w1[2]) + cimag(w1[3])*I;
				LUT3[i][j]   = creal(w3[0]) + creal(w3[1])*I;
				LUT3[i][j+1] = creal(w3[2]) + creal(w3[3])*I;
				LUT3[i][j+2] = cimag(w3[0]) + cimag(w3[1])*I;
				LUT3[i][j+3] = cimag(w3[2]) + cimag(w3[3])*I;
			}
		}
	}
}
 
Listing 4: Conjugate-pair FFT with vectorized loops
#include <math.h>
#include <complex.h>
#include <stdio.h>
#include <stdlib.h>
#include <xmmintrin.h>
 
typedef complex float data_t;
 
#define W(N,k) (cexp(-2.0f * M_PI * I * (float)(k) / (float)(N)))
data_t **LUT1;
 
#include "vecmath.h"
 
data_t *base;
int TN;
 
void conjfft(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) {
		conjfft(in, out, log2stride+1, stride << 1, N >> 1);
		conjfft(in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
	  conjfft(in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);
	
			data_t temp0 = out[0]  + (out[2] + out[3]);
			data_t temp1 = out[0]  - (out[2] + out[3]);
			data_t temp2 = out[1] - I*(out[2] - out[3]);
			data_t temp3 = out[1] + I*(out[2] - out[3]);
    if(log2stride) {
  		out[0] = creal(temp0) + creal(temp2)*I;
  		out[1] = creal(temp1) + creal(temp3)*I;
  		out[2] = cimag(temp0) + cimag(temp2)*I;
  		out[3] = cimag(temp1) + cimag(temp3)*I;
  	}else{
			out[0] = temp0;
			out[2] = temp1;
			out[1] = temp2;
			out[3] = temp3;
  	}
	
	}else if(N == 8) {
		conjfft(in, out, log2stride+1, stride << 1, N >> 1);
		conjfft(in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
	  conjfft(in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);
	
		data_t o[8];
		{
			data_t Uk  = creal(out[0]) + creal(out[2])*I;
			data_t Zk  = out[4];
			data_t Uk2  = creal(out[1]) + creal(out[3])*I;
			data_t Zdk = out[6];
			
			o[0] = Uk  + (Zk + Zdk);
			o[4] = Uk  - (Zk + Zdk);
			o[2] = Uk2 - I*(Zk - Zdk);
			o[6] = Uk2 + I*(Zk - Zdk);
		}
		{
			data_t Uk = cimag(out[0]) + cimag(out[2])*I;
			data_t Zk  = out[5];
			data_t Uk2 = cimag(out[1]) + cimag(out[3])*I;
			data_t Zdk = out[7];
			data_t w1 = LUT1[log2stride][1];
			
			o[1] = Uk  + (w1*Zk + conj(w1)*Zdk);
			o[5] = Uk  - (w1*Zk + conj(w1)*Zdk);
			o[3] = Uk2 - I*(w1*Zk - conj(w1)*Zdk);
			o[7] = Uk2 + I*(w1*Zk - conj(w1)*Zdk);
		}
		if(log2stride) {
			out[0] = creal(o[0]) + creal(o[1])*I;
			out[1] = creal(o[2]) + creal(o[3])*I;
			out[2] = cimag(o[0]) + cimag(o[1])*I;
			out[3] = cimag(o[2]) + cimag(o[3])*I;
			out[4] = creal(o[4]) + creal(o[5])*I;
			out[5] = creal(o[6]) + creal(o[7])*I;
			out[6] = cimag(o[4]) + cimag(o[5])*I;
			out[7] = cimag(o[6]) + cimag(o[7])*I;
		}else{
			int i;
			for(i=0;i<8;i++) out[i] = o[i];
		}
	}else if(!log2stride){
 
		conjfft(in, out, log2stride+1, stride << 1, N >> 1);
		conjfft(in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
	  conjfft(in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);
		int k;	
		for(k=0;k<N/4;k+=4) {
			reg_t Uk  = LOAD((float *)&out[k]);
			reg_t Zk  = LOAD((float *)&out[k+N/2]);
			reg_t Uk2 = LOAD((float *)&out[k+N/4]);
			reg_t Zdk = LOAD((float *)&out[k+3*N/4]);
			reg_t w1 = LOAD((float *)&LUT1[log2stride][k]);
			
			reg_t w3Zdk = MULJ(Zdk, w1);
			reg_t w1Zk = MUL(w1, Zk);
			reg_t sum = ADD(w1Zk, w3Zdk);
			reg_t dif = SUB(w1Zk, w3Zdk);
			
			STOREIL((float *)&out[k], ADD(Uk, sum));
			STOREIL((float *)&out[k+N/2], SUB(Uk, sum));
			STOREIL((float *)&out[k+N/4], SUB_I(Uk2, dif));
			STOREIL((float *)&out[k+3*N/4], ADD_I(Uk2, dif));
		}
 
	}else{
		conjfft(in, out, log2stride+1, stride << 1, N >> 1);
		conjfft(in+stride, out+N/2, log2stride+2, stride << 2, N >> 2);
	  conjfft(in-stride, out+3*N/4, log2stride+2, stride << 2, N >> 2);
		
		int k;
		for(k=0;k<N/4;k+=4) {
			reg_t Uk  = LOAD((float *)&out[k]);
			reg_t Zk  = LOAD((float *)&out[k+N/2]);
			reg_t Uk2 = LOAD((float *)&out[k+N/4]);
			reg_t Zdk = LOAD((float *)&out[k+3*N/4]);
			reg_t w1 = LOAD((float *)&LUT1[log2stride][k]);
			
			reg_t w3Zdk = MULJ(Zdk, w1);
			reg_t w1Zk = MUL(w1, Zk);
			reg_t sum = ADD(w1Zk, w3Zdk);
			reg_t dif = SUB(w1Zk, w3Zdk);
			
			STORE((float *)&out[k], ADD(Uk, sum));
			STORE((float *)&out[k+N/2], SUB(Uk, sum));
			STORE((float *)&out[k+N/4], SUB_I(Uk2, dif));
			STORE((float *)&out[k+3*N/4], ADD_I(Uk2, dif));
		}
	}
}
 
void fft_init(int N) {
	#define log2(x) ((int)(log(x)/log(2)))
	int n_luts = log2(N)-1;
	LUT1 = malloc(n_luts * sizeof(data_t *));
	int i;
	for(i=0;i<n_luts;i++) {
		int n = N / pow(2,i);
		LUT1[i] = _mm_malloc(n/4 * sizeof(data_t),16);
		
		if(n == 8) {
			int j;
			for(j=0;j<n/4;j++) {
				LUT1[i][j] = W(n,j);
			}
		}else{
			int j;		
			for(j=0;j<n/4;j+=4) {
				data_t w1[4];
				int k;
				for(k=0;k<4;k++) w1[k] = W(n,j+k);
 
				LUT1[i][j]   = creal(w1[0]) + creal(w1[1])*I;
				LUT1[i][j+1] = creal(w1[2]) + creal(w1[3])*I;
				LUT1[i][j+2] = cimag(w1[0]) + cimag(w1[1])*I;
				LUT1[i][j+3] = cimag(w1[2]) + cimag(w1[3])*I;
			}
		}
	}
	TN = N;
 
}
 

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