经典傅里叶算法小集合 附完整c代码
原创
前面写过关于傅里叶算法的应用例子。
当然也就是举个例子，主要是学习傅里叶变换。
这个重采样思路还有点瑕疵，
稍微改一下，就可以支持多通道，以及提升性能。
当然思路很简单，就是切分，合并。
留个作业哈。
本文不讲过多的算法思路，傅里叶变换的各种变种，
绝大多数是为提升性能，支持任意长度而作。
当然各有所长，
当时提到参阅整理的算法：
https://github.com/cpuimage/StockhamFFT
https://github.com/cpuimage/uFFT
https://github.com/cpuimage/BluesteinCrz
https://github.com/cpuimage/fftw3
例如 :
Stockham 是优化速度,
BluesteinCrz 是支持任意长度，
uFFT是经典实现。
当然，各有利弊，精度也不一。
最近一直对傅里叶算法没放手。
还是想要抽点时间，不依赖第三方库，实现一份不差于fftw的算法，
既要保证精度，又要保证性能，同时还要支持任意长度。
目前还在进行中，目前项目完成了45%左右。
越是学习，看的资料林林总总，越觉得傅里叶变换的应用面很广。
花点时间，采用纯c ，实现了经典的傅里叶算法，
调整代码逻辑，慢慢开始有点清晰了。
前人栽树后人乘凉，为了学习方便，
把本人用纯c实现的经典傅里叶算法开源出来给大家学习。
算法逻辑写得简洁明了，我也是尽力了。
当然，可能还有更好的实现思路，更多的改进算法。
不过，我的目的更多是便于学习和理解算法。
希望能帮助到一些也在学习傅里叶变换算法的同学。
贴上完整算法代码：
#include <stdio.h>
#include <stdbool.h>
#include <math.h>
#include <stddef.h>
#include <string.h>
#include <stdlib.h>
#ifndef M_PI
#define M_PI 3.14159265358979323846f
#endif
typedef struct {
float real, imag;
} cmplx;
cmplx cmplx_mul_add(const cmplx c, const cmplx a, const cmplx b) {
const cmplx ret = {
(a.real * b.real) + c.real - (a.imag * b.imag),
(a.imag * b.real) + (a.real * b.imag) + c.imag
};
return ret;
}
void fft_Stockham(cmplx *input, cmplx *output, size_t n, int flag) {
size_t half = n >> 1;
cmplx *tmp = (cmplx *) calloc(sizeof(cmplx), n);
cmplx *y = (cmplx *) calloc(sizeof(cmplx), n);
memcpy(y, input, sizeof(cmplx) * n);
for (size_t r = half, l = 1; r >= 1; r >>= 1) {
cmplx *tp = y;
y = tmp;
tmp = tp;
float factor_w = -flag * M_PI / l;
cmplx w = {cosf(factor_w), sinf(factor_w)};
cmplx wj = {1, 0};
for (size_t j = 0; j < l; j++) {
size_t jrs = j * (r << 1);
for (size_t k = jrs, m = jrs >> 1; k < jrs + r; k++) {
const cmplx t = {(wj.real * tmp[k + r].real) - (wj.imag * tmp[k + r].imag),
(wj.imag * tmp[k + r].real) + (wj.real * tmp[k + r].imag)};
y[m].real = tmp[k].real + t.real;
y[m].imag = tmp[k].imag + t.imag;
y[m + half].real = tmp[k].real - t.real;
y[m + half].imag = tmp[k].imag - t.imag;
m++;
}
const float t = wj.real;
wj.real = (t * w.real) - (wj.imag * w.imag);
wj.imag = (wj.imag * w.real) + (t * w.imag);
}
l <<= 1;
}
memcpy(output, y, sizeof(cmplx) * n);
free(tmp);
free(y);
}
void fft_radix3(cmplx *in, cmplx *result, size_t n, int flag) {
if (n < 2) {
memcpy(result, in, sizeof(cmplx) * n);
return;
}
size_t radix = 3;
size_t np = n / radix;
cmplx *res = (cmplx *) malloc(sizeof(cmplx) * n);
cmplx *f0 = res;
cmplx *f1 = f0 + np;
cmplx *f2 = f1 + np;
for (size_t i = 0; i < np; i++) {
for (size_t j = 0; j < radix; j++) {
res[i + j * np] = in[radix * i + j];
}
}
fft_radix3(f0, f0, np, flag);
fft_radix3(f1, f1, np, flag);
fft_radix3(f2, f2, np, flag);
float wexp0 = -2 * (float) M_PI * (flag) / (float) (n);
cmplx wt = {cosf(wexp0), sinf(wexp0)};
cmplx w0 = {1, 0};
for (size_t i = 0; i < np; i++) {
const float w0r = w0.real;
w0.real = (w0r * wt.real) - (w0.imag * wt.imag);
w0.imag = (w0.imag * wt.real) + (w0r * wt.imag);
}
cmplx w = {1, 0};
for (size_t j = 0; j < radix; j++) {
cmplx wj = w;
for (size_t k = 0; k < np; k++) {
result[k + j * np] = cmplx_mul_add(f0[k], cmplx_mul_add(f1[k], f2[k], wj), wj);
const float wjr = wj.real;
wj.real = (wjr * wt.real) - (wj.imag * wt.imag);
wj.imag = (wj.imag * wt.real) + (wjr * wt.imag);
}
const float wr = w.real;
w.real = (wr * w0.real) - (w.imag * w0.imag);
w.imag = (w.imag * w0.real) + (wr * w0.imag);
}
free(res);
}
void fft_radix5(cmplx *x, cmplx *result, size_t n, int flag) {
if (n < 2) {
memcpy(result, x, sizeof(cmplx) * n);
return;
}
size_t radix = 5;
size_t np = n / radix;
cmplx *res = (cmplx *) calloc(sizeof(cmplx), n);
cmplx *f0 = res;
cmplx *f1 = f0 + np;
cmplx *f2 = f1 + np;
cmplx *f3 = f2 + np;
cmplx *f4 = f3 + np;
for (size_t i = 0; i < np; i++) {
for (size_t j = 0; j < radix; j++) {
res[i + j * np] = x[radix * i + j];
}
}
fft_radix5(f0, f0, np, flag);
fft_radix5(f1, f1, np, flag);
fft_radix5(f2, f2, np, flag);
fft_radix5(f3, f3, np, flag);
fft_radix5(f4, f4, np, flag);
float wexp0 = -2 * (float) M_PI * (flag) / (float) (n);
cmplx wt = {cosf(wexp0), sinf(wexp0)};
cmplx w0 = {1, 0};
for (size_t i = 0; i < np; i++) {
const float w0r = w0.real;
w0.real = (w0r * wt.real) - (w0.imag * wt.imag);
w0.imag = (w0.imag * wt.real) + (w0r * wt.imag);
}
cmplx w = {1, 0};
for (size_t j = 0; j < radix; j++) {
cmplx wj = w;
for (size_t k = 0; k < np; k++) {
result[k + j * np] = cmplx_mul_add(f0[k], cmplx_mul_add(f1[k], cmplx_mul_add(f2[k],
cmplx_mul_add(f3[k], f4[k],
wj), wj), wj),
wj);
const float wjr = wj.real;
wj.real = (wjr * wt.real) - (wj.imag * wt.imag);
wj.imag = (wj.imag * wt.real) + (wjr * wt.imag);
}
const float wr = w.real;
w.real = (wr * w0.real) - (w.imag * w0.imag);
w.imag = (w.imag * w0.real) + (wr * w0.imag);
}
free(res);
}
void fft_radix6(cmplx *input, cmplx *output, size_t n, int flag) {
if (n < 2) {
memcpy(output, input, sizeof(cmplx) * n);
return;
}
size_t radix = 6;
size_t np = n / radix;
cmplx *res = (cmplx *) calloc(sizeof(cmplx), n);
cmplx *f0 = res;
cmplx *f1 = f0 + np;
cmplx *f2 = f1 + np;
cmplx *f3 = f2 + np;
cmplx *f4 = f3 + np;
cmplx *f5 = f4 + np;
for (size_t i = 0; i < np; i++) {
for (size_t j = 0; j < radix; j++) {
res[i + j * np] = input[radix * i + j];
}
}
fft_radix6(f0, f0, np, flag);
fft_radix6(f1, f1, np, flag);
fft_radix6(f2, f2, np, flag);
fft_radix6(f3, f3, np, flag);
fft_radix6(f4, f4, np, flag);
fft_radix6(f5, f5, np, flag);
float wexp0 = -2 * (float) M_PI * (flag) / (float) (n);
cmplx wt = {cosf(wexp0), sinf(wexp0)};
cmplx w0 = {1, 0};
for (size_t i = 0; i < np; i++) {
const float w0r = w0.real;
w0.real = (w0r * wt.real) - (w0.imag * wt.imag);
w0.imag = (w0.imag * wt.real) + (w0r * wt.imag);
}
cmplx w = {1, 0};
for (size_t j = 0; j < radix; j++) {
cmplx wj = w;
for (size_t k = 0; k < np; k++) {
output[k + j * np] = cmplx_mul_add(f0[k], cmplx_mul_add(f1[k], cmplx_mul_add(f2[k],
cmplx_mul_add(f3[k],
cmplx_mul_add(
f4[k],
f5[k],
wj), wj),
wj), wj), wj);
const float wjr = wj.real;
wj.real = (wjr * wt.real) - (wj.imag * wt.imag);
wj.imag = (wj.imag * wt.real) + (wjr * wt.imag);
}
const float wr = w.real;
w.real = (wr * w0.real) - (w.imag * w0.imag);
w.imag = (w.imag * w0.real) + (wr * w0.imag);
}
free(res);
}
void fft_radix7(cmplx *x, cmplx *result, size_t n, int flag) {
if (n < 2) {
memcpy(result, x, sizeof(cmplx) * n);
return;
}
size_t radix = 7;
size_t np = n / radix;
cmplx *res = (cmplx *) calloc(sizeof(cmplx), n);
cmplx *f0 = res;
cmplx *f1 = f0 + np;
cmplx *f2 = f1 + np;
cmplx *f3 = f2 + np;
cmplx *f4 = f3 + np;
cmplx *f5 = f4 + np;
cmplx *f6 = f5 + np;
for (size_t i = 0; i < np; i++) {
for (size_t j = 0; j < radix; j++) {
res[i + j * np] = x[radix * i + j];
}
}
fft_radix7(f0, f0, np, flag);
fft_radix7(f1, f1, np, flag);
fft_radix7(f2, f2, np, flag);
fft_radix7(f3, f3, np, flag);
fft_radix7(f4, f4, np, flag);
fft_radix7(f5, f5, np, flag);
fft_radix7(f6, f6, np, flag);
float wexp0 = -2 * (float) M_PI * (flag) / (float) (n);
cmplx wt = {cosf(wexp0), sinf(wexp0)};
cmplx w0 = {1, 0};
for (size_t i = 0; i < np; i++) {
const float w0r = w0.real;
w0.real = (w0r * wt.real) - (w0.imag * wt.imag);
w0.imag = (w0.imag * wt.real) + (w0r * wt.imag);
}
cmplx w = {1, 0};
for (size_t j = 0; j < radix; j++) {
cmplx wj = w;
for (size_t k = 0; k < np; k++) {
result[k + j * np] = cmplx_mul_add(f0[k], cmplx_mul_add(f1[k], cmplx_mul_add(f2[k],
cmplx_mul_add(f3[k],
cmplx_mul_add(
f4[k],
cmplx_mul_add(
f5[k],
f6[k],
wj),
wj), wj),
wj), wj), wj);
const float wjr = wj.real;
wj.real = (wjr * wt.real) - (wj.imag * wt.imag);
wj.imag = (wj.imag * wt.real) + (wjr * wt.imag);
}
const float wr = w.real;
w.real = (wr * w0.real) - (w.imag * w0.imag);
w.imag = (w.imag * w0.real) + (wr * w0.imag);
}
free(res);
}
void fft_Bluestein(cmplx *input, cmplx *output, size_t n, int flag) {
size_t m = 1 << ((unsigned int) (ilogbf((float) (2 * n - 1))));
if (m < 2 * n - 1) {
m <<= 1;
}
cmplx *y = (cmplx *) calloc(sizeof(cmplx), 3 * m);
cmplx *w = y + m;
cmplx *ww = w + m;
float a0 = (float) M_PI / n;
w[0].real = 1;
if (flag == -1) {
y[0].real = input[0].real;
y[0].imag = -input[0].imag;
for (size_t i = 1; i < n; i++) {
const float wexp = a0 * i * i;
w[i].real = cosf(wexp);
w[i].imag = sinf(wexp);
w[m - i] = w[i];
y[i].real = (input[i].real * w[i].real) - (input[i].imag * w[i].imag);
y[i].imag = (-input[i].imag * w[i].real) - (input[i].real * w[i].imag);
}
} else {
y[0].real = input[0].real;
y[0].imag = input[0].imag;
for (size_t i = 1; i < n; i++) {
const float wexp = a0 * i * i;
w[i].real = cosf(wexp);
w[i].imag = sinf(wexp);
w[m - i] = w[i];
y[i].real = (input[i].real * w[i].real) + (input[i].imag * w[i].imag);
y[i].imag = (input[i].imag * w[i].real) - (input[i].real * w[i].imag);
}
}
fft_Stockham(y, y, m, 1);
fft_Stockham(w, ww, m, 1);
for (size_t i = 0; i < m; i++) {
const float r = y[i].real;
y[i].real = (r * ww[i].real) - (y[i].imag * ww[i].imag);
y[i].imag = (y[i].imag * ww[i].real) + (r * ww[i].imag);
}
fft_Stockham(y, y, m, -1);
float scale = 1.0f / m;
if (flag == -1) {
for (size_t i = 0; i < n; i++) {
output[i].real = ((y[i].real * w[i].real) + (y[i].imag * w[i].imag)) * scale;
output[i].imag = -((y[i].imag * w[i].real) - (y[i].real * w[i].imag)) * scale;
}
} else {
for (size_t i = 0; i < n; i++) {
output[i].real = ((y[i].real * w[i].real) + (y[i].imag * w[i].imag)) * scale;
output[i].imag = ((y[i].imag * w[i].real) - (y[i].real * w[i].imag)) * scale;
}
}
free(y);
}
size_t base(size_t n) {
size_t t = n & (n - 1);
if (t == 0) {
return 2;
}
for (size_t i = 3; i <= 7; i++) {
size_t n2 = n;
while (n2 % i == 0) {
n2 /= i;
}
if (n2 == 1) {
return i;
}
}
return n;
}
void FFT(cmplx *input, cmplx *output, size_t n) {
memset(output, 0, sizeof(cmplx) * n);
if (n < 2) {
memcpy(output, input, sizeof(cmplx) * n);
return;
}
size_t p = base(n);
switch (p) {
case 2:
fft_Stockham(input, output, n, 1);
break;
case 3:
fft_radix3(input, output, n, 1);
break;
case 5:
fft_radix5(input, output, n, 1);
break;
case 6:
fft_radix6(input, output, n, 1);
break;
case 7:
fft_radix7(input, output, n, 1);
break;
default:
fft_Bluestein(input, output, n, 1);
break;
}
}
void IFFT(cmplx *input, cmplx *output, size_t n) {
memset(output, 0, sizeof(cmplx) * n);
if (n < 2) {
memcpy(output, input, sizeof(cmplx) * n);
return;
}
size_t p = base(n);
switch (p) {
case 2:
fft_Stockham(input, output, n, -1);
break;
case 3:
fft_radix3(input, output, n, -1);
break;
case 5:
fft_radix5(input, output, n, -1);
break;
case 6:
fft_radix6(input, output, n, -1);
break;
case 7:
fft_radix7(input, output, n, -1);
break;
default: {
fft_Bluestein(input, output, n, -1);
break;
}
}
float scale = 1.0f / n;
for (size_t i = 0; i < n; i++) {
output[i].real = output[i].real * scale;
output[i].imag = output[i].imag * scale;
}
}
int main() {
printf("Fast Fourier Transform\n");
printf("blog: http://cpuimage.cnblogs.com/\n");
printf("A Simple and Efficient FFT Implementation in C");
size_t N = 513;
cmplx *input = (cmplx *) calloc(sizeof(cmplx), N);
cmplx *output = (cmplx *) calloc(sizeof(cmplx), N);
for (size_t i = 0; i < N; ++i) {
input[i].real = i;
input[i].imag = 0;
}
for (size_t i = 0; i < N; ++i) {
printf("(%f %f) \t", input[i].real, input[i].imag);
}
for (int i = 0; i < 100; i++) {
FFT(input, output, N);
}
printf("\n");
IFFT(output, input, N);
for (size_t i = 0; i < N; ++i) {
printf("(%f %f) \t", input[i].real, input[i].imag);
}
free(input);
free(output);
getchar();
return 0;
}项目地址:
https://github.com/cpuimage/cpuFFT
想了好久都没想到取啥名字好，最后还是选择了cpu这个前缀。
以上，权当抛砖引玉。
若有其他相关问题或者需求也可以邮件联系俺探讨。
邮箱地址是: gaozhihan@vip.qq.com
原创声明：本文系作者授权腾讯云开发者社区发表，未经许可，不得转载。
如有侵权，请联系 cloudcommunity@tencent.com 删除。
原创声明：本文系作者授权腾讯云开发者社区发表，未经许可，不得转载。
如有侵权，请联系 cloudcommunity@tencent.com 删除。
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