前面写过关于傅里叶算法的应用例子。 《基于傅里叶变换的音频重采样算法 (附完整c代码)》 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 #include #include #include #include <string.h> #include #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 "); printf("blog: http://cpuimage.cnblogs.com/ "); 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) ", input[i].real, input[i].imag); } for (int i = 0; i < 100; i++) { FFT(input, output, N); } printf(" "); IFFT(output, input, N); for (size_t i = 0; i < N; ++i) { printf("(%f %f) ", 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