FFT算法的完整DSP实现
DFT计算X(k)需要N^2次复数乘法和N(N-1)次复数加法
FFT算法的流程图如下图，总结为3过程3循环：
（1）3过程：单点时域分解（倒位序过程） + 单点时域计算单点频谱 + 频域合成
（2）3循环：外循环——分解次数，中循环——sub-DFT运算，内循环——2点蝶形算法

分解过程或者说倒位序的获得参考下图理解：

FFT的DSP实现 下面为使用C语言实现的FFT及IFFT算法实例，能计算任意以2为对数底的采样点数的FFT，算法参考上面给的流程图。 /* * zx_fft.h * * Created on: 2013-8-5 * Author: monkeyzx */ #ifndef ZX_FFT_H_ #define ZX_FFT_H_ typedef float FFT_TYPE; #ifndef PI #define PI (3.14159265f) #endif typedef struct complex_st { FFT_TYPE real; FFT_TYPE img; } complex; int fft(complex *x, int N); int ifft(complex *x, int N); void zx_fft(void); #endif /* ZX_FFT_H_ */ [cpp] view plain copy /* * zx_fft.c * * Implementation of Fast Fourier Transform(FFT) * and reversal Fast Fourier Transform(IFFT) * * Created on: 2013-8-5 * Author: monkeyzx */ #include "zx_fft.h" #include #include /* * Bit Reverse * === Input === * x : complex numbers * n : nodes of FFT. @N should be power of 2, that is 2^(*) * l : count by bit of binary format, @l=CEIL{log2(n)} * === Output === * r : results after reversed. * Note: I use a local variable @temp that result @r can be set * to @x and won't overlap. */ static void BitReverse(complex *x, complex *r, int n, int l) { int i = 0; int j = 0; short stk = 0; static complex *temp = 0; temp = (complex *)malloc(sizeof(complex) * n); if (!temp) { return; } for(i=0; i0; j = 0; do { stk |= (i>>(j++)) & 0x01; if(j1; } }while(jif(stk < n) { /* 满足倒位序输出 */ temp[stk] = x[i]; } } /* copy @temp to @r */ for (i=0; ifree(temp); } /* * FFT Algorithm * === Inputs === * x : complex numbers * N : nodes of FFT. @N should be power of 2, that is 2^(*) * === Output === * the @x contains the result of FFT algorithm, so the original data * in @x is destroyed, please store them before using FFT. */ int fft(complex *x, int N) { int i,j,l,ip; static int M = 0; static int le,le2; static FFT_TYPE sR,sI,tR,tI,uR,uI; M = (int)(log(N) / log(2)); /* * bit reversal sorting */ BitReverse(x,x,N,M); /* * For Loops */ for (l=1; l<=M; l++) { /* loop for ceil{log2(N)} */ le = (int)pow(2,l); le2 = (int)(le / 2); uR = 1; uI = 0; sR = cos(PI / le2); sI = -sin(PI / le2); for (j=1; j<=le2; j++) { /* loop for each sub DFT */ //jm1 = j - 1; for (i=j-1; i<=N-1; i+=le) { /* loop for each butterfly */ ip = i + le2; tR = x[ip].real * uR - x[ip].img * uI; tI = x[ip].real * uI + x[ip].img * uR; x[ip].real = x[i].real - tR; x[ip].img = x[i].img - tI; x[i].real += tR; x[i].img += tI; } /* Next i */ tR = uR; uR = tR * sR - uI * sI; uI = tR * sI + uI *sR; } /* Next j */ } /* Next l */ return 0; } /* * Inverse FFT Algorithm * === Inputs === * x : complex numbers * N : nodes of FFT. @N should be power of 2, that is 2^(*) * === Output === * the @x contains the result of FFT algorithm, so the original data * in @x is destroyed, please store them before using FFT. */ int ifft(complex *x, int N) { int k = 0; for (k=0; k<=N-1; k++) { x[k].img = -x[k].img; } fft(x, N); /* using FFT */ for (k=0; k<=N-1; k++) { x[k].real = x[k].real / N; x[k].img = -x[k].img / N; } return 0; } /* * Code below is an example of using FFT and IFFT. */ #define SAMPLE_NODES (128) complex x[SAMPLE_NODES]; int INPUT[SAMPLE_NODES]; int OUTPUT[SAMPLE_NODES]; static void MakeInput() { int i; for ( i=0;isin(PI*2*i/SAMPLE_NODES); x[i].img = 0.0f; INPUT[i]=sin(PI*2*i/SAMPLE_NODES)*1024; } } static void MakeOutput() { int i; for ( i=0;isqrt(x[i].real*x[i].real + x[i].img*x[i].img)*1024; } } void zx_fft(void) { MakeInput(); fft(x,128); MakeOutput(); ifft(x,128); MakeOutput(); } 程序在TMS320C6713上实验，主函数中调用zx_fft()函数即可。 FFT的采样点数为128，输入信号的实数域为正弦信号，虚数域为0，数据精度定义FFT_TYPE为float类型，MakeInput和MakeOutput函数分别用于产生输入数据INPUT和输出数据OUTPUT的函数，便于使用CCS 的Graph功能绘制波形图。
输入波形

输入信号的频域幅值表示

FFT运算结果

对FFT运算结果逆变换(IFFT)

如何检验运算结果是否正确呢？
使用matlab验证，下面为相同情况的matlab图形验证代码 SAMPLE_NODES = 128; i = 1:SAMPLE_NODES; x = sin(pi*2*i / SAMPLE_NODES); subplot(2,2,1); plot(x);title('Inputs'); axis([0 128 -1 1]); y = fft(x, SAMPLE_NODES); subplot(2,2,2); plot(abs(y));title('FFT'); axis([0 128 0 80]); z = ifft(y, SAMPLE_NODES); subplot(2,2,3); plot(abs(z));title('IFFT'); axis([0 128 0 1]);