傅里叶变换或者FFT的理论参考:
[1] http://www.dspguide.com/ch12/2.htm
The Scientist and Engineer's Guide to Digital Signal Processing, By Steven W. Smith, Ph.D.
[2] http://blog.csdn.net/v_JULY_v/article/details/6196862,可当作[1]的中文参考
[3] 任意一本数字信号处理教材,上面都有详细的推导DCT求解转换为FFT求解的过程
[4] TI文档:基于TMS320C64x+DSP的FFT实现。 使用baidu/google可以搜索到。
另外,FFT的开源代码可参考:
[1] FFTW: http://www.fftw.org/ 最快,最好的开源FFT。
[2] FFTReal: http://ldesoras.free.fr/prod.html#src_fftreal 轻量级FFT算法实现
[3] KISS FFT: http://sourceforge.net/projects/kissfft/ 简单易用的FFT的C语言实现
关于FFT这里只想提到两点:
(1)DFT变换对的表达式(必须记住)
—— 称旋转因子
(2)FFT用途——目标只有一个,加速DFT的计算效率。
DFT计算X(k)需要N^2次复数乘法和N(N-1)次复数加法;FFT将N^2的计算量降为。
“FFT其实是很难的东西,即使常年在这个领域下打拼的科学家也未必能很好的写出FFT的算法。”——摘自参考上面提供的参考文献[1]
因此,我们不必太过纠结于细节,当明白FFT理论后,将已有的算法挪过来用就OK了,不必为闭着教材写不出FFT而郁闷不堪。
FFT的BASIC程序伪代码如下:
IFFT的BASIC程序伪代码如下(IFFT通过调用FFT计算):
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
下面为本人使用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_ */
/* * 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 <math.h> #include <stdlib.h> /* * 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; i<n; i++) { stk = 0; j = 0; do { stk |= (i>>(j++)) & 0x01; if(j<l) { stk <<= 1; } }while(j<l); if(stk < n) { /* 满足倒位序输出 */ temp[stk] = x[i]; } } /* copy @temp to @r */ for (i=0; i<n; i++) { r[i] = temp[i]; } free(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;i<SAMPLE_NODES;i++ ) { x[i].real = sin(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;i<SAMPLE_NODES;i++ ) { OUTPUT[i] = sqrt(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(); }
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。这里调试时使用CCS v5中的Tools -> Graph功能得到下面的波形图(怎么用自己琢磨,不会的使用CCS 的Help)。
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?有几种方法:
(1)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]);
(2)使用IFFT验证:输入信号的FFT获得的信号再IFFT,则的到的信号与原信号相同
可能大家发现输入信号上面的最后IFFT的信号似乎不同,这是因为FFT和IFFT存在精度截断误差(也叫数据截断噪声,意思就是说,我们使用的float数据类型数据位数有限,没法完全保留原始信号的信息)。因此,IFFT之后是复数(数据截断噪声引入了虚数域,只不过值很小),所以在绘图时使用了计算幅值的方法,
C代码中:
OUTPUT[i] = sqrt(x[i].real*x[i].real + x[i].img*x[i].img)*1024;
matlab代码中:
subplot(2,2,3); plot(abs(z));title('IFFT');
所以IFFT的结果将sin函数的负y轴数据翻到了正y轴。另外,在CCS v5的图形中我们将显示信号的幅度放大了1024倍便于观察,而matlab中没有放大。
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更正 更正 。。。
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上面程序中的BitReverse函数由于使用了malloc函数,当要分配的n比较大时,在DSP上运行会出现一定的问题,因此改用伪代码中提供的倒位序方法更可靠。
修正后的完整FFT代码文件粘贴如下,在实际的DSP项目中测试通过,可直接拷贝复用。
/* * zx_fft.h * * Created on: 2013-8-5 * Author: monkeyzx */ #ifndef _ZX_FFT_H #define _ZX_FFT_H #include "../Headers/Global.h" #define TYPE_FFT_E float /* Type is the same with COMPLEX member */ #ifndef PI #define PI (3.14159265f) #endif typedef COMPLEX TYPE_FFT; /* Define COMPLEX in Config.h */ extern int fft(TYPE_FFT *x, int N); extern int ifft(TYPE_FFT *x, int N); #endif /* ZX_FFT_H_ */
/* * zx_fft.c * * Implementation of Fast Fourier Transform(FFT) * and reversal Fast Fourier Transform(IFFT) * * Created on: 2013-8-5 * Author: monkeyzx * * TEST OK 2014.01.14 * == 2014.01.14 * Replace @BitReverse(x,x,N,M) by refrence to * <The Scientist and Engineer's Guide to Digital Signal Processing> */ #include "zx_fft.h" /* * 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(TYPE_FFT *x, int N) { int i,j,l,k,ip; static int M = 0; static int le,le2; static TYPE_FFT_E sR,sI,tR,tI,uR,uI; M = (int)(log(N) / log(2)); /* * bit reversal sorting */ l = N / 2; j = l; //BitReverse(x,x,N,M); for (i=1; i<=N-2; i++) { if (i < j) { tR = x[j].real; tI = x[j].imag; x[j].real = x[i].real; x[j].imag = x[i].imag; x[i].real = tR; x[i].imag = tI; } k = l; while (k <= j) { j = j - k; k = k / 2; } j = j + k; } /* * 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].imag * uI; tI = x[ip].real * uI + x[ip].imag * uR; x[ip].real = x[i].real - tR; x[ip].imag = x[i].imag - tI; x[i].real += tR; x[i].imag += 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(TYPE_FFT *x, int N) { int k = 0; for (k=0; k<=N-1; k++) { x[k].imag = -x[k].imag; } fft(x, N); /* using FFT */ for (k=0; k<=N-1; k++) { x[k].real = x[k].real / N; x[k].imag = -x[k].imag / N; } return 0; }
typedef struct { float real; float imag; } COMPLEX;
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增加:FFT频谱结果显示
=====================
clc; clear; % Read a real audio signal [fname,pname]=uigetfile(... {'*.wav';'*.*'},... 'Input wav File'); [x fs nbits] = wavread(fullfile(pname,fname)); % Window % N = 1024; N = size(x,1); x = x(1:N, 1); % FFT y = fft(x); % 频率对称,取N/2 y = y(1:N/2); % FFT频率与物理频率转换 x1 = 1:N/2; x2 = (1:N/2)*fs/(N/2); % /1000表示KHz log_x2 = log2(x2); % plot figure, subplot(2,2,1);plot(x); xlabel('Time/N');ylabel('Mag');grid on title('原始信号'); subplot(2,2,2);plot(x1, abs(y)); xlabel('Freq/N');ylabel('Mag');grid on title('FFT信号/横坐标为点数'); subplot(2,2,3);plot(x2,abs(y)); xlabel('Freq/Hz');ylabel('Mag');grid on title('FFT信号/横坐标为物理频率'); subplot(2,2,4);plot(log_x2,abs(y)); xlabel('Freq/log2(Hz)');ylabel('Mag');grid on title('FFT信号/横坐标为物理频率取log'); % 更常见是将幅值取log y = log2(abs(y)); figure, plot(x2,y); xlabel('Freq/Hz');ylabel('Mag/log2');grid on title('幅值取log2');
最终部分优化的源代码我放在https://github.com/xiahouzuoxin/fft。