CZT变换（chirp z-transform）

作者：桂。

时间：2018-05-20  12:04:24

链接：http://www.cnblogs.com/xingshansi/p/9063131.html 

---

前言

相比DFT，CZT是完成频谱细化的一种思路，本文主要记录CZT的C代码实现。

一、代码实现

原理主要参考MATLAB接口：

对应C代码实现：

Complex.c

/*=============================
                          Chirp-Z Transform
=============================*/
#include 
#include 
#include 
#include "Complex.h"
#include "FFT.h"

void CZT(comp* x, int  N, comp A, comp W, comp *xCZT, int  M);
void main()
{
    int i;
    int N,M;

    double PI;
    double A0,Theta0;
    double W0,Phi0;

    comp* x;
    comp* xCZT;
    comp A,W;

    PI = 3.1415926;

    N = 5;              //信号长度
    M = 10;            //chirp-z 变换输出长度
    x = (comp *)calloc(N,sizeof(comp));
    xCZT  = (comp *)calloc(M,sizeof(comp));
    for (i = 0;i < N; i++)
    {
        x[i].re=(float)(i-3);
        x[i].im = 0.0;
    }

    A0 = 1.0;                   //起始抽样点z0的矢量半径长度
    Theta0 = 0.0;             //起始抽样点z0的相角
    A.re = (float)(A0*cos(Theta0));
    A.im = (float)(A0*sin(Theta0));

    Phi0 = 2.0*PI/M;      //两相邻抽样点之间的角度差
    W0 = 1.0;                  //螺线的伸展率
    W.re = (float)(W0*cos(-Phi0));
    W.im = (float)(W0*sin(-Phi0));


    CZT(x,N,A,W,xCZT,M);

    printf("The Original Signal:\n");
    for (i = 0; i)
    {
        printf("%10.4f",x[i].re);
        printf("%10.4f\n",x[i].im);
    }

    printf("The Chirp-Z Transfrom:\n");
    for (i = 0 ;i)
    {
        printf("%10.4f",xCZT[i].re);
        printf("%10.4f\n",xCZT[i].im);
    }

}


/*----------------函数说明----------------------
Name: CZT
Function: Chirp-Z Transform
Para:  x[in][out]:待变换信号                            N[in]:信号长度
         A[in]:                                                       W[in]:
         M[in]:Chirp-Z变换输出长度
--------------------------------------------*/
void CZT(comp* x, int  N, comp A, comp W, comp* xCZT, int  M)
{
    int i;
    int L;

    comp* h;
    comp* g;
    comp* pComp;
    comp tmp,tmp1,tmp2;

    i=1;
    do 
    {
        i*=2;
    } while (i1);
    L = i;

    h = (comp*)calloc(L,sizeof(comp));
    g = (comp*)calloc(L,sizeof(comp));
    pComp = (comp*)calloc(L,sizeof(comp));

    for (i = 0; i)
    {
        tmp1 = cpow(A,-i);
        tmp2 = cpow(W, i*i/2.0);
        tmp = cmul(tmp1,tmp2);
        g[i] = cmul(tmp,x[i]);
    }
    for (i = N;i)
    {
        g[i] =czero();
    }

    FFT(g,L,1);

    for (i = 0;i<=M-1;i++)
    {
        h[i] = cpow(W, -i*i/2.0);
    }
    for (i=M; i<=L-N;i++)
    {
        h[i] =czero();
    }
    for (i = L-N+1; i<=L;i++)
    {
        h[i] = cpow(W,-(L-i)*(L-i)/2.0);
    }

    FFT(h,L,1);

    for (i = 0; i)
    {
        pComp[i] = cmul(h[i],g[i]);
    }

    FFT(pComp,L,-1);         //IDFT

    for (i = 0; i)
    {
        tmp = cpow(W,i*i/2.0);
        xCZT[i] = cmul(tmp,pComp[i]);
    }


}

View Code

Complex.h

/*===========================
Define comp as complex type
cmplx     c = (a,b)
cmul     c=a*b
conjg    c=a'
cabs1    f=|a|
cabs2    f=|a|**2
cadd     c=a+b
csub     c=a-b
czero    c=(0.0,0.0)
===========================*/
#ifndef COMPLEX_H
#define COMPLEX_H
#include 
typedef  struct xy
{
    float re;
    float im;
}comp;
comp cmplx(float a,float b);
comp cmul(comp a,comp b);
comp conjg(comp a);
float cabs1(comp a);
float cabs2(comp a);
comp cadd(comp a,comp b);
comp csub(comp a,comp b);
comp czero();
comp cpow(comp a,double n);
float arg(comp a);

#endif

View Code

CZT.c

/*=============================
                          Chirp-Z Transform
=============================*/
#include 
#include 
#include 
#include "Complex.h"
#include "FFT.h"

void CZT(comp* x, int  N, comp A, comp W, comp *xCZT, int  M);
void main()
{
    int i;
    int N,M;

    double PI;
    double A0,Theta0;
    double W0,Phi0;

    comp* x;
    comp* xCZT;
    comp A,W;

    PI = 3.1415926;

    N = 5;              //信号长度
    M = 10;            //chirp-z 变换输出长度
    x = (comp *)calloc(N,sizeof(comp));
    xCZT  = (comp *)calloc(M,sizeof(comp));
    for (i = 0;i < N; i++)
    {
        x[i].re=(float)(i-3);
        x[i].im = 0.0;
    }

    A0 = 1.0;                   //起始抽样点z0的矢量半径长度
    Theta0 = 0.0;             //起始抽样点z0的相角
    A.re = (float)(A0*cos(Theta0));
    A.im = (float)(A0*sin(Theta0));

    Phi0 = 2.0*PI/M;      //两相邻抽样点之间的角度差
    W0 = 1.0;                  //螺线的伸展率
    W.re = (float)(W0*cos(-Phi0));
    W.im = (float)(W0*sin(-Phi0));


    CZT(x,N,A,W,xCZT,M);

    printf("The Original Signal:\n");
    for (i = 0; i)
    {
        printf("%10.4f",x[i].re);
        printf("%10.4f\n",x[i].im);
    }

    printf("The Chirp-Z Transfrom:\n");
    for (i = 0 ;i)
    {
        printf("%10.4f",xCZT[i].re);
        printf("%10.4f\n",xCZT[i].im);
    }

}


/*----------------函数说明----------------------
Name: CZT
Function: Chirp-Z Transform
Para:  x[in][out]:待变换信号                            N[in]:信号长度
         A[in]:                                                       W[in]:
         M[in]:Chirp-Z变换输出长度
--------------------------------------------*/
void CZT(comp* x, int  N, comp A, comp W, comp* xCZT, int  M)
{
    int i;
    int L;

    comp* h;
    comp* g;
    comp* pComp;
    comp tmp,tmp1,tmp2;

    i=1;
    do 
    {
        i*=2;
    } while (i1);
    L = i;

    h = (comp*)calloc(L,sizeof(comp));
    g = (comp*)calloc(L,sizeof(comp));
    pComp = (comp*)calloc(L,sizeof(comp));

    for (i = 0; i)
    {
        tmp1 = cpow(A,-i);
        tmp2 = cpow(W, i*i/2.0);
        tmp = cmul(tmp1,tmp2);
        g[i] = cmul(tmp,x[i]);
    }
    for (i = N;i)
    {
        g[i] =czero();
    }

    FFT(g,L,1);

    for (i = 0;i<=M-1;i++)
    {
        h[i] = cpow(W, -i*i/2.0);
    }
    for (i=M; i<=L-N;i++)
    {
        h[i] =czero();
    }
    for (i = L-N+1; i<=L;i++)
    {
        h[i] = cpow(W,-(L-i)*(L-i)/2.0);
    }

    FFT(h,L,1);

    for (i = 0; i)
    {
        pComp[i] = cmul(h[i],g[i]);
    }

    FFT(pComp,L,-1);         //IDFT

    for (i = 0; i)
    {
        tmp = cpow(W,i*i/2.0);
        xCZT[i] = cmul(tmp,pComp[i]);
    }


}

View Code

FFT.c

#include "FFT.h"

void FFT(comp *data,int FFTn,int inverse)
{
    comp      u,w,t;
    double     temp1,temp2;
    double     pi;
    int       i,j,k,l,ip;
    int       le,le1,m;

    pi=3.1415926f;
    m=0;
    while(FFTn != (0x0001<;
    for(i=0,j=0; i1; i++)
    {
        if(i<j)
        {
            t=data[j];
            data[j]=data[i];
            data[i]=t;
        }
        k=FFTn/2;
        while(k<=j)
        {
            j-=k;
            k/=2;
        }
        j+=k;
    }
    le=1;
    for(l=0; l )
    {
        le*=2;
        le1=le/2;
        u.re=(float)1.0;
        u.im=(float)0.0;
        w.re=(float)cos(pi/le1);
        w.im=(float)(inverse*sin(pi/le1));
        for(j=0; j)
        {
            for(i=j; ile)
            {
                ip=i+le1;
                t.re=data[ip].re*u.re-data[ip].im*u.im;
                t.im=data[ip].re*u.im+data[ip].im*u.re;
                data[ip].re=data[i].re-t.re;
                data[ip].im=data[i].im-t.im;
                data[i].re+=t.re;
                data[i].im+=t.im;
             }
            temp1=u.re;
            temp2=u.im;
            u.re=(float)(temp1*w.re-temp2*w.im);
            u.im=(float)(temp1*w.im+temp2*w.re);
        }
    }
    if(inverse>0) return;
    for(i=0; i)
    {
        data[i].re/=FFTn;
        data[i].im/=FFTn;
    }
    return;
}

View Code

FFT.h

#ifndef    _FFT1_H_
#define    _FFT1_H_

#include "Complex.h"
//========================================
//功能:    实现FFT
//输入:    data[in][out]: 数据指针;  FFTn[in]:FFT点数; 
//            inverse[in]:正反FFT标志位:1,正FFT;-1,逆FFT
void FFT(comp *data,int FFTn,int inverse);

#endif

View Code

二、仿真测试

参数设置：

结果对比：

• matlab

• C

 

利用CZT对信号进行频率估计：

 频率细化直接查找：

仿真code:

clc;clear all;close all;
fs = 1000;
f0 = 201.3;
t = [0:199]/fs;
sig = (sin(2*pi*t*f0));
% %存入txt
% fp=fopen('data.txt','a');%'A.txt'为文件名；'a'为打开方式：在打开的文件末端添加数据，若文件不存在则创建。
% fprintf(fp,'%f ',sig);%fp为文件句柄，指定要写入数据的文件。注意：%d后有空格。
% fclose(fp);%关闭文件。

f1 = 190;
f2 = 210;
m = 100;
w = exp(-1j*2*pi*(f2-f1)/(m*fs));
a = exp(1j*2*pi*f1/fs);
y = czt(sig,m,w,a);
[val,pos] = max(abs(y));
fre_est = (pos-1)*(f2-f1)/m+f1;
y1 = fft(sig);
figure()
subplot 211
plot(linspace(f1,f2,length(y)),abs(y));
hold on;
scatter(fre_est,abs(y(pos)),'r*');
subplot 212
plot(t/max(t)*fs,abs(y1))

　　C读取txt数据：

//
#include   
int n,r;  
double d;  
FILE *f;  
void main() {  
    f=fopen("data.txt","r");  
    n=0;  
    while (1) {  
        r=fscanf(f,"%lf",&d);  
        if (1==r) {  
            n++;  
            printf("[%d]==%lg\n",n,d);  
        } else if (0==r) {  
            fscanf(f,"%*c");  
        } else break;  
    }  
    fclose(f);  
    system("pause");
}  

C语言仿真：

/*=============================
                          Chirp-Z Transform
=============================*/
#include 
#include 
#include 
#include "Complex.h"
#include "FFT.h"

#define Size 100 //信号长度

void CZT(comp* x, int  N, comp A, comp W, comp *xCZT, int  M);

double max_ab(double x,double y);//求最大值

void main()
{
    int i;
    int r;
    int N,M;
    float da;
    double fre_est;
    FILE *f; 

    double PI;
    double A0,Theta0;
    double W0,Phi0;
    int pos;
    double maxdata;
    double maxcache;
    float fs;
    float fstart;
    float fend;
    comp* x;
    double absx[Size];//存储绝对值
    comp* xCZT;
    comp A,W;
    fs = 1000.0;//frequency sample
    fstart = 200.0;
    fend = 215.0;
    PI = 3.1415926;
    f=fopen("data.txt","r");  
    N = Size;              //信号长度
    M = Size;            //chirp-z 变换输出长度
    x = (comp *)calloc(N,sizeof(comp));
    xCZT  = (comp *)calloc(M,sizeof(comp));
     for (i = 0; i) {  
        r=fscanf(f,"%f",&da);  //读取数据
        x[i].re = da;
        x[i].im = 0.0;
        printf("n = %f\n",x[i].re );
    }  
    fclose(f); 

    //

    A0 = 1.0;                   //起始抽样点z0的矢量半径长度
    Theta0 = 2.0*PI*fstart/fs;             //起始抽样点z0的相角
    A.re = (float)(A0*cos(Theta0));
    A.im = (float)(A0*sin(Theta0));

    Phi0 = (fend-fstart)*2.0*PI/M/fs;      //两相邻抽样点之间的角度差
    W0 = 1.0;                  //螺线的伸展率
    W.re = (float)(W0*cos(-Phi0));
    W.im = (float)(W0*sin(-Phi0));


    CZT(x,N,A,W,xCZT,M);

    printf("The Original Signal:\n");
    for (i = 0; i)
    {
        absx[i] = sqrt(xCZT[i].re*xCZT[i].re + xCZT[i].im*xCZT[i].im);//频谱
        printf("%10.4f",x[i].re);
        printf("%10.4f\n",x[i].im);
    }

    printf("The Chirp-Z Transfrom:\n");
    for (i = 0 ;i)
    {
        printf("%10.4f",xCZT[i].re);
        printf("%10.4f\n",xCZT[i].im);
    }
    printf("The Chirp-Z Transfrom square:\n");
    for (i = 0 ;i)
    {
        printf("%lf\n",absx[i]);
    }
    //找出频谱峰值
    maxcache = 0;
    pos = 0;
    for (i = 0 ;i)
    {
        maxdata = max_ab(maxcache,absx[i]);
        if (maxdata == absx[i]) pos++;
        maxcache = maxdata;
    }
    //频率估计
    fre_est = fstart + (pos-1.0)*(fend-fstart)/M;
    printf("frequency estiamton: %lf",fre_est);
    //绘制频谱
    //
    system("pause");
}

/*
Function: max
*/
double max_ab(double x,double y)
{
    return(x>y?x:y); 
} 


/*----------------函数说明----------------------
Name: CZT
Function: Chirp-Z Transform
Para:  x[in][out]:待变换信号                            N[in]:信号长度
         A[in]:                                                       W[in]:
         M[in]:Chirp-Z变换输出长度
--------------------------------------------*/
void CZT(comp* x, int  N, comp A, comp W, comp* xCZT, int  M)
{
    int i;
    int L;

    comp* h;
    comp* g;
    comp* pComp;
    comp tmp,tmp1,tmp2;

    i=1;
    do 
    {
        i*=2;
    } while (i1);
    L = i;

    h = (comp*)calloc(L,sizeof(comp));
    g = (comp*)calloc(L,sizeof(comp));
    pComp = (comp*)calloc(L,sizeof(comp));

    for (i = 0; i)
    {
        tmp1 = cpow(A,-i);
        tmp2 = cpow(W, i*i/2.0);
        tmp = cmul(tmp1,tmp2);
        g[i] = cmul(tmp,x[i]);
    }
    for (i = N;i)
    {
        g[i] =czero();
    }

    FFT(g,L,1);

    for (i = 0;i<=M-1;i++)
    {
        h[i] = cpow(W, -i*i/2.0);
    }
    for (i=M; i<=L-N;i++)
    {
        h[i] =czero();
    }
    for (i = L-N+1; i<=L;i++)
    {
        h[i] = cpow(W,-(L-i)*(L-i)/2.0);
    }

    FFT(h,L,1);

    for (i = 0; i)
    {
        pComp[i] = cmul(h[i],g[i]);
    }

    FFT(pComp,L,-1);         //IDFT

    for (i = 0; i)
    {
        tmp = cpow(W,i*i/2.0);
        xCZT[i] = cmul(tmp,pComp[i]);
    }


}

原始数据：

估计结果：

与MATLAB结果一致：

如果存在两个信号呢？

MATLAB

clc;clear all;close all;
fs = 256;
% f0 = 202;
% f1 = 208;
t = [0:625]/fs;
sig = single(2*cos(2*pi*102*t)+5*cos(2*pi*105*t));
%存入txt
fp=fopen('data.txt','a');%'A.txt'为文件名；'a'为打开方式：在打开的文件末端添加数据，若文件不存在则创建。
fprintf(fp,'%f\n',sig);%fp为文件句柄，指定要写入数据的文件。注意：%d后有空格。
fclose(fp);%关闭文件。

f1 = 100;
f2 = 108;
m = 625;
w = exp(-1j*2*pi*(f2-f1)/(m*fs));
a = exp(1j*2*pi*f1/fs);
y = czt(sig,m,w,a);
data_cache = diff(abs(y));
vals = abs(y);
dataall = [];
posall = [];
flag = 0;
for i = 1:length(y)-2
    if ((data_cache(1,i)*data_cache(1,i+1))<0)
        flag = flag + 1;
        dataall = [ dataall vals(i)];
        posall = [posall,i];
    end
end

[val,pos] = sort(dataall,'descend');
pos = posall(pos);
fre_est = (pos-1)*(f2-f1)/m+f1;
figure()

y1 = fft(sig);

subplot 211
plot(linspace(f1,f2,length(y)),abs(y));
axis([95,110,0,1500]);
hold on;
scatter(fre_est(1:2),ones(1,2),'r*');
hold on;
% scatter(fre_est,abs(y(pos)),'r*');
subplot 212
plot(t/max(t)*fs,abs(y1))
axis([95,110,0,1500]);

　　C：

运行中如果报错：

 

需要将fopen的data.txt添加绝对路径，注意：\\而不是\。

/*=============================
                          Chirp-Z Transform
=============================*/
#include 
#include 
#include 
#include "Complex.h"
#include "FFT.h"

#define Size 626 //信号长度

void CZT(comp* x, int  N, comp A, comp W, comp *xCZT, int  M);
double max_ab(double x,double y);//求最大值

void main()
{
    int i;
    int r;
    int flag = 0;
    int N,M;
    float da;
    double fre_est1, fre_est2;
    FILE *f; 
    double datall[Size] = {0};//新建足够大的数据
    int posall[Size] = {0};
    double diffdata [Size-1];
    double PI;
    double A0,Theta0;
    double W0,Phi0;
    int pos1,pos2;
    double maxdata;
    double maxcache;
    float fs;
    float fstart;
    float fend;
    comp* x;
    double absx[Size];//存储绝对值
    comp* xCZT;
    comp A,W;
    fs = 256.0;//frequency sample
    fstart = 100.0;
    fend = 108.0;
    PI = 3.1415926;
    f=fopen("data.txt","r");  
    N = Size;              //信号长度
    M = Size;            //chirp-z 变换输出长度
    x = (comp *)calloc(N,sizeof(comp));
    xCZT  = (comp *)calloc(M,sizeof(comp));
     for (i = 0; i) {  
        r=fscanf(f,"%f",&da);  //读取数据
        x[i].re = da;
        x[i].im = 0.0;
        printf("n = %f\n",x[i].re );
    }  
    fclose(f); 

    //

    A0 = 1.0;                   //起始抽样点z0的矢量半径长度
    Theta0 = 2.0*PI*fstart/fs;             //起始抽样点z0的相角
    A.re = (float)(A0*cos(Theta0));
    A.im = (float)(A0*sin(Theta0));

    Phi0 = (fend-fstart)*2.0*PI/M/fs;      //两相邻抽样点之间的角度差
    W0 = 1.0;                  //螺线的伸展率
    W.re = (float)(W0*cos(-Phi0));
    W.im = (float)(W0*sin(-Phi0));


    CZT(x,N,A,W,xCZT,M);

    printf("The Original Signal:\n");
    for (i = 0; i)
    {
        absx[i] = sqrt(xCZT[i].re*xCZT[i].re + xCZT[i].im*xCZT[i].im);//频谱
        printf("%10.4f",x[i].re);
        printf("%10.4f\n",x[i].im);
    }

    printf("The Chirp-Z Transfrom:\n");
    for (i = 0 ;i)
    {
        printf("%10.4f",xCZT[i].re);
        printf("%10.4f\n",xCZT[i].im);
    }
    printf("The Chirp-Z Transfrom square:\n");
    //for (i = 0 ;i//{
    //    printf("%lf\n",absx[i]);
    //}
    //找出频谱峰值
    //maxcache = 0;
    //pos = 0;
    //for (i = 0 ;i//   {
    //    maxdata = max_ab(maxcache,absx[i]);
    //    if (maxdata == absx[i]) pos++;
    //    maxcache = maxdata;
 //   }
    ////频率估计
    //fre_est = fstart + (pos-1.0)*(fend-fstart)/M;
    //printf("frequency estiamton: %lf",fre_est);
    //找第二个频率分量
    for (i = 0 ;i < M-1 ;i++)
    {
        diffdata[i] = absx[i+1]-absx[i];
    }
    for (i = 0;i < M-2 ;i++)
    {
        if((diffdata[i]*diffdata[i+1])<0.0)
        {
            datall[flag] = absx[i];
            posall[flag] = i;
            flag ++;
        }
    }
    maxcache = -1;
    pos1 = 0;
    for (i = 0 ;i)
    {
        maxdata = max_ab(maxcache,datall[i]);
        if (maxdata == datall[i]) pos1 = i;
        maxcache = maxdata;
    }
    
    datall[pos1] = 0;
    maxcache = -1;
    pos2 = 0;
    for (i = 0 ;i)
    {
        maxdata = max_ab(maxcache,datall[i]);
        if (maxdata == datall[i]) pos2 = i;
        maxcache = maxdata;
    }
    fre_est1 = fstart + (posall[pos1]-1.0)*(fend-fstart)/M;
    fre_est2 = fstart + (posall[pos2]-1.0)*(fend-fstart)/M;
    printf("frequency estiamton: %lf and %lf",fre_est1,fre_est2);
    //绘制频谱
    //
    system("pause");
}

/*
Function: max
*/
double max_ab(double x,double y)
{
    return(x>y?x:y); 
} 


/*----------------函数说明----------------------
Name: CZT
Function: Chirp-Z Transform
Para:  
x[in][out]:待变换信号                            
N[in]:信号长度
         A[in]:                                                       W[in]:
         M[in]:Chirp-Z变换输出长度
--------------------------------------------*/
void CZT(comp* x, int  N, comp A, comp W, comp* xCZT, int  M)
{
    int i;
    int L;

    comp* h;
    comp* g;
    comp* pComp;
    comp tmp,tmp1,tmp2;

    i=1;
    do 
    {
        i*=2;
    } while (i1);
    L = i;

    h = (comp*)calloc(L,sizeof(comp));
    g = (comp*)calloc(L,sizeof(comp));
    pComp = (comp*)calloc(L,sizeof(comp));

    for (i = 0; i)
    {
        tmp1 = cpow(A,-i);
        tmp2 = cpow(W, i*i/2.0);
        tmp = cmul(tmp1,tmp2);
        g[i] = cmul(tmp,x[i]);
    }
    for (i = N;i)
    {
        g[i] =czero();
    }

    FFT(g,L,1);

    for (i = 0;i<=M-1;i++)
    {
        h[i] = cpow(W, -i*i/2.0);
    }
    for (i=M; i<=L-N;i++)
    {
        h[i] =czero();
    }
    for (i = L-N+1; i<=L;i++)
    {
        h[i] = cpow(W,-(L-i)*(L-i)/2.0);
    }

    FFT(h,L,1);

    for (i = 0; i)
    {
        pComp[i] = cmul(h[i],g[i]);
    }

    FFT(pComp,L,-1);         //IDFT

    for (i = 0; i)
    {
        tmp = cpow(W,i*i/2.0);
        xCZT[i] = cmul(tmp,pComp[i]);
    }


}

View Code

另外：quantusII有对应的DSP builder，Vivado/ISE是否有类似工具，直接代码转HDL？ HLS算是一个，其他呢？有时间可以将*.c转成*.v试验。

一个基本应用，频率精确测量：一种卫星信号载波频率精确估计算法.pdf