数字信号处理实验（二） —— 利用FFT实现快速卷积

Love The Way You Lie 2023-02-15 03:34 33阅读 0赞

## 前言 ##

由第一个实验，我们大致知道了，可以通过卷积脉冲响应的方式，得到输出，得到经过一个系统处理过后的输出。如果这个系统是数字滤波器，那么输出就是滤波后的结果

所以 这一节介绍的FFT实现快速卷积意义不小，这意味着DSP的很多应用，比如快速滤波，快速陷波，都是可以实现的。

## 实践 ##

数字滤波器脉冲响应为 h ( n ) = ( 1 2 ) n R N 2 ( n ) h(n)=(\\frac\{1\}\{2\})^nR\_\{N\_2\}(n) h(n)=(21)nRN2(n)， N 1 = 16 , N 2 = 17 N\_1=16 ,N\_2=17 N1=16,N2=17，输入序列有如下几种情况，我们要设计matlab程序得到滤波后的结果（就是和滤波器脉冲响应卷积后的结果）

*   x ( n ) = R N 1 ( n ) x(n)=R\_\{N\_1\}(n) x(n)=RN1(n)
 *   x ( n ) = c o s ( 2 π N 1 n ) R N 1 ( n ) x(n)=cos(\\frac\{2\\pi\}\{N\_1\}n)R\_\{N\_1\}(n) x(n)=cos(N12πn)RN1(n)
 *   x ( n ) = ( 1 3 ) n R N 1 ( n ) x(n)=(\\frac\{1\}\{3\})^nR\_\{N\_1\}(n) x(n)=(31)nRN1(n)

## 第一题 ##

x ( n ) = R N 1 ( n ) , N 1 = 16 x(n)=R\_\{N\_1\}(n) , N1=16 x(n)=RN1(n),N1=16

n=[0:1:16];
    m=[0:1:17];
    N1=length(n);
    N2=length(m);
    
    %输入Xn Hn
    Xn=ones(1,N1);
    Hn=0.5.^m.*ones(1,N2);
    
    %修剪序列长度
    N=N1+N2-1;
    XK=fft(Xn,N);
    HK=fft(Hn,N);
    YK=XK.*HK;
    Yn=ifft(YK,N);
    
    if all(imag(Xn)==0)&(all(imag(Hn)==0))
        Yn=real(Yn);
    end
    
    %修剪横坐标序列长度
    y=0:N-1;
    h=0:N2-1;
    x=0:N1-1;
    
    subplot(1,4,1);
    stem(x,Xn,'r*');
    title({ '$$ Xn = R_{N1}(n) $$' },'Interpreter','latex');
    axis([0,length(x),0,1.1]);
    
    subplot(1,4,2);
    stem(h,Hn,'r*');
    title({ '$$ Hn $$' },'Interpreter','latex');
    axis([0,length(h),0,1.1]);
    
    subplot(1,4,3);
    stem(y,Yn,'r*');
    title({ '$$ Yn $$' },'Interpreter','latex');
    axis([0,length(y),0,2.1]);
    
    subplot(1,4,4);
    Yline=conv(Xn,Hn);
    stem(y,Yline,'r*');
    title({ '$$ Xn*Hn $$' },'Interpreter','latex');
    axis([0,length(y),0,2.1]);

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzE3ODgyOA_size_16_color_FFFFFF_t_70]  
第一第二个分别就是源信号以及脉冲响应，后两个的对比，我们可以发现利用FFT进行的卷积和之前直接卷积的效果几乎相同  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzE3ODgyOA_size_16_color_FFFFFF_t_70 1]

## 第二题 ##

% 修改序列长度 N1=16 N2=17
    n=[0:1:16];
    m=[0:1:17];
    N1=length(n);
    N2=length(m);
    % Xn Hn表达
    Xn=cos(2*pi*n/N1);
    Hn=0.5.^m.*ones(1,N2);
    %修剪序列长度
    N=N1+N2-1;
    
    XK=fft(Xn,N);
    HK=fft(Hn,N);
    YK=XK.*HK;
    Yn=ifft(YK,N);
    
    %实序列的循环卷积 = 实序列
    if all(imag(Xn)==0)&(all(imag(Hn)==0))
        Yn=real(Yn);
    end
    
    %修剪横坐标序列长度
    y=0:N-1;
    h=0:N2-1;
    x=0:N1-1;
    subplot(1,4,1);
    stem(x,Xn,'r*');
    title({ '$$ Xn = cos(\frac{2\pi}{N1}n) R_{N1}(n) $$' },'Interpreter','latex');
    axis([0,length(x),-1.1,1.1]);
    
    subplot(1,4,2);
    stem(h,Hn,'r*');
    title({ '$$ Hn $$' },'Interpreter','latex');
    axis([0,length(y),0,1.1]);
    
    subplot(1,4,3);
    stem(y,Yn,'r*');
    title({ '$$ Yn $$' },'Interpreter','latex');
    axis([0,length(h),-2.1,2.1]);
    
    subplot(1,4,4);
    Yline=conv(Xn,Hn);
    stem(y,Yline,'r*');
    title({ '$$ Xn*Hn $$' },'Interpreter','latex');
    axis([0,length(h),-2.1,2.1]);

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzE3ODgyOA_size_16_color_FFFFFF_t_70 2]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzE3ODgyOA_size_16_color_FFFFFF_t_70 3]

## 第三题 ##

clear all;
    n=[0:1:16];
    m=[0:1:17];
    
    N1=length(n);
    N2=length(m);
    
    %生成Xn Hn
    Xn=(1/3).^n;
    Hn=0.5.^m.*ones(1,N2);
    %修剪序列长度
    N=N1+N2-1;
    XK=fft(Xn,N);
    HK=fft(Hn,N);
    YK=XK.*HK;
    Yn=ifft(YK,N);
    %实序列的循环卷积 = 实序列
    if all(imag(Xn)==0)&(all(imag(Hn)==0))
        Yn=real(Yn);
    end
    
    y=0:N-1;
    h=0:N2-1;
    x=0:N1-1;
    subplot(1,4,1);
    stem(x,Xn,'r*');
    title({ '$$ Xn = (\frac{1}{3})^n R_{N1}(n) $$' },'Interpreter','latex');
    axis([0,length(x),0,1.1]);
    
    subplot(1,4,2);
    stem(h,Hn,'r*');
    title({ '$$ Hn $$' },'Interpreter','latex');
    axis([0,length(y),0,1.1]);
    
    subplot(1,4,3);
    stem(y,Yn,'r*');
    title({ '$$ Yn $$' },'Interpreter','latex');
    axis([0,length(h),0,1.1]);
    
    subplot(1,4,4);
    Yline=conv(Xn,Hn);
    stem(y,Yline,'r*');
    title({ '$$ Xn*Hn $$' },'Interpreter','latex');
    axis([0,length(h),0,1.1]);

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzE3ODgyOA_size_16_color_FFFFFF_t_70 4]  
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