数字信号仿真实验——实验二离散时间信号与系统的频域分析

不念不忘少年蓝@ 2023-10-05 12:24 16阅读 0赞

## 1.实验目的 ##

（1）进一步加深DTFT、DFT和z 變化的算法原理和基本性質的理解。  
（2）熟悉系統的頻率响应和传输函数。  
（3）学习用FFT对时域离散信号进行谱分析的方法，了解可能出现的问题，以便在实际中正确应用FFT。

## 2.实验原理 ##

（1）信号的频域表示  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 1]  
![在这里插入图片描述][20201025104237274.bmp_pic_center]  
（2）频率响应  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 2]  
（3）传输函数  
![在这里插入图片描述][20201025104452716.bmp_pic_center]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 3]

## 3.实验内容 ##

（1）编写程序产生以下典型信号：  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 4]  
（2）N = 8和N= 16分别对（1）中所给出的信号逐个进行频谱分析，观察幅频特性。  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 5]

## 4.实验报告要求 ##

(1)在实验报告中简述实验目的和实验原理要点。  
(2)在实验报告中附上实验过程记录的各典型信号的幅频特性曲线，与理论结果比较并分析误差产生的原因以及用FFT谱分析时有关参数的选择方法。  
(3)总结实验中的主要结论。

## Matlab程序1\_1,2： ##

Problem1_1,2
    clear
    x1=[];m=0:7;x2=[];
    for n=0:3;
        xa=n+1;
        xb=4-n;
        x1=[x1 xa];
        x2=[x2 xb];
    end
    for n=4:7;
        xa=8-n;
        xb=n-3;
        x1=[x1 xa];
        x2=[x2 xb];
    end
    figure(1)
    subplot(211)
    stem(m,x1);%%绘制火柴梗，产生离散信号
    xlabel('Time index n');
    ylabel('Amplitude');
    title('Original time-domain sequence x1 LEI');
    subplot(212)
    stem(m,x2);%%绘制火柴梗，产生离散信号
    xlabel('Time index n');
    ylabel('Amplitude');
    title('Original time-domain sequence x2 LEI');
    %DFT
    N=input('N=');
    k=0:N-1;
    X1=fft(x1,N);%%傅里叶变换
    X2=fft(x2,N);%%傅里叶变换
    figure(2)
    subplot(211)
    stem(k,abs(X1));%%绘制火柴梗，产生离散信号
    xlabel('Frequency index k');
    ylabel('Amplitude');
    title('Magnitude of the DFT samples X1 LEI');
    subplot(212)
    stem(k,abs(X2));%%绘制火柴梗，产生离散信号
    xlabel('Frequency index k');
    ylabel('Amplitude');
    title('Magnitude of the DFT samples X2 LEI');

## 运行结果1\_1,2： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 6]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 7]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 8]

## Matlab程序1\_3,4： ##

Problem1_3,4
    clear
    N=input('N=');
    n=0:N-1;
    x3=cos(pi*n/4);
    x4=sin(pi*n/4);
    figure(1)
    subplot(211)
    stem(n,x3);%%绘制火柴梗，产生离散信号
    xlabel('Time index n');
    ylabel('Amplitude');
    title('Original time-domain sequence x3 LEI');
    subplot(212)
    stem(n,x4);%%绘制火柴梗，产生离散信号
    xlabel('Time index n');
    ylabel('Amplitude');
    title('Original time-domain sequence x4 LEI');
    %DFT
    X3=fft(x3,N);%%傅里叶变换
    X4=fft(x4,N);%%傅里叶变换
    k=0:N-1;
    figure(2)
    subplot(211)
    stem(k,abs(X3));%%绘制火柴梗，产生离散信号
    xlabel('Frequency index k');
    ylabel('Amplitude');
    title('Magnitude of the DFT samples X3 LEI');
    subplot(212)
    stem(k,abs(X4));%%绘制火柴梗，产生离散信号
    xlabel('Frequency index k');
    ylabel('Amplitude');
    title('Magnitude of the DFT samples X4 LEI');

## 运行结果1\_3,4： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 9]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 10]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 11]

## Matlab程序3： ##

Problem3
    clear
    N=input('N=');
    n=0:N-1;
    k=0:N-1;
    x3=cos(pi*n/4);
    x4=sin(pi*n/4);
    x=x3+x4;
    X=fft(x,N);%%傅里叶变换
    X31=real(X);%%复数的实部
    X41=imag(X);%%复数的虚部
    figure(1)
    subplot(211)
    stem(k,abs(X31));%%绘制火柴梗，产生离散信号
    xlabel('Time index k');
    ylabel('Amplitude');
    title('Original time-domain sequence X31 LEI');
    subplot(212)
    stem(k,abs(X41));%%绘制火柴梗，产生离散信号
    xlabel('Time index k');
    ylabel('Amplitude');
    title('Original time-domain sequence X41 LEI');

## 运行结果3： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 12]

## Matlab程序4： ##

Problem4
    clear
    N=input('N=');
    n=0:N-1;
    k=0:N-1;
    x3=cos(pi*n/4);
    x4=sin(pi*n/4);
    xa=x3+i*x4;
    xb=x3-i*x4;
    Xa=fft(xa,N);%%傅里叶变换
    Xb=fft(xb,N);%%傅里叶变换
    X32=0.5*(Xa+Xb);
    X42=0.5*(Xa-Xb);
    figure(1)
    subplot(211)
    stem(k,abs(X32));%%绘制火柴梗，产生离散信号
    xlabel('Time index k');
    ylabel('Amplitude');
    title('Original time-domain sequence X32 LEI');
    subplot(212)
    stem(k,abs(X42));%%绘制火柴梗，产生离散信号
    xlabel('Time index k');
    ylabel('Amplitude');
    title('Original time-domain sequence X42 LEI');

## 运行结果4： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 13]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 14]

## Matlab程序5\_a,c： ##

Problem5_ac
    clear
    k=256;
    num=[0.9 -0.45 0.35 0.002];
    den=[1 0.71 -0.46 -0.62];
    w=0:pi/(k-1):pi;
    h=freqz(num,den,w);%%计算频率响应
    figure(1)
    subplot(211)
    plot(w/pi,abs(h));%%幅频特性曲线
    title('f(t)的幅频特性曲线 LEI');
    xlabel('w');
    ylabel('幅度');
    subplot(212)
    plot(w/pi,angle(h));%%相频特性曲线
    title('f(t)的相频曲线 LEI');
    xlabel('w');
    ylabel('相位(弧度)');
    figure(2)
    [z,p,k]=tf2zp(num,den);%%求得有理分式形式的系统转移函数的零、极点
    zplane(num,den);%%绘制零极点
    title('零极点图 LEI');

## 运行结果5\_a,c： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 15]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 16]

## Matlab程序5\_b： ##

Problem5_b
    clear
    n=40;
    k=0:1:n-1;
    num=[0.9 -0.45 0.35 0.002];
    den=[1 0.71 -0.46 -0.62];
    y1=impz(num,den,n);%%系统冲激响应
    x1=[1 2 0 -0.5];
    y=conv(y1,x1);%%计算卷积
    subplot(211)
    stem(y)
    title('线性卷积 LEI');
    l=length(x1)+length(y1)-1;
    X=fft(x1,1);%%傅里叶变换
    H=fft(y1',1);
    Y=X.*H;
    yy=real(ifft(Y,1));%%FFT逆变换
    subplot(212)
    stem(yy)
    xlabel('n');
    ylabel('y(n)');
    title('Made by LEI');

## 运行结果5\_b： ##

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQyMDc4OTM0_size_16_color_FFFFFF_t_70_pic_center 17]  
**本人能力有限，解释尚不清楚明了，如遇任何问题，大家可留言或私信。后续将程序文件打包上传，供大家学习使用。**

**本文希望对大家有帮助，当然上文若有不妥之处，欢迎指正。**

**分享决定高度，学习拉开差距**

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