第二章实验离散时间系统的时域分析

1、 南昌大学实验报告学生姓名：* 学 号: 6103413001 专业班级： * 实验类型： 验证 综合 设计 创新 实验日期： 实验成绩： 第二章：离散时间信号的时域分析一、实验目的：1、学会用MATLAB在时域中产生一些基本的离散时间信号，并对这些信号进行一些基本的运算。2、学会使用基本的MATLAB命令，并将它们应用到简单的数字信号处理问题中。二、实验要求：1、学习并调试本章所给的例子。2、回答书后给出的问题。3、实验报告仅回答奇数信号的例子。三、实验程序及结果Q2.1 对M=2，运行上述程序，生成输入xn=s1n+s2n的输出信号。输入xn的哪个分量被该离散时间系统抑制？Project

2、2.1滑动平均系统 % 程序 P2_1% 一个M点滑动平均滤波器的仿真% 产生输入信号n = 0:100;s1 = cos(2*pi*0.05*n); % 一个低频正弦s2 = cos(2*pi*0.47*n); % 一个高频正弦x = s1+s2;% M点滑动平均滤波器的实现M = input('滤波器所需的长度 = ');num = ones(1,M);y = filter(num,1,x)/M;clf;subplot(2,2,1);plot(n, s1);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅

3、');title('低频正弦');subplot(2,2,2);plot(n, s2);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('高频正弦');subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输入信号');subplot(2,2,4);plot(n, y);axis(0, 100, -2,

4、 2);xlabel('时间序号n'); ylabel('振幅');title('输出信号'); axis;仿真结果如图所示：输入部分高频分量被抑制了。Q2.3对滤波器长度M和正弦信号s1n和s2n的频率取其他值，运行程序P2.1,算出结果。n = 0:100;s1=cos(2*pi*0.02*n);s2=cos(2*pi*0.46*n);x = s1+s2;% M点滑动平均滤波器的实现M = input('滤波器所需的长度 = ');num = ones(1,M);y = filter(num,1,x)/M;clf;figure

5、,subplot(2,2,1);plot(n, s1);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('低频正弦');subplot(2,2,2);plot(n, s2);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('高频正弦');subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel('时间序号n')

6、; ylabel('振幅');title('输入信号');subplot(2,2,4);plot(n, y);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输出信号'); axis;num =1,-ones(1,M-1);y = filter(num,1,x)/M;figure,subplot(2,2,1);plot(n, s1);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅

7、');title('低频正弦');subplot(2,2,2);plot(n, s2);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('高频正弦');subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输入信号');subplot(2,2,4);plot(n, y);axis(0, 100, -2,

8、 2);xlabel('时间序号n'); ylabel('振幅');title('输出信号'); axis;输入部分的高频成分成分被抑制了。close all n = 0:100;s1=cos(2*pi*0.02*n);s2=cos(2*pi*0.46*n);x = s1+s2;% M点滑动平均滤波器的实现M = input('滤波器所需的长度 = ');num = ones(1,M);y = filter(num,1,x)/M;clf;figure,subplot(2,2,1);plot(n, s1);axis(0, 100,

9、-2, 2);xlabel('时间序号n'); ylabel('振幅');title('低频正弦');subplot(2,2,2);plot(n, s2);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('高频正弦');subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输入信号

10、9;);subplot(2,2,4);plot(n, y);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输出信号'); axis;num =1,-ones(1,M-1);y = filter(num,1,x)/M;figure,subplot(2,2,1);plot(n, s1);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('低频正弦');subplot(2,2,

11、2);plot(n, s2);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('高频正弦');subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('振幅');title('输入信号');subplot(2,2,4);plot(n, y);axis(0, 100, -2, 2);xlabel('时间序号n'); ylabel('

12、振幅');title('输出信号'); axis;仿真结果如下图所示：修改程序中的num=ones(1,M)为num=1,-ones(1,M-1)。  运行结果如下图，可以看出输出信号保留了输入信号xn的高频分量，即保留了s2n分量，低频部分s1n被抑制了。Q2.5 用不同频率的正弦信号作为输入信号，计算每个输入信号的输出信号。输出信号是如何受到输入信号频率的影响的？从数学上对你的结论加以证明。% 程序P2_2% 产生一个正弦输入信号clf;n = 0:200;f=input('Please input the value of f:'

13、;)x = cos(2*pi*f*n);% 计算输出信号x1 = x 0 0; % x1n = xn+1 x2 = 0 x 0; % x2n = xnx3 = 0 0 x; % x3n = xn-1y = x2.*x2-x1.*x3;y = y(2:202);% 画出输入和输出信号subplot(2,1,1)plot(n, x)xlabel('时间序列n');ylabel('振幅');title('输入信号')subplot(2,1,2)plot(n,y)xlabel('时间信号n');ylabel('振幅');t

14、itle('输出信号');分别取F=0.05，F=0.47，F=0.5以及F=0，仿真结果如下所示：F=0.05F=0.47F=0.5F=0证明：设输入信号为，则以及。则Q2.7 运行程序P2.3，对由加权输入得到的yn在与相同权系数下输出y1n和y2n相加得到的ytn进行比较，这两个序列是否相等？该系统是线性系统么？程序P2.3clf;n = 0:40;a = 2;b = -3;x1 = cos(2*pi*0.1*n);x2 = cos(2*pi*0.4*n);x = a*x1 + b*x2;num=2.2403 2.4908 2.2403;den = 1 -0.4 0.75

15、;ic = 0 0;y1 = filter(num,den,x1,ic);y2 = filter(num,den,x2,ic); y = filter(num,den,x,ic);yt = a*y1 + b*y2; d = y - yt; subplot(3,1,1)stem(n,y);ylabel('振幅');title('加权输入: a cdot x_1n + b cdot x_2n的输出');subplot(3,1,2)stem(n,yt);ylabel('这幅');title('加权输出: a cdot y_1n + b cdot

16、 y_2n');subplot(3,1,3)stem(n,d);xlabel('时间序号 n');ylabel('振幅');title('差信号');仿真结果如下图所示：该仿真结果说明这两个序列相等，该系统是线性系统。Q2.9 当初始条件非零时重做习题Q2.7。解：令ic = 10 20；则仿真结论如下所示：这两个序列相等，该系统为线性系统。Q2.11 假定另一个系统为yn=xnxn-1,修改程序P2.3，计算这个系统的输出序列y1n,y2n和yn。比较yn和ytn。这两个序列是否相等？该系统是线性系统吗？解：这两个序列不相等。该系统是非

17、线性系统。n=0:40;a=2;b=-3;x11=0 cos(2*pi*f1*n) 0;x12=0 0 cos(2*pi*f1*n);x21=0 cos(2*pi*f2*n) 0; x22=0 0 cos(2*pi*f2*n);y1=x11.*x12;y2=x21.*x22;yt=a*y1+b*y2;y=(a*x11+b*x21).*(a*x12+b*x22);d=y-yt;subplot(3,1,1)stem(0 n 0,y);ylabel('振幅');title('加权输入: a cdot x_1n + b cdot x_2n的输出');subplot(3,

18、1,2)stem(0 n 0,yt);ylabel('振幅');title('加权输出: a cdot y_1n + b cdot y_2n');subplot(3,1,3)stem(0 n 0,d);xlabel('时间序号 n');ylabel('振幅');title('差信号');仿真结果如下所示：这两个序列不相等，该系统不是线性系统。Q2.13 采用三个不同的延时变量D的值重做习题Q2.12。解：三个不同的延时D分别为2 6 12，图形分别见下图：D=2；D=6;D=12;该系统是时不变系统，满足yn-D=

19、ydn。Q2.15 在非零的初始条件下重做习题Q2.12，该系统是时不变系统吗？解：设ic = 5 10; 仿真结果如下所示：该仿真结果说明该系统是时变系统。Q2.17 考虑另一个系统：yn=nxn+xn-1,修改程序P2.4，以仿真上面的系统并确定该系统是否为时不变系统。解：% 程序 P2_4clf;n = 0:40; D = 10;a = 3.0;b = -2;x = a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n);xd = zeros(1,D) x;nd=0:length(xd)-1;y=(n.*x)+0 x(1:40);yd=(nd.*xd)+0 xd(1:

20、length(xd)-1);d = y - yd(1+D:41+D);subplot(3,1,1)stem(n,y);ylabel('振幅'); title('输出 yn'); grid;subplot(3,1,2)stem(n,yd(1:41);ylabel('振幅');title('由于延时输入 xn-10的输出'); grid;subplot(3,1,3)stem(n,d);xlabel('时间序号 n'); ylabel('振幅');title('差值信号');grid;仿真

21、结果如下所示：从仿真结果看，该系统是时变系统。Q2.19 运行程序P2.5，生成式（2.15）所给的离散系统的冲激响应。解：clf;N=40;num=2.2403 2.4908 2.2403;den=1 -0.4 0.75;y=impz(num,den,N);%画出冲激相应stem(y);xlabel('时间序号n');ylabel('振幅');>> title('冲激响应');grid;仿真结果如下所示：Q2.21 利用filter命令编写一个MATLB程序，生成式（2.17）给出的因果线性时不变系统的冲激响应，计算并画出前40个的

22、样本。把你的结果和习题Q2.20中得到的结果相比较。% Program Q2_21% Compute the impulse response yclf;N = 40;num = 0.9 -0.45 0.35 0.002;den = 1.0 0.71 -0.46 -0.62;% input: unit pulsex = 1 zeros(1,N-1);% outputy = filter(num,den,x);% Plot the impulse response% NOTE: the time axis will be WRONG; h0 will% be plotted at n=1; bu

23、t this will agree with% the INCORRECT plotting that was also done% by program P2_5.stem(y);xlabel('Time index n'); ylabel('Amplitude');title('Impulse Response'); grid;仿真结果如下所示：Q2.23 运行程序P2.6，计算输出序列yn和y2n以及差值信号dn。yn和y2n相等吗？% Program P2_6% Cascade Realizationclf;x = 1 zeros(1,4

24、0); % Generate the inputn = 0:40;% Coefficients of 4th order systemden = 1 1.6 2.28 1.325 0.68;num = 0.06 -0.19 0.27 -0.26 0.12;% Compute the output of 4th order systemy = filter(num,den,x);% Coefficients of the two 2nd order systemsnum1 = 0.3 -0.2 0.4;den1 = 1 0.9 0.8;num2 = 0.2 -0.5 0.3;den2 = 1 0

25、.7 0.85;% Output y1n of the first stage in the cascadey1 = filter(num1,den1,x);% Output y2n of the second stage in the cascadey2 = filter(num2,den2,y1);% Difference between yn and y2nd = y - y2;% Plot output and difference signalssubplot(3,1,1);stem(n,y);ylabel('Amplitude');title('Output

26、 of 4th order Realization'); grid;subplot(3,1,2);stem(n,y2)ylabel('Amplitude');title('Output of Cascade Realization'); grid;subplot(3,1,3);stem(n,d)xlabel('Time index n');ylabel('Amplitude');title('Difference Signal'); grid;仿真结果如下所示：y(n)和y2(n)相等。Q2.25 用任意的

27、非零初始向量ic，ic1和ic2来重做习题Q2.23。clf;x=sin(2*pi*0.2*n); % Generate the inputn = 0:40;% Coefficients of 4th order systemden = 1 1.6 2.28 1.325 0.68;num = 0.06 -0.19 0.27 -0.26 0.12;xi=1 2 3 4;% Compute the output of 4th order systemy = filter(num,den,x,xi);% Coefficients of the two 2nd order systemsnum1 =

28、0.3 -0.2 0.4;den1 = 1 0.9 0.8;num2 = 0.2 -0.5 0.3;den2 = 1 0.7 0.85;xi1=1 2;% Output y1n of the first stage in the cascadey1 = filter(num1,den1,x,xi1);xi2=3 4;% Output y2n of the second stage in the cascadey2 = filter(num2,den2,y1,xi2);% Difference between yn and y2nd = y - y2;% Plot output and diff

29、erence signalssubplot(3,1,1);stem(n,y);ylabel('Amplitude');title('Output of 4th order Realization'); grid;subplot(3,1,2);stem(n,y2)ylabel('Amplitude');title('Output of Cascade Realization'); grid;subplot(3,1,3);stem(n,d)xlabel('Time index n');ylabel('Ampli

30、tude');title('Difference Signal'); grid;仿真结果如下图所示：Q2.27 用任意非零初始向量ic，ic1和ic2来重做习题Q2.26。% Program P2.27% Cascade Realizationclf;x = 1 zeros(1,40); % Generate the inputn = 0:40;% Coefficients of 4th order systemden = 1 1.6 2.28 1.325 0.68;num = 0.06 -0.19 0.27 -0.26 0.12;ic=4 10 2 12% Compu

31、te the output of 4th order systemy = filter(num,den,x,ic);% Coefficients of the two 2nd order systemsnum1 = 0.3 -0.2 0.4;den1 = 1 0.9 0.8;num2 = 0.2 -0.5 0.3;den2 = 1 0.7 0.85;% Output y1n of the first stage in the cascadey1 = filter(num2,den2,x,ic(1:2);% Output y2n of the second stage in the cascad

32、ey2 = filter(num1,den1,y1,ic(3:4);% Difference between yn and y2nd = y - y2;% Plot output and difference signalssubplot(3,1,1);stem(n,y);ylabel('Amplitude');title('Output of 4th order Realization'); grid;subplot(3,1,2);stem(n,y2)ylabel('Amplitude');title('Output of Cascad

33、e Realization'); grid;subplot(3,1,3);stem(n,d)xlabel('Time index n');ylabel('Amplitude');title('Difference Signal'); grid;仿真结果如下所示：Q2.29 修改程序P2.7，计算长度为15的序列hn和长度为10的序列xn的卷积，重做问题Q2.28。hn和xn的样本值你自己给定。% Program P2.29clf;h = 3 2 1 -2 1 0 -4 0 3 1 5 4 0 3 5; % impulse response

34、x = 1 -2 3 -4 3 2 1 5 6 1; % input sequencey = conv(h,x);n = 0:23;subplot(2,1,1);stem(n,y);xlabel('Time index n'); ylabel('Amplitude');title('Output Obtained by Convolution'); grid;x1 = x zeros(1,14);y1 = filter(h,1,x1);subplot(2,1,2);stem(n,y1);xlabel('Time index n'); ylabel('Amplitude');title('Output Generated by Filtering'); grid;仿真结果如下所示：xn后面补零数应为x(n)和hn序列的长度之和减一，为14.Q2.31 使用命令break的目的是什么？使用命令break是使当在k未到最后一个数值是此时的值已经小于时，跳出for循环。Q2.33 考虑用差分方程描述的离散时间系统