Experiment ExportName:Student No:Institute:Dec 26, 2011Experiment Purposes1. Be familiar with the software Environment and Programming flow in MATLAB5.3.2. Learn how to draw the signal waveform and determine the signal properties.3. Calculate the convolution, frequency response and system output by using the functions: conv, freqz, freqs and filter.Experiment Contents实验项目一：MATLAB编程基础及典型实例①画出离散时间正弦信号并确定基波周期(注：pi 表示圆周率)1 x1[n]=sin(pi*4/4)*cos(pi*n/4)2 x2[n]=cos(pi*n/4)*cos(pi*n/4)3 x3[n]=sin(pi*n/4)*cos(pi*n/8)program for matlabn=0:31;x1=sin(pi*n/4).*cos(pi*n/4);x2=cos(pi*n/4).*cos(pi*n/4);x3=sin(pi*n/4).*cos(pi*n/8);subplot(3,1,1);stem(n,x1);title('x1');subplot(3,1,2);stem(n,x2);title('x2');subplot(3,1,3);stem(n,x3);title('x3');grid on;Conclusion: These signals is periodic, the first and second signal’s peri od are 4. The third signal’s period is 16.②离散时间系统性质：离散时间系统往往是用几个性质来表征，如线性、时不变性、稳定性、因果性及可逆性等。

MATLAB可用来构成一些反例证明某些性质不满足。

(a) 系统y[n]=sin((pi/2)x[n])不是线性的。

利用信号x1[n]=δ[n]和x2=2δ[n]来证明该系统不满足线性性质。

(b) 系统y[n]=x[n]+x[n+1]不是因果的。

利用信号x[n]=u[n]证明它。

定义向量x 和y分别代表在-5<=n<=9上的输入和在-6<=n<=9上的输出。

Program for matlab1.4(a)n=[0:20];x1=[1 zeros(1,20)]; x2=2*x1;x=x1+x2;y1=sin((pi/2)*x1); y2=sin((pi/2)*x2); y=sin((pi/2)*x); figure(1),stem(n,y1) figure(2),stem(n,y2) figure(3),stem(n,y)1.4(b)x1=[zeros(1,5) ones(1,10)];x2=[zeros(1,4) ones(1,11)];y=x1+x2;n1=[-5:9];n2=[-5:9];figure(1),stem(n1,x1)figure(2),stem(n2,y)Conclusion: y[n]=sin((pi/2)x[n]) is not linear and y[n]=x[n]+x[n+1] is not cuasal and the result is shown in the chart above.○3卷积计算：有限长信和(1) 用解析方法计算y[n]=x[n]*h[n](2) 用conv计算y。

步骤：a. 定义0≤n≤5区间上的向量xb. 定义0≤n≤5区间上的向量hc. 用y=conv(x,h)计算yd. 构造y的标号向量nye. 用stem(ny,y)画出结果f. 验证此结果与解析导出的结果是否一致？Program for matlabN=6;M=6;L=N+M-1;x=[1,1,1,1,1,1];h=[0,1,2,3,4,5];y=conv(x,h);nx=0:N-1;nh=0:M-1;ny=0:L-1;stem(ny,y);xlabel('n');xlabel('y');Conclusion: y=ans =3 and the result is show in the picture above.实验项目2一、实验项目名称：周期信号傅里叶分析及其MATLAB实现二、上机实验题目：特征函数在LTI系统傅里叶分析中的应用1．实验项目的目的和任务：掌握特征函数在系统响应分析中的作用，正确理解滤波的概念。

2．上机实验容：1 函数Filter、Freqz和Freqs的使用：2.2节(g)、3.2节、4.1节2 计算离散时间傅里叶级数：3.1节3 LTI系统的特征函数：3.4节(a),(b),(c)4 用离散时间傅里叶级数综合信号：3.5节(d),(e),(f),(h).5 吉布斯现象：根据英文教材Example 3.5验证Fig3.9的吉布斯现象(a)~(d).1 函数Filter、Freqz和Freqs的使用：2.2节(g)、3.2节、4.1节filter：计算由线性常系数差分方程表征的因果LTI系统的输出若x是在nx≤n≤nx+Nx-1上的输入向量，而向量a和b 包含系数ak和bm，那么y=filter(b,a,x)就得到在nx≤n≤nx+Nx-1上的系统输出y[n]+2y[n-1]=x[n]-3x[n-1]则a=[1 2],b=[1 –3]%2.2gMain Programx=[1,1,1,1,1,1,0,0,0,0,0];h=[0,1,2,3,4,5];y=filter(h,1,x);ny=[0:10];stem(ny,y);xlabel('n');xlabel('y');%3.2a1=[1 -0.8 0];b1=[2 0 -1];N=4;[h1 omega1]=freqz(b1,a1,N)[h2 omega2]=freqz(b1,a1,N,'whole')%4.1a=[1 3];b=3;figure;freqs(b,a); w=linspace(0,3*pi);h=freqs(b,a,w);figure;plot(w,abs(h));a1=[3 4 1];b1=[1 0 5];figure;freqs(b1,a1);②计算离散时间傅里叶级数：3.1节x=[1 1 zeros(1,30)];a=(1/32)*fft(x)n=[0:31];figure(1),stem(n,a)figure(2),stem(n,imag(a))y=32*ifft(a);figure(3),stem(n,y)figure(4),stem(n,imag(y))③LTI系统的特征函数：3.4节(a),(b),(c)%3.4(a)n=-20:100;x1=exp(j*(pi/4)*n);x2=sin(pi*n/8+pi/16);x3=(9/10).^n;x4=n+1;subplot(5,1,1),stem(n,real(x1));title('real(x1)');axis([0,100,-5,5]) subplot(5,1,2),stem(n,imag(x1));title('imag(x1)'); axis([0,100,-5,5])subplot(5,1,3),stem(n,x2);title('x2'); axis([0,100,-5,5]) subplot(5,1,4),stem(n,x3);title('x3')subplot(5,1,5),stem(n,x4);title('x4')%3.4(b)n=0:100;x1=exp(j*(pi/4)*n);x2=sin(pi*n/8+pi/16);x3=(9/10).^n;x4=n+1;a=[1 -0.25];b=[1 0.9];y1=filter(b,a,x1);y2=filter(b,a,x2);y3=filter(b,a,x3);y4=filter(b,a,x4);figure;subplot(5,1,1),stem(n,real(y1));title('real(y1)') subplot(5,1,2),stem(n,imag(y1));title('imag(y1)')subplot(5,1,3),stem(n,y2);title('y2')subplot(5,1,4),stem(n,y3);title('y3')subplot(5,1,5),stem(n,y4);title('y4')%3.4(c)h1=y1./x1;h2=y2./x2;h3=y3./x3;h4=y4./x4;figure; subplot(5,1,1),stem(n,real(h1));title('real(h1)') subplot(5,1,2),stem(n,imag(h1));title('imag(h1)') subplot(5,1,3),stem(n,h2);title('h2')subplot(5,1,4),stem(n,h3);title('h3')subplot(5,1,5),stem(n,h4);title('h4')%3.5(d)function x=period(xn)x=zeros(1,64);[a,b]=size(xn);for k=1:(64/b)x(1,b*(k-1)+1:b*k)=xn; endx1=ones(1,8);x2=[x1,zeros(1,8)]x3=[x1,zeros(1,24)];x11=period(x1);x22=period(x2);x33=period(x3);n=[0:63];subplot(3,1,1);stem(n,x11)subplot(3,1,2);stem(n,x22)subplot(3,1,3);stem(n,x33)x2 =Columns 1 through 131 1 1 1 1 1 1 1 0 0 0 0 0Columns 14 through 160 0 0%3.5(e)x11=[ones(1,8) ones(1,8) ones(1,8) ones(1,8) ones(1,8) ones(1,8) ones(1,8) ones(1,8)];x22=[ones(1,8) zeros(1,8) ones(1,8) zeros(1,8) ones(1,8) zeros(1,8) ones(1,8) zeros(1,8)];x33=[ones(1,8) zeros(1,24) ones(1,8) zeros(1,24)];a1=(1/64)*fft(x11);a2=(1/64)*fft(x22);a3=(1/64)*fft(x33);n=[0:63];subplot(3,1,1);figure(1),stem(n,a1) subplot(3,1,2);figure(2),stem(n,a2) subplot(3,1,3);figure(3),stem(n,a3) figure(4),stem(n,imag(a1)) figure(5),stem(n,imag(a2)) figure(6),stem(n,imag(a3))%3.5(f)function x=func(ak,n)x=0;for k=-31:31if k<=0ak(k)= ak(-k) x= x+ak(k)*exp(i*k*(2*pi/32)*n) ;elsex=x+ak(k)*exp(i*k*(2*pi/32)*n) ;endendsubplot(3,1,1);n=[0:31];x3_2=zeros(1,32);for k=1:32x3_2(k)=real(func(a3,k));endstem(n,x3_2)%3.5(h)function x=dtfs(ak,n)x=0;for k=-31:31if k<=0ak(k)=ak(-k) x= x+ak(k)*exp(i*k*(2*pi/32)*n) ;elsex=x+ak(k)*exp(i*k*(2*pi/32)*n) ;endend⑤吉布斯现象：根据英文教材Example 3.5验证Fig3.9的吉布斯现象(a)~(d)%Gibbsx=1/2;k=0;t=-2:0.01:2;x1=[-2 -1 -1 1 1 2];y1=[0 0 1 1 0 0];for i=1:2:49k=i-k;x=x+(-1)^(k+1)*2/(i*pi)*cos(i*pi*t/2);if(i==1)figure; plot(x1,y1,t,x); title('N=1')elseif(i==3)figure; plot(x1,y1,t,x);title('N=3')elseif(i==7)figure; plot(x1,y1,t,x);title('N=7') elseif(i==9)figure; plot(x1,y1,t,x);title('N=9') elseif(i==49)figure; plot(x1,y1,t,x);title('N=49') endend实验项目3:一、实验项目名称：非周期信号傅里叶分析的MATLAB实现二、上机实验题目：傅里叶变换的基本性质及其在系统分析中的应用1．实验项目的目的和任务：熟练掌握连续时间傅里叶变换的基本性质及其在系统分析中应用。