数字信号处理实验 3：DTFT、频率响应和系统函数

%1.设置差分方程y(n)-0.8y(n-1) = x(n)

a = 0.8;

%初值y(-1) =0

ys = 0;

N = 30;

xn1 = [1,zeros(1,N-1)];

n =[1:N];

%输入序列xn2

xn2 =0.8.^n;

B = 1;A= [1, -0.8];

xi =filtic(B,A,ys);

%2.求解系统的单位脉冲相应和输入序列的单位脉冲相应

yn1 =filter(B,A,xn1);

yn2 =filter(B,A,xn2);

n1 =0:length(yn1)-1;

n2 =0:length(yn2)-1;

subplot(3,2,1);stem(n1,yn1,'.');

title('系统单位脉冲响应');xlabel('n');ylabel('h(n)');

subplot(3,2,2);stem(n2,yn2,'.');

title('输入序列响应（函数库法)');xlabel('n');ylabel('s(n)');

%3.dtft

w = -2*pi:0.01:2*pi;

X = dtft(xn2, n, w);

subplot(3,2,3); plot(n, xn2);title('signal = 0.8^n');

subplot(3,2,4); plot(w, X); title('DTFT');

%4.幅频相频特性

X1 = abs(X)

X2 = angle(X)

subplot(3,2,5); plot(w,X1); title('幅频特性');

subplot(3,2,6); plot(w,X2); title('相频特性');

%%%%%%%%%%%%%%%%%%%%%%%%根据系统

%H(Z)=Z/(Z-0.8)

num = [1];

den = [1,-0.8];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%零点极点分布

subplot(3,1,3);zplane(num,den);

%%%%%%%%%%%%%%%%改变零点极点的分布带来的变化

w = -2*pi:0.01:2*pi;

%H(Z)=(Z-1)/(Z-0.8) 零点在单位圆上,谷值为0

num = [1,-1];

den = [1,-0.8];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%零点极点分布

subplot(3,1,3);zplane(num,den);

%H(Z)=(Z-0.7)/(Z-0.8) 1.零点越靠近单位圆，谷值越深

num = [1,-0.7];

den = [1,-0.8];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%零点极点分布

subplot(3,1,3);zplane(num,den);

%H(Z)=(Z-0.2)/(Z-0.8) 1.零点越靠近单位圆，谷值越深

num = [1,-0.2];

den = [1,-0.8];

%响应分布

g = freqz(num,den,w');

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%零点极点分布

subplot(3,1,3);zplane(num,den);

%H(Z)=Z/(Z-0.3) 极点越靠近单位圆，峰值越尖锐

num = [1];

den = [1,-0.3];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%H(Z)=Z/(Z-0.9) 极点越靠近单位圆，峰值越尖锐

num = [1];

den = [1,-0.9];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

%H(Z)=Z/Z-1 极点在单位圆上

num = [1];

den = [1,-1];

%响应分布

g = freqz(num,den,w);

figure;

g1 = abs(g)

g2 = angle(g)

subplot(3,1,1); plot(w,g1); title('幅频特性');grid on;

subplot(3,1,2); plot(w,g2); title('相频特性');grid on;

DTFT.m

function [ X ] = dtft(x,n,w)

% [X] = dtft（x，n，w）

% X =在w频率计算的DTFT值

% x = n上的有限持续时间序列

% n =样本位置向量

% w =频率位置矢量

temp = w' * n;

temp = - 1i  * temp;

e = exp(temp);

X = e * x' ;

End

Ztrans.m

clear all;close all;clc;

syms k

xn3 = 0.8^k;

Fz = ztrans(xn3)