Question

Consider a periodic signal
x[n] = sin(0.1πn) + 13sin(0.3πn) + 15sin(0.5πn).
For each of the following systems, determine if the system imparts (i) no distortion, (ii) magnitude distortion, and/or (iii) phase (or delay) distortion. In each case, graph the input and the steady state response for 0 ≤ n ≤ 60.

(a) H(z) = 1/9(1 + 2z^(−1) + 3z^(−2) + 2z^(−3) + z^(−4)).

(b) h[n] = {1(n=0),−1.1756,1}

(c) H(z) = {1+1.778z^(−2)+3.1605z^(−4) }/{1+0.5625z^(−2)+0.3164z^(−4)}

Answer 1

in each case the syste, imparts both magnitude distorion and phase distorion. MATLAB code is given below.

clc;
close all;
clear all;

% define n
n = 0:60;
% define x
x = sin(0.1*pi*n) + 13*sin(0.3*pi*n) + 15*sin(0.5*pi*n);

% filter coefficiets
Ha = 1/9*[1 2 3 2 1] ;
Hb = [1 -1.1756 1] ;
Hc_num = [1 0 1.778 0 3.1605];
Hc_den = [1 0 0.5625 0 0.3164];

figure;
freqz(Ha);
title('frequency response of system a')

figure;
freqz(Hb);
title('frequency response of system b')

figure;
freqz(Hc_num,Hc_den);
title('frequency response of system c')

% now filter the signal
ya = filter(Ha,1,x); % filter the signal x with system 1
yb = filter(Hb,1,x); % filter the signal x with system 2
yc = filter(Hc_num,Hc_den,x); % filter the signal x with system 3

% plot the input and outputs
figure;
subplot(211);
plot(n,x);grid on;xlabel('n');ylabel('Stem Height');
title('input');

subplot(212);
plot(n,ya);grid on;xlabel('n');ylabel('Stem Height');
title('respone of system in problem a to input x');

figure;
subplot(211);
plot(n,x);grid on;xlabel('n');ylabel('Stem Height');
title('input');

subplot(212);
plot(n,yb);grid on;xlabel('n');ylabel('Stem Height');
title('respone of system in problem b to input x');

figure;
subplot(211);
plot(n,x);grid on;xlabel('n');ylabel('Stem Height');
title('input');

subplot(212);
plot(n,yc);grid on;xlabel('n');ylabel('Stem Height');
title('respone of system in problem c to input x');