Question

Realize signal processing systems described by the difference equation: y1(n)=1/2 [x(n)+x(n-1)] and y2(n)=1/2 [x(n)-x(n-1)] using Matlab. Assuming same input signal x(n)=sin(ωn) for various values of ω ={0,/6, ,3/2, 1.9/2} applied to both systems find the following: Obtain stem plots of y1(n) and y2(n) in each case. (15 Marks) Critically analyze y1(n) and y2(n) in terms of type of filter, maximum gain and cut off frequency. (Hint : The system can be tested by computing frequency response of the system).

Answer 1

MATLAB code is given below.

clc;
close all;
clear all;

% define w
w = [0 pi/6 3*pi/2 1.9*pi/2];

% define n
n = 1 : 50;

% define x1 and x2 and compute y1 and y2 using for loop for
% different values of w

for k = 1:length(w)
  
% define x1 and x2
x1 = sin(w(k)*n);
x2 = cos(w(k)*n);
  
% now compute y1 and y2
y1 = 0.5*([zeros(1,length(x1)+1) x1]+[fliplr(x1) zeros(1,length(x1)+1)]);
y2 = 0.5*([zeros(1,length(x1)+1) x1]-[fliplr(x1) zeros(1,length(x1)+1)]);
  
% now plot the signals
figure;
subplot(211);

stem(-max(n):max(n),y1,'fill'); grid on;
xlabel('n');ylabel('Amplitude');title(['y1[n] for w = ',num2str(w(k))]);
  

subplot(212);
stem(-max(n):max(n),y2,'fill'); grid on;
xlabel('n');ylabel('Amplitude');title(['y2[n] for w = ',num2str(w(k))]);
end

The plots are given below

yi [n] for w = 0 0.5 -0.5ト 50 40 30 -20 -10 0 10 2030 40 50 y2[n] for w = 0 0.5 0.5 50 40 30 20 1010 20 30 0 50

y1In] for w 0.5236 0.5 0.5 50 40 30 20 1010 20 30 40 50 y2[n] for w 0.5236 0.5 -0.5 50 40 30 20 10 10 20 30 0 50

y1[n] for w 4.7124 0.5 0.5* 50 40 32010 0 10 20 304 50 y2[n] for w4.7124 0.5 -0.5 50 40 30 20 10 0 10 20 30 50

y1In] for w 2.9845 0.5 0.5 50 40 30 20 1010 20 30 40 50 y2[n] for w 2.9845 -0.5 50 40 30 20 1010 20 30 0 50

from the above fiures it is observed that y1[n] is symmeric about y axis which means that y1[n] is an even part of the signal x1. likewise y2[n] is anti-symmeric about y axis which means that y2[n] is an odd part of the signal x1