Question

2. A system is described by the following difference equation n]1.5y[n0.56y[n -2]+x{n-0.2x{n-] a) Find the transfer function of the system b) Let xn]un]. Compute an analytical expression for the response y[n]. Use Matlab to calculate the coefficients c) Simulate and plot the response using Matlab.(stem plot) Generate 50 points. (Matlab: x ones(1,50));

Answer 1

2) a) Transfer function:

Apply z transform to the difference equation,

Y(z)=1.5 z^-1 Y(z) - 0.56 z^-2 Y(z) + X(z) -0.2 z^-1X(z)

Y(z)[1-1.5 z^-1+0.56 z^-2 ]= X(z)[1 -0.2 z^-1]

Y(z)   [1 -0.2 z^-1]

H(z)= ------ = _______________

X(z)   [1-1.5 z^-1+0.56 z^-2 ]

Analyjcal erpreut Xl2) C1-0.22) C-08-) - (1- 02/07) C1-0 yt2)A 0-714 1-os72 Co.22 (1-0.75 - 1- 0.2/08) (1-070i-) -0125x025 - o428x-0.85 A- 24 20.7 5

0.22 8.833 (i-0)1-0)2xo3

MATLAB:

clc;close all;clear all;
b=[1 -0.2]
a=[1 -1.5 0.56]
N=50
n=0:1:N-1
figure;
x=ones(1,N)
y=filter(b,a,x)

subplot(211)
stem(n,y,'r')
xlabel('n')
ylabel('y(n)')
title('Response using filter command')

subplot(212)
%from manually calculated analytical expression of y(n)
y=((-24*(0.8.^n))+(11.66 *(0.7.^n))+13.33).*(n>=0)
stem(n,y,'m');xlabel('n')
ylabel('y(n)')
title('from manually calculated analytical expression of y(n)')

Response using filter command 14 12 10 6 4 2 40 50 10 20 30 n from manually calculated analytical expression of y(n) 14 12 10 8 6 4 2 0 10 30 40 50 n 20 T O N O (u)A (u)A