Question

(42)1+ (z-0.5)z-0.9)(z-0.8) 3. The transfer function of a system is H(z) = a) Compute an analytical expression for the response y[n] if x[n] = u[n]. . Use Matlab to calculate the coefficients b) Simulate the response using Matlab (stem plot). Generate 50 points. (enter transfer function into Matlab and apply step input)

Answer 1

Analytical expression for y(n)

(2+ (2+0.S) Z. (2 0.2-0.D C2-018)(2. A 2-0°8 z-o.9 Z-o.S [Z+) (2+016 A y2 (z-0.5 2 o.S (2-0.9)2-08) z-i) (o+2) Gta (z-i) 20.5)2-0.0 94 0. 4 x O lx-o. -65

-0.2 2. 1_0 2 O & O.2XO 3 390 Y(2)(2-1) 2 os (tos O.SX0 xo 2

MATLAB:

clc;close all;clear all;

b=conv([1,1],[1 0.5])
a=conv(conv([1 -0.5],[1,-0.9]),[1,-0.8])
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 the analytical expression of y(n)
y=(((-25*(0.5.^n))+(-665 *(0.9.^n))+(390 *(0.8.^n))+(300 *(1.^n))).*(n>=0))
stem(n,y/max(y),'m');xlabel('n')
ylabel('y(n)')
title('from the analytical expression of y(n)')

Response using filter command 300 250 200 150 100 50 50 10 20 30 40 0 n from the analytical expression of y(n) 1 0.8 0.6 0.4 0.2 0 10 30 40 50 n 20 (u)A (u)A