Question

Consider the system:y[n]-0.5y[n-1]-0.25y[n-2]=x[n]+2x[n-1]+x[n-2] • Plot, using MATLAB, the impulse and step responses of the system. Highlight the response characteristics in your plots • Assume initial conditions y(-1) = 1, y(-2) = 0 and that the input signal to the system is a discrete-time unit step. Determine the formula for the Z-transform of the solution, Y(z). Subsequently, determine the formula for the solution, y[n], itself.

Answer 1

Script:

clc;close all;clear all;

%Impulse response

b=[ 1 2 1]; %coefficients of x(n)

a = [1 -1/2 1/4]; %coefficients of y(n)

n = 0:100;

x = (n==0);%Unit impulse function

figure;

y = filter(b, a, x, filtic(b, a, [1 0], [0 0]));

subplot(211)

stem(n,y)

title('Impulse response')

xlabel('n');grid;

ylabel('h(n)')

%step response

x = (n>=0);%Unit step function

subplot(212)

y = filter(b, a, x, filtic(b, a, [1 0], [0 0]));

stem(n,y);grid;

title('step response')

xlabel('n')

ylabel('s(n)')