Question

Section 1: Given a system y[n]-y[n-1]+y[n-2]=x[n] (refer to M3.2 on textbook)
In class, we analytically derived the solutions of second order difference equations, including zero-input response, unit impulse response, zero-state response and total response. The Matlab has imbedded commands to do the same job. Get familiar with the following commends, and use them to get (0≤n≤40)

a) unit impulse response and plot it
b) zero-input response and plot it, with initial conditions of y[-1]=1 and y[-2]=2 c) zero-state response and plot it, x[n]=2cos(2πn/6)u[n]
d) total response and plot it

inline
filter
filtic
stem
xlabel, ylabel

Answer 1

****** Matlab Code ********

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% finding ZIR and ZSR for discrete systems

H = tf([1 0 0],[1 -1 1],1); % given (or derived) transfer function
n = 0:1:40; % time vector
imp = double(n==0); % impulse signal
b = 1;
a = [1 -1 1];
y = filter(b,a,imp); % impulse response
subplot(2,2,1);
stem(n,y);
title('Impulse response');
xlabel('n');
ylabel('Amplitude')
%%%%%%%% creating state space model for finding ZSR and ZIR
sys_ss = ss(H);
x0 = [1;2]; % initial conditions
[y1,n,x] = initial(sys_ss,x0,n); % ZIR
subplot(2,2,2);
stem(n,y1);
title('Zero Input response');
xlabel('n');
ylabel('Amplitude')

u = 2*cos(2*pi*n/6);
subplot(2,2,3);
y2 = lsim(sys_ss,u,n); % ZSR
stem(n,y2);
title('Zero State response');
xlabel('n');
ylabel('Amplitude')

subplot(2,2,4);
stem(n,y1+y2);
title('Total response');
xlabel('n');
ylabel('Amplitude')

**************************

Output::-