Question

Solve using MATLAB

(1) Consider the difference equation given follows as у(п) — — 0.9у(п - 1) — 0.81у(n — 2) + 1.62г(п) + 1.82(п — 1) + 2ar (п — 2), п > 0. Find and plot x(n) and y(n) and hence find the energy of y(n) for the following cases (а) 2(п) — 8(п — 2) (b) 2(п) — и(п) 6(п — 3) _ (с) г(п) — и(п)u(-п + 23) _ (d) x(n)r(n) - r(n - 25), where r(n) is the unit-ramp sequence.

Answer 1

Given difference equation,

By applying Z-Transform on both sides we get

Transfer function is given as follows

H(z) Y(z) X(z) 1.62 1.8z 2z2 10.z 0.81z

Y(z) H(z)X(z)

Now we will solve for y[n] using MATLAB

(a) MATLAB CODE:

clc
clear all
close all

syms n z
H(z) = (1.62+1.8*z^(-1)+2*z^(-2))/(1+0.9*z^(-1)+0.81*z^(-2));
x(n) = kroneckerDelta(n,1); % del[n-1]
X(z) = ztrans(x(n),n,z);    % Z-Transform of x[n]
Y(z) = H(z)*X(z);
y(n) = iztrans(Y(z),z,n);

fprintf('Output y[n]:\n')
disp(y(n))

n = -10:10;

subplot(2,1,1)   % Plot of x[n]
stem(n,x(n))
xlabel('n')
title('x[n]')

subplot(2,1,2)   % Plot of y[n]
stem(n,y(n))
xlabel('n')
title('y[n]')

OUTPUT:

Output y [n] (200*kroneckerDelta(n - 1, 0))/ (380*kroneckerDelta(n, 0))/ 729 1)^n*3^(1/2)*100^ (1 3^ (1/2)*45i)^(n - - ((-1)^n*3^ (1/2)*100^(1 3^ (1/2)*45i)^(n - 1)*2489i)/6075 (6878* ( -1)^n*81^n*cos ( (pi*n)/3))/ (3645*90^n) 81 n) * (45 1) *2489i)/6075 - n)*(45

x[n] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 8 8 6 -10 y[n] 2 1.5 1 0.5 10 -0.5 -1 -1.5 2 -2.5 10 10 8 4 6 2 4 -6 -8