Overall Impulse Response (a) Implement the system in Fig. 1 using MATLAB to get

Question

Overall Impulse Response (a) Implement the system in Fig. 1 using MATLAB to get the impulse response of the overall cascaded system for the case where q = 0.85, r = 0.85 and M = 15. Here you will use two calls to filter. Plot the impulse response of the overall cascaded system. (b) Work out the impulse response h[n] of the cascaded system by hand to verify that your MATLAB result in part (a) is correct. (c) In a deconvolution application, the second system (FIR FILTER-2) tries to undo the convolutional effect of the first. Perfect deconvolution would require that the cascade combination of the two systems be equivalent to the identity system: y[n = x[n]. If the impulse responses of the two systems are h_1[n] and h_2[n], state the condition on h_1[n] = h_2[n] to achieve perfect de convolution, i.e., what must h_1[n] * h_2[n] look like if h_2[n] achieves perfect deconvolution? Does the cascade of FIR FIITER-1 and FulTER-2 perform perfect deconvolution?

Explanation / Answer

clc;
close all;
clear all;

% QUUESTION A
N = 30;
M = 15;
n = 0:1:N;
k = 0:1:M;
q = 0.85;
r = 0.85;

un = (n>=0)*1;

B = [1 -0.85];
A = [1];
wn = filter(B,A,un);

rk = zeros(size(k));

for k = 0:1:M;
    rk(k+1) = r.^k;
end

yn =filter(rk,A,wn)

figure
subplot(3,1,1);
stem(un);

subplot(3,1,2);
stem(wn);

subplot(3,1,3);
stem(yn);

% QUESTION C
%TO PERFORM PERFECT CONVOLUTION, h1(n)*h2(n) = u(n)
%IN the above case, we observe almost perfect deconvolution

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