This is the solution in MATLAB, please translate to Python 3. 1. (5 points) Prog

Question

This is the solution in MATLAB, please translate to Python 3.

1. (5 points) Programming exercise.] Consider an LTI system with a transfer function .98 sin(?/24): 22-1.96cos(T/24) + .9604 Its impulse response is given by h[n] - .98" sin(Tn/24)u[n] Suppose that this system has the following signal as input: r[n] = ?(99)" cos(nn/48)u [n] Since both h[n] and r[n] decay to zero, we can approximate them as L-point signals (i.e., keep their first L samples) when L is sufficiently large. For this problem, let us choose L 512 (so, r[n]| and h[n| are truncated and represented by their respective samples for 0 S n 511) (a) (1 point) Directly co mpute the convolution of an] and hin] and plot the resulting ydir [n In Python, you may want to use 'np.convolve(x,h)'; you will also need to import numpy scipy and matplotlib.pyplot. (b) (3 points) Implement the convolution of x[n] and hm] using the following scheme Padding L zeros h[n] FFT Yfastln] IFFT Paddingx[n] L zeros xIn] FFT To compute the 1024-point FFT of h[n] in Python, you could use 'np.fft.fft (y)/N' (same for x[n]; N = 1024). IFFT is called in a similar way ?npft.ift(Y)'). Plot the resulting yfast[n] on the same graph as ydir [n

Explanation / Answer

clc;

close all;

clear all;

%QUESTION a figure(1);

n = 0:1:511;

hn = (0.98).^n .* sin(pi*n/24);

subplot(3,1,1);

stem(n,hn); xn = (n.^2).*(0.99.^n).*cos(pi*n/48);

subplot(3,1,2);

stem(n,xn);

m = 0: 1 : length(xn)+length(hn)-2;

ydirn = conv(xn,hn);

subplot(3,1,3);

stem(m,ydirn);

%QUESTION b n = 0:1:1023;

figure(2); hn = (0 < n < 511).*(((0.98).^n) .*

sin(pi*n/24)) + (n > 511).*0;

Hfft = fft(hn,1024);

subplot(3,1,1);

stem(n,Hfft);

xn = (0 < n < 511).*(n.^2).*(0.99.^n).

*cos(pi*n/48) + (n > 511).*0

; Xfft = fft(xn,1024);

subplot(3,1,2);

stem(n,Xfft);

Yconv = Hfft.*Xfft;

yfastn = ifft(Yconv);

m1 = 0: 1 : length(Yconv)-1;

subplot(3,1,3);

plot(m1,yfastn,'r',m,ydirn,'b');

%QUESTION c L = 1:1:1000

; figure(3); ndir = 2*(L.^2);

nfast = 12*L.*log2(2*L) + 8*L + 4;

plot(L,ndir,'r',L,nfast,'b');

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