Question

1. For each of the following choices of r(n) and N, you will perform the five tasks stated below (a) x(n (b) r(n)-2-a(n), Л-16, (d) x(n) is same as in part (c) with N = 8 otherwise Task 4: Compute DTFT of y. You may not be able to obtain a closed form expression for the DTFT of y. However since y has finite duration of length N, you can just code the analysis equation in Matlab. Let y(k 1) = y(k) forS k1,... .N. Specifically, since the signal y(n) is 0 outside the interval 0

Answer 1

MATLAB code is given below in bold letters.

clc;
clear all;
close all;

% Question a
N = 16;
n = 0:N-1;
x = 1./(2.^abs(n));
% 1000 points for frequency axis
k = 1:1000;
w = 2*pi*(k-1)/1000;clear k;
% initialize DTFTy
DTFTy = zeros(size(w));
for k = 1:length(w)
DTFTy(k) = sum(x.*exp(-1j*w(k)*n));
end

figure;plot(w/(pi),DTFTy,'linewidth',2);grid;xlabel('w/pi');ylabel('Amplitude');
title('Spectrum of x[n]=1/2^|^n^|');

% Question b
N = 16;
n = 0:N-1;
x = 1./(2.^n);
% 1000 points for frequency axis
k = 1:1000;
w = 2*pi*(k-1)/1000;clear k;
% initialize DTFTy
DTFTy2 = zeros(size(w));
for k = 1:length(w)
DTFTy2(k) = sum(x.*exp(-1j*w(k)*n));
end

figure;plot(w/(pi),DTFTy2,'linewidth',2);grid;xlabel('w/pi');ylabel('Amplitude');
title('Spectrum of x[n]=1/2^n');

% Question c
N = 16;
n = 0:N-1;
x = (n+1).*(n<=6) + (13-n).*(n>=7).*(n<=13);
% 1000 points for frequency axis
k = 1:1000;
w = 2*pi*(k-1)/1000;clear k;
% initialize DTFTy
DTFTy3 = zeros(size(w));
for k = 1:length(w)
DTFTy3(k) = sum(x.*exp(-1j*w(k)*n));
end

figure;plot(w/(pi),DTFTy3,'linewidth',2);grid;xlabel('w/pi');ylabel('Amplitude');
title('Spectrum of x_3[n]');

% Question d
N = 8;
n = 0:N-1;
x = (n+1).*(n<=6) + (13-n).*(n>=7).*(n<=13);
% 1000 points for frequency axis
k = 1:1000;
w = 2*pi*(k-1)/1000;clear k;
% initialize DTFTy
DTFTy4 = zeros(size(w));
for k = 1:length(w)
DTFTy4(k) = sum(x.*exp(-1j*w(k)*n));
end

figure;plot(w/(pi),DTFTy4,'linewidth',2);grid;xlabel('w/pi');ylabel('Amplitude');
title('Spectrum of x_4[n]');

Spectrum of x[n]-1/2 nl 2 1.8 1.6 1.4 1.2 0.8 0.6 0 0.2 0.40.6 0.8 1 1.2 1.4 16 1.8 2 w/pi

Spectrum of x[n]-1/2" 2 1.8 1.6 1.4 0.8 0.6 0 0.2 0.40.6 0.8 1 1.2 1.4 16 1.8 2 w/pi

Spectrum of xzln] 50 40 30 ) 20 T 10 0 -10 -20 0 0.2 0.40.6 0.8 1 1.2 1.4 16 1.8 2 wlpi

Spectrum of x4In] 40 30 20 10 0 -10 -20 0 0.2 0.40.6 0.8 1 1.2 1.4 16 1.8 2 w/pi