Question

MATLAB CODE IS REQUIRED**

Let x[n] = n(0.9)ⁿu[n].

(a) Determine the DTFT X ̃ (e^jω) of x[n].

(b) Choose first N = 20 samples of x[n] and compute the approximate DTFT X ̃N(e^jω) using the fft function. Plot magnitudes of X ̃(e^jω) and X ̃N(e^jω) in one plot and compare your results.

(c) Repeat part (b) using N = 50.

(d) Repeat part (b) using N = 100.

Answer 1

(a)

MATLAB CODE :

clc
clear all
close all

syms n w
u(n) = heaviside(n) ;      % Unit step signal
x(n) = n*(0.9^n)*u(n) ;    % Given x[n]
X(w) = symsum(x(n)*exp(-1i*w*n),n,-inf,inf) ; % DTFT of x[n]

fplot(w,abs(X(w)))
xlabel('w')
ylabel('|X(e^{jw})|')
title('Magnitude using DTFT')
grid on

OUTPUT :

Magnitude using DTFT X(el) -5 -4 -3 -2 - 1 1 2 3 4 5

(b)

MATLAB CODE :

clc
clear all
close all

N = 20 ;            % Given N value
n = 0:N-1 ;
u = n >= 0 ;        % Unit Step sequence
x = n.*(0.9.^n).*u; % x[n]
X = fft(x) ;        % FFT of x[n]

plot(abs(X))
title('Magnitude using FFT for N = 20')
grid on

OUTPUT :

Magnitude using FFT for N = 20 0 2 4 6 8 10 12 14 16 18 20

(c)

MATLAB CODE :

clc
clear all
close all

N = 50 ;            % Given N value
n = 0:N-1 ;
u = n >= 0 ;        % Unit Step sequence
x = n.*(0.9.^n).*u; % x[n]
X = fft(x) ;        % FFT of x[n]

plot(abs(X))
title('Magnitude using FFT for N = 50')
grid on

OUTPUT :

Magnitude using FFT for N = 50 201 0 5 10 15 20 25 30 35 40 45 50

(d)

MATLAB CODE :

clc
clear all
close all

N = 100 ;            % Given N value
n = 0:N-1 ;
u = n >= 0 ;        % Unit Step sequence
x = n.*(0.9.^n).*u; % x[n]
X = fft(x) ;        % FFT of x[n]

plot(abs(X))
title('Magnitude using FFT for N = 100')
grid on

OUTPUT :

Magnitude using FFT for N = 100 90 80 0 10 20 30 40 50 60 70 80 90 100