How can I run Code for Discrete time fourier transform of x[n] = a^u[n] where |a|<1

kindly provide MATLAB code for this queston.

Answer 1

[TED MOSBY]

Hi,

For the specific signal , where u[n] is the unit step function (meaning u[n]=1 for n≥0 and u[n]=0 for n<0), the sum starts from n=0:

This is a geometric series. A geometric series converges to 1/(1-r)​ if ∣r∣<1. In our case, . The series converges when . Since for any real ω, this condition simplifies to ∣a∣<1. If this condition is met, the sum of the series is:

a = 0.8; % Choose a value for 'a' such that |a| < 1.0
N = 50;  
omega = -pi:0.01:pi; 
% 1. Generate the discrete-time signal x[n] = a^n u[n]
n = 0:N-1;
x_n = a.^n;
% 2. Compute the DTFT 
X_jw_analytical = 1 ./ (1 - a * exp(-1i * omega));
% 3. Approximate the DTFT using Fast Fourier Transform
L = 1024; % Length of the DFT. Should be greater than N.
X_k = fft(x_n, L);
f_dft = (0:L-1) * (2*pi/L); % Frequency axis 
f_dft_shifted = fftshift(f_dft - pi); % Shift to -pi to pi
X_k_shifted = fftshift(X_k);
% 4. You can now proceed to plotting the Magnitude and Phase Responses

https://www.mathworks.com/help/matlab/ref/fft.html?s_tid=srchtitle_support_results_1_fft

https://www.mathworks.com/help/matlab/ref/fftshift.html?s_tid=srchtitle_support_results_1_fftshift

Hope this helps!​

[Paul]

The expression for f_dft_shifted appears to be incorrect.

a = 0.8; % Choose a value for 'a' such that |a| < 1.0
N = 50;  
omega = -pi:0.01:pi; 
% 1. Generate the discrete-time signal x[n] = a^n u[n]
n = 0:N-1;
x_n = a.^n;
% 2. Compute the DTFT 
X_jw_analytical = 1 ./ (1 - a * exp(-1i * omega));
% 3. Approximate the DTFT using Fast Fourier Transform
L = 1024; % Length of the DFT. Should be greater than N.
X_k = fft(x_n, L);
f_dft = (0:L-1) * (2*pi/L); % Frequency axis 
f_dft_shifted = fftshift(f_dft - pi); % Shift to -pi to pi
X_k_shifted = fftshift(X_k);
% 4. You can now proceed to plotting the Magnitude and Phase Responses
plot(omega,abs(X_jw_analytical),f_dft_shifted,abs(X_k_shifted))

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