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

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Ayesha Latif
Ayesha Latif 2020 年 7 月 3 日
コメント済み: Paul 2025 年 6 月 25 日
kindly provide MATLAB code for this queston.

回答 (1 件)

TED MOSBY
TED MOSBY 2025 年 6 月 25 日
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
Hope this helps!​
  1 件のコメント
Paul
Paul 2025 年 6 月 25 日
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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