필터 지우기
필터 지우기

Expression for the Fourier Transform of a signal with finite support

조회 수: 4 (최근 30일)
Nicolae-Eusebiu IFTIMI
Nicolae-Eusebiu IFTIMI 2023년 10월 28일
답변: Sudarsanan A K 2023년 11월 6일
Hello! I want to determinate the expresion of Fourier Transformation for x[n] = e^(j*w0*n), n ∈ 0, N-1 , ( w - omega ) , w = pi/8. I just know that the Fourier Transformation sould look like this X(w), But i don't know how to do it
I tried this but it's not working as i wish
N = 10;
n = 0:0.01:N-1;
omega = -pi:0.01:pi;
j = sqrt(-1);
w = 0:0.01:pi;
n = 0:0.01:N-1;
x = exp (j*n*omega0);
X = x * exp(-j * n' * w);
  댓글 수: 3
Nicolae-Eusebiu IFTIMI
Nicolae-Eusebiu IFTIMI 2023년 10월 28일
My desired output is fourier fransform, X(omega) , is in the photo
Walter Roberson
Walter Roberson 2023년 10월 28일
Should n be integer? The [] notation is used for discrete signal processing

댓글을 달려면 로그인하십시오.

답변 (1개)

Sudarsanan A K
Sudarsanan A K 2023년 11월 6일
Hello Nicolae,
I understand that you are trying to compute the Discrete-Time Fourier Transform (DTFT) of the signal , where rad and N is the total number of samples. I note that you are facing issue while computing the DTFT of the signal using the formula.
The formula for computing DTFT of a signal is:
This can be coded and compared with the result you are having as follows:
N = 32; % Number of points
n = 0:N-1;
w0 = pi/8;
j = sqrt(-1);
% Generate the time-domain signal
x = exp(j*w0*n);
% DTFT using the provided expression (The result you are having)
omega = -pi:0.01:pi;
X_expr = exp(-j * (omega - w0) * N/2) ./ exp(-j * (omega - w0)/2) .* sin((omega - w0) * N/2) ./ sin((omega - w0)/2);
% Compute the DTFT using the DTFT formula
X_dtft = zeros(size(omega));
for k = 1:N
X_dtft = X_dtft + exp(-j * omega * n(k)) * x(k);
end
% Plotting the time-domain signal
subplot(2,1,1);
stem(n, real(x), 'b', 'LineWidth', 1.5);
hold on;
stem(n, imag(x), 'r', 'LineWidth', 1.5);
hold off;
xlabel('n');
ylabel('x[n]');
title('Time-Domain Signal');
legend('Real Part', 'Imaginary Part');
grid on;
% Plotting the DTFT using the formula and expression
subplot(2,1,2);
plot(omega, abs(X_expr), 'g-', 'LineWidth', 2);
hold on;
plot(omega, abs(X_dtft), 'r--', 'LineWidth', 1.5);
hold off;
xlabel('Angular Frequency (\omega)');
ylabel('|X(\omega)|');
title('DTFT using Formula and Expression');
legend('Using Expression', 'Using Formula');
grid on;
If you are looking for the simplified expression for the DTFT of your input signal , you can consider the following code using the Symbolic Math Toolbox as follows:
syms n w w0 N;
% Define the sequence x[n]
x = exp(1i * w0 * n);
% Define the Fourier Transform X(w)
X = symsum(x * exp(-1i * w * n), n, 0, N-1);
% Simplify the expression
X = simplify(X);
% Display the result
X
X = 
The MathWorks documentations of the "symsum()" and "simplify()" functions can be found at:
Additionally, you can refer to the following blog that describes the relationship between DFT and DTFT where you can utilize the "fft()" function to compute DTFT:
I hope this clarifies your query.

카테고리

Help CenterFile Exchange에서 Discrete Fourier and Cosine Transforms에 대해 자세히 알아보기

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by