Expression for the Fourier Transform of a signal with finite support

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);

[Mann Baidi]

What is your desired output?

[Nicolae-Eusebiu IFTIMI]

My desired output is fourier fransform, X(omega) , is in the photo

[Walter Roberson]

Should n be integer? The [] notation is used for discrete signal processing

Answer 1

[Sudarsanan A K]

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:

https://mathworks.com/help/symbolic/sym.symsum.html

https://mathworks.com/help/symbolic/simplify.html

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:

https://blogs.mathworks.com/steve/2010/06/25/plotting-the-dtft-using-the-output-of-fft/

I hope this clarifies your query.