Workthrough

Taking us through the math on the fourier transform of a Gaussian (Bell-shaped)

function:

                                                                     

                                                                               

Below, the sinusC ie sinus_cardinale function, which is what the square signal

transforms to:
                                                                                

Here, the gausian function transforms to another gaussian function. From Matlab:

                                                                                   

                                                                                   

from Meta ai:

Here's the updated code with normalization:

Matlab

% Define the function f(t)
t_start = -40;  % start time
t_end = 40;    % end time
dt = 0.01;     % time step
t = t_start:dt:t_end;  % time axis

% Define f(t) 
f = exp(-t.^2);  % Gaussian function

% Normalize f(t) to have unit energy
f = f ./ sqrt(sum(f.^2)*dt);

% Compute Fourier Transform using FFT
F = fftshift(fft(f));

% Compute frequencies corresponding to F
w = (2*pi)/(dt*length(t))*(-length(t)/2:length(t)/2-1);

% Normalize Fourier Transform to have unit energy
F = F ./ sqrt(sum(abs(F).^2));

% Plot time-domain function
figure;
plot(t, f);
xlabel('t (s)');
ylabel('f(t)');
title('Normalized Time-Domain Function f(t)');

% Plot Fourier Transform
figure;
plot(w, abs(F));
xlabel('\omega (rad/s)');
ylabel('|F(\omega)|');
title('Normalized Fourier Transform of f(t)');

% Optional: Plot phase spectrum
figure;
plot(w, angle(F));
xlabel('\omega (rad/s)');
ylabel('Phase of F(\omega)');
title('Phase Spectrum of F(\omega)');

Changes:

Normalized f(t) to have unit energy using f = f ./ sqrt(sum(f.^2)*dt).
Normalized F(ω) to have unit energy using F = F ./ sqrt(sum(abs(F).^2)).

Now, the energy of f(t) and F(ω) will be:

Matlab

% Verify unit energy
disp('Energy of f(t):'); disp(sum(f.^2)*dt);
disp('Energy of F(ω):'); disp(sum(abs(F).^2));

This should output:

Energy of f(t): 
     1

Energy of F(ω): 
     1

Indicating that both f(t) and F(ω) have unit energy.