Why ifft results of analytic signal spectrum don't match with theoretical time formula?

Dear all,

I need to convert a spectrum data back to time domain. Before it, I use a simple Gaussian pulse as a test. However, I found that results computed by ifft don't match with theoretical time equation. Why it happened? How to resolve it?

Fs=200;
T=1/Fs;
L=1200;
tau=0.4;
t0=2.8;
mag=10;
t=(0:L-1)*T;

f=Fs*(-L/2:L/2-1)/L;
Xf=@(x)sqrt(pi).*mag.*tau.*exp(-(2.*pi.*tau.*x).^2).*exp(-1i.*2.*pi.*x.*t0);
figure;
plot(f,real(Xf(f)));

% shift spectrum starting from 0 frequency
signal_comp=ifft(fftshift(Xf(f)).*L);
signal_theoretical=mag.*exp(-(t-t0).^2/tau^2);
figure;
plot(t,signal_theoretical);
hold on;
plot(t,real(signal_comp),'r--');

Answer 1

[Matt J]

The IFFT and the continuous inverse Fourier transform are not the same thing.

tau=0.4;
t0=2.8;
mag=10;

Fs=200;
dT=1/Fs;
L=2000;
dF=1/dT/L;



f=dF*(-L/2:L/2-1);
t=dT*(-L/2:L/2-1);


Xf=@(x)sqrt(pi).*mag.*tau.*exp(-(2.*pi.*tau.*x).^2).*exp(-1i.*2.*pi.*x.*t0);


% shift spectrum starting from 0 frequency
signal_comp= fftshift(ifft(ifftshift(Xf(f))) ) /dT; %note factor of 1/dT

signal_theoretical=mag.*exp(-((t-t0)/2).^2/tau^2)/2; %fix the theoretical formula


figure;
plot(t(1:20:end),signal_theoretical(1:20:end),'o',...
      t,real(signal_comp),'r--');
legend('Theoretical','Computed',location='northwest')

[Jiali]

Thank you for your clarification! I do have misunderstanding between discrete Fourier transform and continuous Fourier transform although the concepts were taught before.