Look here is an example, the time domain lasts for a ms and the Fourier response extends from 0 to some tens of KHz. Plug it in Matlab and tweak it as you like

% Define time-domain signal
dt = 0.000001; % time step (s)
t = 0:dt:0.001; % time vector (s)
x = cos(50000.*t-0.0004).*exp(-(t-0.0003).^2/0.00006^2); % time-domain signal
% Perform FFT
X = fft(x); % FFT of time-domain signal
f = (0:length(X)-1)/(dt*length(X)); % frequency vector (Hz)
% Plot results
figure;
subplot(2,1,1);
plot(t,x);
xlabel('Time (s)');
ylabel('Amplitude');
title('Time-domain signal');
subplot(2,1,2);
Amplitude=abs(X);
semilogy(f(1:length(X)/20),Amplitude(1:length(X)/20));
xlabel('Frequency (Hz)');
ylabel('Amplitude');
title('Frequency-domain signal (FFT)');

In the extreme case? Infinitesimally short time. The FT of a Dirac pulse is a flat, constant response from 0 Hz to infinity Hz (or, in the case of DFT, Nyquist frequency).

As I said, the FT of a Gaussian is a Gaussian. Really sharp in the time domain means really wide in the frequency domain. If you do not understand why that is, your intuition about FTs will be flawed.

Maybe this helps: The impulse response in OPs post image basically does not have to be windowed at all, it’s decently close to zero at the edges. Plug it into a FT and you’ll get a decently accurate FR.