Suppose we have 2 identical gaussian pulse signals in time domain, offset by time delay $\tau$. After taking fourier transform of both into frequency domain, I want to phase shift one of them such that when I do an inverse FT back, the two pulses are now matching. I know I can do this using fourier shift theorem,
$f(t-\tau) = e^{-i \tau \omega} * \mathcal{F}(f)$.
If I multiply $e^{i \tau \omega}$ then IFT, I'll get the offset pulse to no longer be offset.
I've tried this on 2 offset sinusoids, it works. Question is, If I have two single gaussian pulses, what will $\omega$ be? For sinusoid example, it's straightforward but what would be the "frequency of a gaussian pulse"?
If you want to obtain $f(t - \tau)$ form the fourier transfor of $f$, say $\hat f$, you should multiply it by $\hat g(\omega) = e^{i\omega \tau}$ in the transformed domain. In this case, you should multiply the two functions $\hat g, \hat f$ pointwise. Obviously the product will be zero outside the support of $\hat f$. For the case of a sinusoid the support is only its frequency.