Suppose I have 2 identical Gaussian pulses, but separated by some phase offset. If I take the Fourier transform of it to move from time domain to frequency domain, how can I manipulate the phase terms (imaginary part) such that when I inverse Fourier transform, the two pulses are now aligned?
Thank in advance.
Best Answer
Recall that translation in time (or space, if you prefer) corresponds to a modulation in the frequency domain; i.e.
$$\mathcal{F} (T_\alpha f(x)) = \mathcal{F}( f(x-\alpha)) = E_\alpha \mathcal{F}(f)(\xi) = e^{-2\pi i \alpha \xi} \mathcal{F}(f)(\xi)$$
where I have denoted $T_\alpha$ to be the translation operator, $T_\alpha f(x) = f(x-a)$ and $E_\alpha$ to be the modulation operator, $E_\alpha g(\xi) = e^{-2\pi i \alpha \xi}g(\xi)$ and the Fourier transform $\mathcal{F}(f) = \int_\mathbb{R} e^{-2\pi i \xi x}f(x)dx$. So if you want the signals to be offset in time, modulate in the frequency domain.