Say I have two non-periodic signals, $f_1(t)$ and $f_2(t)$, with Fourier transforms $F_1(\omega)$ and $F_2(\omega)$. Basically, I need to line up $f_1(t)$ and $f_2(t)$ as close as possible, and I am allowed to shift the signals in time and multiply their Fourier transforms by a constant phase.
What is the most efficient method to find the values of the phase shift $\Delta \phi$ and the time delay $\Delta t$ such that, when the Fourier transform of $f_2$ is modified into
$$ F_2(\omega) e^{i (\Delta \phi + \omega \Delta t)}, $$
the modified $f_2$ best "lines up" with $f_1$ according to their cross-correlation (or some other measurement)?
I am interested in simple and computationally efficient methods, even if they are not necessarily perfect.