How to reconstruct signal of its phase and magnitude functions?

Question

I have two continuous periodic (a period of $2\pi$) functions which belong to the phase and magnitude of Fourier transform of a signal, how can I reconstruct the original signal? What kind of transform should I use? (Basic Fourier transform or DTFT)

Answer 1

You can look at the domains covered by the Fourier transform (FT) or the DTFT (Discrete Time Fourier Transform). The DTFT applies to discrete signals, and yields a (possibly $2\pi$) periodic transformation. Here, $\omega$ denotes the continuous normalized radian frequency variable. So:

$$ (s[k])_{k\in \mathbb{Z}} \stackrel{\mathrm{DTFT}}{\longrightarrow} S(\omega) = \sum_{k=-\infty}^{\infty} s[k] z^{-k}, \quad z=e^{\imath \omega}$$ while for the FT:

$$s(t) \stackrel{\mathrm{FT}}{\longrightarrow} S(\omega) = \int_{-\infty}^{\infty} s(t) e^{- \imath \omega t}dt$$ with no specific periodicity in general.

So most likely, you are asked to recover a discrete signal. The complex quantity $S(\omega)$ can be recovered with the amplitude and the phase you have. You can then recover the discrete signal using the inverse DTFT over one period:

$$s[k] = \int_{-\pi}^{\pi} S(w)e^{\imath \omega k} d\omega \,.$$

You definitely need to check the Discrete Time Fourier Transform (DTFT) and the associated pages.