If I have a known discrete values for the characteristic function (I know $a_n$ values for specific values of $\omega_n = \frac{2\pi i n}{d}$):
$$a_n = \phi_x\left(\omega = \frac{2\pi in}{d}\right),$$
How do I find the probability distribution for this characteristic function?
I've seen some stuff online about possibly needing an inverse discrete-time Fourier transform, but I haven't had much luck unfortunately. (Edit: I believe its actually related to the discrete-frequency Fourier Transform (DFFT), but this is much less well-documented on the internet. A paper discussing the topic is here: The discrete frequency Fourier transform. The DFFT differs from the DTFT in that it's defined for discrete values of frequency, giving a continuous function of time, whereas the DTFT is defined for discrete values of time, giving a continuous function of frequency)
Alternatively, I'm just trying to find the variance in this probability distribution, so if there's a way to get the variance directly from discrete values of the characteristic function, I'd be happy to hear that :).