When is the Fourier transform of a signal periodic?

Question

Also , is the inverse Fourier transform of a periodic signal also periodic ?

Answer 1

I'm going to assume that we can start with a signal for which a Fourier transform exists (such as an absolutely integrable function). If we construct another signal by sampling this original signal at regular time intervals, then the Fourier transform of that newly constructed signal would corresponds to the Discrete-Time Fourier Transform (DTFT) which would be periodic. Note that if the original signal's bandwidth is less than the Nyquist frequency then the time-sampled signal can also be recovered from this frequency-domain representation.

Conversely, if we evenly sample a continuous frequency-domain function, the corresponding inverse transform would be a periodic signal in the time domain. Correspondingly an evenly time-sampled and periodic signal would have a periodic evenly frequency sampled signal (ie. the inverse Fourier transform of that periodic signal would also be periodic).