Is it true that the frequency domain representations of signals are always periodic? If so, is there intuition as to why?
I'm having some trouble understanding what periodicity in the frequency domain means, especially in relationship to the time domain. The time domain signal can be aperiodic which make sense because if a signal is a function of time you wouldn't necessarily expect it to follow a pattern. What am I missing that would relate the two?
Thanks!
Your time signal consists of a number of samples in a limited time interval. The Fourier transform approximates that signal with a set of sines and cosines. The result is that the approximated signal is indeed periodic.
Effectively the signal is treated as if it repeats itself after the measured time interval, whether that is really the case or not. Only if the time interval extends to infinity, which is not possible with a sampled signal, can the Fourier transform be non-periodic.
There are several methods (windowing) to reduce unwanted edge effects. They basically rely on the idea that the amplitude of the signal to transform should be zero at the boundaries to suppress undesired frequencies due to the periodic characteristic of the Fourier transform.