I am trying to figure out something about folding in the DFT. Let's suppose we do the DFT of a function with a specific sampling rate of $\delta t$. Now I know that folding will occur, in other words all components in the power spectrum for $\omega > \nu_N$ with $\nu_N = \pi/\delta t$ will be folded to be in the range $[-\nu_N, \nu_N]$. If I indicate the Discrete Fourier transform with $F(\nu)$, then I can write that the DFT as $$ F(\nu)+F(-\nu+2\nu_N)+F(-\nu-2\nu_N) $$ for the frequency above $\nu_N$ and below $-\nu_N$. Now my question is what about frequencies above $3\nu_n$ for example? They are not zero so that would be folded too. Therefore I need theoretically to write down an infinite series of terms. Is that correct? I hope my question is clear enough...
In the picture I calculated the DFT of a Lorentian function.
The blue line is the exponential function (the FT of the Lorentian), the orange line is the folded one and the red one is the sum of the two. I would say that one need an infinite sum of terms (although in this case two would probably be enough numerically to get a good approximation).
Am I right?
Thanks, Umberto
I researched a bit and the answer lies in the Continuous-Time Aliasing theorem.
https://ccrma.stanford.edu/~jos/st/Continuous_Time_Aliasing_Theorem.html
I would appreciate if anyone had any information on what book I can find a discussion of it. Thanks!