Can you show the steps to get $f[m,n] = f[m,n]$ for this?
$$ f[m',n'] = IFFT \: ( FFT \: ( \: f[m',n'] \: ) \: ) $$
$$ f[m',n'] = \sum_{k=0}^{M-1}\sum_{l=0}^{N-1} \left[\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}(f[m,n]e^{-j2\pi(\frac{km}{M}+\frac{ln}{N})}) \right]e^{j2\pi(\frac{km'}{M}+\frac{ln'}{N})}$$
It is a discrete time fourier transform. The reason I'm asking this is I will add an additional $e^{something}$ term after the FFT and take it out of the summation term.