Fourier coefficients relation to Power Spectral Density

[Skaiserollz89]

TL;DR

Help deriving a result found in "Numerical Simulation of Optical Wave Propagation" by Jason Schmidt. I'm trying to work out by hand an equation stating that the ensemble average of the squared fourier coefficients of a 2D phase function equals the Power Spectral Density( Phi(fx,fy) multiplied by 1/A, where A is the domain area ( either delta_fx*delta_fy in frequency space, or 1/(L_x*L_y) in real space). I am having trouble seeing how to get this result. Please assist in the derivation.

 

[osilmag]

https://mathworld.wolfram.com/ParsevalsTheorem.html

Wolfram provides a good explanation of Parsevals Theorem. It applies a trig identity to simplify it.