Fourier coefficients relation to Power Spectral Density

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Thread starter Skaiserollz89
Start date May 31, 2023
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Coefficients Fourier Power

In summary, the Fourier coefficients represent the frequency components of a signal, while the Power Spectral Density (PSD) describes the power of the signal at each frequency. The PSD is calculated using the squared magnitudes of the Fourier coefficients. The Fourier coefficients influence the shape of the PSD by determining the amplitudes of the different frequency components in the signal. A larger coefficient at a specific frequency results in a higher power value at that frequency in the PSD. The PSD can be calculated directly from the coefficients, but it is more commonly done using FFT algorithms for efficiency. Fourier coefficients and PSD are commonly used in signal analysis in various fields and can provide valuable information about a signal's frequency components, patterns, and anomalies. There is a direct relationship

May 31, 2023

Skaiserollz89

TL;DR Summary: 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.

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Jun 22, 2023

osilmag

Gold Member

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

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