A Fourier coefficients relation to Power Spectral Density

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

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Fourier coefficients are directly related to Power Spectral Density (PSD) through Parseval's Theorem, which states that the total energy of a signal can be expressed in both time and frequency domains. The theorem demonstrates that the sum of the squares of the Fourier coefficients equals the integral of the square of the signal over time. This relationship highlights the importance of Fourier analysis in signal processing and energy distribution. Wolfram's explanation effectively uses a trigonometric identity to simplify the theorem's application. Understanding this connection is crucial for analyzing signals in various fields, including communications and audio processing.

May 31, 2023

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.

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Jun 22, 2023

osilmag

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

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

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