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.

1. What are Fourier coefficients?

Fourier coefficients are used to represent a periodic signal as a sum of sinusoidal functions. They are calculated by taking the inner product of the signal with the basis functions, which are complex exponentials with different frequencies.

2. How are Fourier coefficients related to Power Spectral Density (PSD)?

The PSD is a measure of the power of a signal at different frequencies. It is calculated by taking the squared magnitude of the Fourier coefficients. Therefore, the Fourier coefficients provide the necessary information to calculate the PSD.

3. What is the significance of the phase of Fourier coefficients?

The phase of the Fourier coefficients represents the relative timing of each sinusoidal component in the signal. It is important in determining the shape and characteristics of the signal, such as its symmetry and sharpness.

4. How do Fourier coefficients help in signal processing?

Fourier coefficients are used in many signal processing techniques, such as filtering, noise reduction, and compression. They allow us to analyze and manipulate signals in the frequency domain, which can be more useful than the time domain in certain applications.

5. Can Fourier coefficients be used for non-periodic signals?

Yes, Fourier coefficients can be extended to non-periodic signals through the use of the Fourier transform. This allows us to analyze and process non-periodic signals in the frequency domain as well.