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.

Related to Fourier coefficients relation to Power Spectral Density

What are Fourier coefficients?

Fourier coefficients are the set of coefficients of the terms in the Fourier series expansion of a periodic function. They represent the amplitude and phase of the sinusoidal components (sines and cosines) that make up the function.

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

The Power Spectral Density (PSD) of a signal provides a measure of its power distribution over frequency. The Fourier coefficients, when squared and normalized, can be used to calculate the PSD of a periodic signal. Specifically, the PSD is proportional to the square of the magnitude of the Fourier coefficients.

Can Fourier coefficients be used to compute the PSD of non-periodic signals?

For non-periodic signals, the Fourier Transform is used instead of Fourier series. The Fourier Transform provides a continuous spectrum, and the Power Spectral Density can be derived from the magnitude squared of the Fourier Transform. The relation is similar in principle to the periodic case but applies to a continuous frequency range.

What is the significance of the magnitude of Fourier coefficients in PSD?

The magnitude of the Fourier coefficients indicates the strength of the corresponding frequency component in the signal. When calculating the PSD, the magnitude squared of these coefficients gives the power contributed by each frequency component, thus providing a detailed representation of how power is distributed across different frequencies.

How do you normalize Fourier coefficients for PSD calculation?

To normalize Fourier coefficients for PSD calculation, you typically divide the squared magnitude of each coefficient by the period of the signal. For discrete signals, this normalization involves dividing by the total number of samples. This ensures that the resulting PSD has the correct units of power per frequency unit.