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Fourier Analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.

The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis has a corresponding inverse transform that can be used for synthesis.

from Fourier Analysis - Wikipedia

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MATLAB Function: ztrans

MathWorks
Intermediate
Software
Application

ztrans(f) finds the Z-Transform of f. By default, the independent variable is n and the transformation variable is z. If f does not contain n, ztrans uses symvar. 

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Discrete Fourier Transform

MathWorks
Intermediate
Article / Blog
Theory

The discrete Fourier transform, or DFT, is the primary tool of digital signal processing. The foundation of the product is the fast Fourier transform (FFT), a method for computing the DFT...

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Parseval's Theorem

Steve Brunton
5 min
Intermediate
Video
Theory

Parseval's theorem is an important result in Fourier analysis that can be used to put guarantees on the accuracy of signal approximation in the Fourier domain.

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Complex Fourier Series

Steve Brunton
12 min
Intermediate
Video
Theory

This video will describe how the Fourier Series can be written efficiently in complex variables.

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