Topic

 

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

This topic includes the following resources and journeys:

 

 

Understanding the Z-Transform

MathWorks - Brian Douglas
20 min
Beginner
Video
Theory

 

This intuitive introduction shows the mathematics behind the Z-transform and compares it to its similar cousin, the discrete-time Fourier transform. Mathematically, the Z-transform is...

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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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The Fourier Transform

Steve Brunton
14 min
Beginner
Video
Theory

This video will discuss the Fourier Transform, which is one of the most important coordinate transformations in all of science and engineering.

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Computing the DFT Matrix

Steve Brunton
7 min
Beginner
Video
Theory

This video discusses how to compute the Discrete Fourier Transform (DFT) matrix in Matlab and Python. In practice, the DFT should usually be computed using the fast Fourier transform (FFT)...

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The Taylor Series

Christopher Lum
84 min
Beginner
Video
Theory

In this video we discuss the Taylor Series (and the closely related Maclaurin Series). These are two specific types of Power Series that allow you to approx...

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