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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Fourier Series: Part 2

Steve Brunton
6 min
Beginner
Video
Theory

This video will show how to approximate a function with a Fourier series, which is an infinite sum of sines and cosines. We will discuss how these sines and cosines form a basis for the...

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The Discrete Fourier Transform (DFT)

Steve Brunton
17 min
Beginner
Video
Application

This video introduces the Discrete Fourier Transform (DFT), which is how to numerically compute the Fourier Transform on a computer. The DFT, along with its fast FFT implementation, is one...

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Fourier Analysis: Overview

Steve Brunton
7 min
Beginner
Video
Theory

This video presents an overview of the Fourier Transform, which is one of the most important transformations in all of mathematical physics and engineering. This series will introduce the...

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The Fourier Transform and Derivatives

Steve Brunton
10 min
Beginner
Video
Theory

This video describes how the Fourier Transform can be used to accurately and efficiently compute derivatives, with implications for the numerical solution of differential equations.

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Fourier Series: Part 1

Steve Brunton
12 min
Beginner
Video
Theory

This video will show how to approximate a function with a Fourier series, which is an infinite sum of sines and cosines. We will discuss how these sines and cosines form a basis for the...

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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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