# 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:

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36 items

## But what is the Fourier Transform? A visual introduction.

20 min
Beginner
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Theory

An animated introduction to the Fourier Transform.

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## Introduction to the Fourier Transform (Part 2)

12 min
Beginner
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This video is the second part of the introduction to the Fourier Transform. I address an error that I made in the first video concerning the scaling term of the transform. I also try to...

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## Frequency domain – tutorial 12: FT of periodic signals

10 min
Beginner
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Theory

In this video, we learn how to find the Fourier transform for periodic signals. The following materials are covered:1) relation between Fourier transform and...

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## Computing the DFT Matrix

7 min
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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 Fast Fourier Transform Algorithm

10 min
Beginner
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Here I discuss the Fast Fourier Transform (FFT) algorithm, one of the most important algorithms of all time.

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## Frequency domain – tutorial 5: Fourier transform

9 min
Beginner
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Theory

In this video, we learn about Fourier transform which enables us to travel from time to frequency domain when a signal is not periodic. The learning objectiv...

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## Fourier Series [Python]

8 min
Beginner
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Application

This video will describe how to compute the Fourier Series in Python.

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## The Taylor Series

84 min
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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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## Introduction to the Fourier Transform (Part 1)

13 min
Beginner
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Theory

This video is an introduction to the Fourier Transform. I try to give a little bit of background into what the transform does and then I go step by step through explaining the Inverse...

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## Frequency domain – tutorial 2: Fourier series

25 min
Beginner
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In this video, we learn Fourier series which enables us to travel from time to the frequency domain when a signal is periodic. The following materials are co...

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## Complex Fourier Series

12 min
Intermediate
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Theory

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

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## The Fast Fourier Transform (FFT)

8 min
Beginner
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Application

Here I introduce the Fast Fourier Transform (FFT), which is how we compute the Fourier Transform on a computer. The FFT is one of the most important algorithms of all time.

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## Frequency domain – tutorial 3: filtering (periodic signals)

13 min
Beginner
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Theory

In this video, we learn about filtering which enables us to manipulate the frequency content of a signal. A common filtering application is to preserve desi...

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## Fourier Series [Matlab]

8 min
Beginner
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Application

This video will describe how to compute the Fourier Series in Matlab.

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## Fourier Series: Part 2

6 min
Beginner
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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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## Frequency domain – tutorial 1: concept of frequency (with Chinese subtitle)

9 min
Beginner
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In this video, the following materials are covered:1) intuitive explanation on the frequency concept 2) what is the relation between time and frequency domai...

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## The Discrete Fourier Transform (DFT)

17 min
Beginner
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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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## Frequency domain – tutorial 11: equalization

11 min
Beginner
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In this video, we learn about equalization technique which is used in communication systems to compensate for the destructive effect of the channel between t...

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## Fourier Analysis: Overview

7 min
Beginner
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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

10 min
Beginner
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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

12 min
Beginner
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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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## Frequency domain – tutorial 6: Fourier transform tables

34 min
Beginner
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In this video, we learn about Fourier transform tables which enable us to quickly travel from time to the frequency domain. The main learning objective is to...

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## Solving the Heat Equation with the Fourier Transform

11 min
Beginner
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This video describes how the Fourier Transform can be used to solve the heat equation. In fact, the Fourier transform is a change of coordinates into the eigenvector coordinates for the...

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## Frequency domain – tutorial 7: Fourier transform examples marathon

46 min
Beginner
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Theory

In this video, we solve lots of lots examples to practice how to quickly find Fourier transform using table of pairs and properties. The learning objective i...

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## Denoising Data with FFT [Python]

10 min
Beginner
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Application

This video describes how to clean data with the Fast Fourier Transform (FFT) in Python.

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