# 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:

Filters
41 items

## MATLAB Function: ztrans

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.

See More

## Discrete Fourier Transform

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...

See More

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

20 min
Beginner
Video
Theory

An animated introduction to the Fourier Transform.

See More

## Understanding the Z-Transform

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...

See More

## Understanding the Discrete Fourier Transform and the FFT

19 min
Beginner
Video
Theory

The discrete Fourier transform (DFT) transforms discrete time-domain signals into the frequency domain. The most efficient way to compute the DFT is using a fast Fourier transform (FFT)...

See More

## Discrete-Time Fourier Transform MATLAB App

Intermediate
App
Demonstration

This is the app that Brian Douglas created for his MATLAB Tech Talk video on the Z-transform.

See More

## Fourier Series and Gibbs Phenomena [Matlab]

7 min
Beginner
Video
Application

This video will describe how to compute the Fourier Series in Matlab and Gibbs Phenomena that appear for discontinuous functions.

See More

## Frequency domain – tutorial 13: sampling (theory of everything in signal pro...

25 min
Beginner
Video
Theory

In this video, we learn about sampling which enables us to travel from analog to digital world. The following materials are covered:1) why traveling to the ...

See More

## Introduction to the Fourier Transform (Part 2)

12 min
Beginner
Video
Theory

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...

See More

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

10 min
Beginner
Video
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...

See More

## Computing the DFT Matrix

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)...

See More

## The Fast Fourier Transform Algorithm

10 min
Beginner
Video
Theory

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

See More

## Frequency domain – tutorial 5: Fourier transform

9 min
Beginner
Video
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...

See More

## Fourier Series [Python]

8 min
Beginner
Video
Application

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

See More

## The Taylor Series

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...

See More

## Introduction to the Fourier Transform (Part 1)

13 min
Beginner
Video
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...

See More

## Frequency domain – tutorial 2: Fourier series

25 min
Beginner
Video
Theory

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...

See More

## Complex Fourier Series

12 min
Intermediate
Video
Theory

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

See More

## The Fast Fourier Transform (FFT)

8 min
Beginner
Video
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.

See More

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

13 min
Beginner
Video
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...

See More

## Fourier Series [Matlab]

8 min
Beginner
Video
Application

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

See More

## Fourier Series: Part 2

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...

See More

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

9 min
Beginner
Video
Theory

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...

See More

## The Discrete Fourier Transform (DFT)

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...

See More

## Frequency domain – tutorial 11: equalization

11 min
Beginner
Video
Theory

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...

See More