to update :1 APR 2016

For Fourier series, see Mathematical equations :Fourier Series

Fourier Transformation :

For satisfying Dirichlet The function of the condition \(f(t)\) Define... At its continuous point

\(F(\omega)=\int_{-\infty}^{+\infty}f(t)e^{-\mathrm{i}\omega t}dt\)

be \(f(t)\) Can be transformed into

\(f(t)=\dfrac{1}{2\pi}\int_{-\infty}^{+\infty}F(\omega)e^{\mathrm{i}\omega t}d \omega\)

This is known as Fourier Transformation , It's a one-to-one mapping in a function space , Write it down as

\(F(\omega)=\mathscr{F}[f(t)],\qquad f(t)=\mathscr{F}^{-1}[F(\omega)]\)


Fourier The basic properties of transformation :

1. linear

\(\mathscr{F}[\alpha f_1(t)+\beta f_2(t)]=\alpha \mathscr{F}[f_1(t)]+\beta \mathscr{F}[f_2(t)]\)

2. Differentiability

(1) \(\mathscr{F}[f’(t)]=\mathrm{i}\omega\mathscr{F}[f(t)]\)

(2) \(\dfrac{d}{d\omega}\mathscr{F}[f(t)]=\mathscr{F}[-\mathrm{i}tf(t)]\)

3. Integrality

Ruodong \(t \rightarrow +\infty\) when ,\(g(t)=\int_{-\infty}^tf(a)da \rightarrow 0\), be




Convolution is a binary operation defined in function space . For the function \(f_1(t)\),\(f_2(t)\), Define convolution operations \(*\)


Convolution operation satisfies commutative law 、 Associative law 、 The distributive law of addition .


Convolution theorem

if \(f_1(t)\),\(f_2(t)\) Can be done Fourier Transformation , be


Transpose convolution and multiplication .

In mathematical equation, it can be used to solve the function which is difficult to inverse transformation —— Decomposing factors to simplify transformations .

Mathematical equations :Fourier Transformation and convolution of more related articles

  1. Mathematical equations :Laplace Transformation & Residue ( Updating )

    to update :25 APR 2016 Laplace Transformation Let's set the function \(f(t)\) stay \(t>0\) There are definitions , integral \(F(s)=\int_0^{+\infty}f(t)e^{-st}dt \qquad ( ...

  2. Using discrete Fourier Transform to solve quadratic equation of one variable

    Let's have a quadratic equation $$x^2+bx+c=0$$ The two roots of are $x_1,x_2$. be $$(x-x_1)(x-x_2)=x^2+bx+c.$$ therefore $$\begin{cases}  x_1+x_2=-b\\x_1 ...

  3. dennis gabor From Fourier (Fourier) Switch to Gabriel (Gabor) Transform to wavelet (Wavelet) Transformation ( Reprint )

    dennis gabor subject : From Fourier (Fourier) Switch to Gabriel (Gabor) Transform to wavelet (Wavelet) Transformation This article is a summary and excerpt of the contents of various references while learning , It's a comprehensive introductory document , The point is to sort out Fu ...

  4. Mathematical equations :Fourier Series

    to update :25 MAR 2016 For periodic functions ( The period is \(2\pi\)) Or defined in \([-\pi,\pi]\) The function on \(f(x)\), It can be expanded to * \(\large f(x)=\dfrac{a_0}{2} ...

  5. Fourier Basis of analysis ( Two )—— From the series, we derive the continuity Fourier Transformation

    The derivation here refers to ( To copy verbatim ) A First Course in Wavelets with Fourier Analysis Second Edition, Albert Boggess& Fran ...

  6. Why? Fourier analysis ?

    The purpose of this paper is to give Fourier Several motivations for analysis . Catalog Wave equation The thermal conductivity equation Lapalce Transformation Summation formula Expressionism Characteristic theory Reference material Wave equation Consider the simplest boundary value problem of one-dimensional wave equation $$u(x,t), x\in ...

  7. Gabor Transformation

    Gabor Transformation Gabor Transform belongs to windowed Fourier transform ,Gabor Functions can be scaled at different scales in the frequency domain . Extract relevant features in different directions . in addition Gabor The function is similar to the biological action of the human eye , So it is often used in texture recognition , And achieved good results .Gab ...

  8. Don't panic ," Convolution " It's very simple ( Next )

        The article comes from my CSDN Blog of the same name , Welcome to scan the code at the end of the article ~   Definition Based on the popular example of the previous article , We understand convolution from the basic concept , So what's the stricter definition ? Mathematically speaking , Convolution is just an operation , For a lot of people who don't have ...

  9. Matlab Image processing series 4——— Images of Fourier transform and inverse transform

    Be careful : The image processing program for this series of experiments , Use Matlab Realize the most important image processing algorithm 1.Fourier exchange (1) Frequency domain enhancement In addition to being able to process images in the spatial domain , We can also transform the image to other space for processing . These parties ...

Random recommendation

  1. About c# stay DataTable Delete a line according to conditions in

    We often put data sources in DataTable Inside , But sometimes it's also necessary to remove unwanted lines , The following code tells you DataTable dts:                DataRow[] foundRow;   ...

  2. Small size, big use , use PrimusUI Navigate your page

    “PrimusUI” I'm learning from a lot of open source resources on the Internet UI library , After finishing a simple code set . Of each function block CSS Very little code , Try to be easy to understand , Low threshold , The code can be easily modified according to the actual situation , Change to fit your own scene ...

  3. Android root + modify host

    1. Use KingRoot Download the mobile version , After installation Root Handle . 2. download RE File manager , After installation , Open application , Get into etc, find host, Check , From the menu Edit as text , After modification , Press the back key , Prompt to save ...

  4. 【BZOJ-1976】 Energy Cube Cube Minimum cut + Black and white

    1976: [BeiJing2010 organize a team ] Energy Cube Cube Time Limit: 10 Sec  Memory Limit: 64 MBSubmit: 884  Solved: 307[Submi ...

  5. Java SE series:1. environment configure and Hello world! [We use compiler and packager to create an application!]

    1. cli (command line interface) and gui (graphic user interface) use javahome path, search classpath ...

  6. PAT 1089. Insert or Merge (25)

    According to Wikipedia: Insertion sort iterates, consuming one input element each repetition, and gr ...

  7. adopt JS Determine the networking type and connection status

    adopt JS Determine the networking type and connection status China's mobile network environment is complex , In order to give users a better access experience , Developers want to know the current networking mode of users , Then give the user a request result that matches the current network environment . W3C A method is given in the specification of ...

  8. docker Service registration

    docker Service registration etcd docker run -d --name etcd -p 4001:4001 -p 7001:7001 elcolio/etcd

  9. Mobile - Popup demo

    <!doctype html> <html> <head> <meta charset="UTF-8"> <meta name ...

  10. IIS install asp Components :JMail Mail sending and receiving components

    JMail brief introduction jmail It's a server-side mail sending component , It's different from personal client email software .jmail It is used to send e-mail to the program on the server , Except for software programmers , Other people don't usually use . jmail It's a third party mail ...