Convolution concepts and applications

Source: Internet
Author: User
Han weiyun ▲

Convolution of two square pulses: the resulting waveform is a triangular pulse. One of the functions (in this caseG) Is first reflected about
And then offsetT, Making it
. The area under the resulting product gives the convolutionT. The horizontal axis is
G, And t.

Convolution of a square pulse (as input signal) with the impulse response of an RC Circuit to obtain the output signal waveform. the integral of their product is the area of the yellow region. in both animations the FunctionGIs using Ric, and so is
Unchanged under reflection.

In functional analysis,Convolution(Zookeeper),RotatingOrZookeeper, Through two functionsF
AndGA mathematical operator that generates the third function to characterize the function.FAnd flipped and movedG. If we regard a function that participates in Convolution as a range indicator function, convolution can also be seen as a promotion of "Moving Average.



  • 1. Brief Introduction
  • 2 Definitions
    • 2.1 fast convolution algorithm
    • 2.2 multivariate function convolution
  • 3 Nature
  • 4 convolution Theorem
  • 5 convolution on a group
  • 6 Applications
  • 7. See
  • 8 External links
[Edit] Brief Introduction

Convolution is an important operation in analytical mathematics. Set:, is the two product functions on, for points:

It can be proved that the above points exist for almost all. In this way
This integral defines a new function called a function.
Convolutional. Easy to verify, and
It is still a product function. That is to say, we replace convolution with multiplication. Space is an algebra, or even a he algebra.

Convolution is closely related to Fourier transformation. Using a single property, that is, the product of the Fourier transformation of the two functions equals the Fourier transformation after their convolution, can simplify the processing of many problems in Fourier analysis.

Functions obtained from convolution are generally compared
And are smooth. Especially for smooth functions with tight support sets,
When it is a partial product, their convolution is also a smooth function. By using this property, a column can be constructed to approximate any product functions.
This method is called smoothing or regularization of functions.

Convolution can also be applied to series, measurements, and generalized functions.

[Edit] Definition

FunctionFAndGIt is the integral of the product of one function after one function is flipped and translated, and is a function of the translation volume.

The credit interval depends onFAndG.

For functions defined in discrete fields, convolution is defined

[Edit] Quick convolution algorithm

If it is a finite length
, Requires an approximate operation. Some quick algorithms can be used to reduce

The most common fast convolution algorithm is the use of Fast Fourier transformation through circular convolution. You can also use
Such as number theory conversion.

[Edit] multivariate function convolution

Based on the definition of flip, translation, and points, you can also define the points on the multivariate functions as follows:

[Edit] nature

All convolution operators meet the following requirements:

Exchange Law
Combination Law
Allocation Law
Number multiplication combination Law

Any real number (or plural number ).

Differential Theorem

DFIndicatesFIf it is in a discrete region, it refers to a difference operator, which includes two types: Forward difference and backward difference:

  • Forward difference:
  • Backward difference:
[Edit] convolution Theorem

Convolution TheoremIt is pointed out that the Fourier transformation of function convolution is the product of function Fourier transformation. That is, convolution in one domain is equivalent to the product in another domain. For example, convolution in the time domain corresponds to the product in the frequency domain.

IndicatesFFourier transformation.

This must be true for Laplace transformation, bilateral Laplace transformation, ztransformation, mellin transformation, and Hartley Transformation (see Mellin
Inversion Theorem) and other Fourier transform variants are also true. In the harmonic analysis, we can also extend the Fourier transformation defined on the local tightening Abel group.

Convolution theorem can be used to simplify the calculation of convolution. For sequences of long characters, perform group-based multiplication based on the definition of convolution. The calculation complexity is, only a set of bitwise multiplication is required. After using the Fast Algorithm of Fourier transformation, the total computational complexity is. This result can be applied in fast multiplication calculation.

[Edit] convolution on a group

Group of measure (for example, the topological group with local closeness under the upper Hal measure in the hosdorf space),G
UpperM-Leberger real or plural functions that can be productF
AndG, You can define their Convolution:

Convolution defined on these groups can also give properties such as convolution theorem, but this requires the expression theory of these groups and the Peter-outer theorem of harmonic analysis.

[Edit] Application

Convolution has many applications in engineering and mathematics:

  • In statistics, weighted moving average is a convolution.
  • In probability theory, the sum of the two independent statistical variables X and Y is the convolution of the probability density functions of X and Y.
  • In acoustics, an echo can be represented by a convolution of the source sound and a function that reflects various reflection effects.
  • In electronic engineering and signal processing, the output of any linear system can be obtained by convolution of the input signal and system function (system impulse response.
  • In physics, any linear system (conforming to the superposition principle) has convolution.
[Edit] See
  • Convolution
  • Fourier transform
[Edit] external link
  • Information about convolution on planetmath.
  • Visual convolution Java Applet
From Title = % E5 % 8d % B7 % E7 % A7 % AF & oldid = 19614366 "4 categories:

  • Functional Analysis
  • Signal Processing
  • Binary operations
  • Bilinear Operators

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