# Convolution

**convolvedconvolvingconvolution kernelkernelconvolutionsconvolution operatordiscrete convolutionconvolveconvolutionalconvolves**

In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) that produces a third function expressing how the shape of one is modified by the other.wikipedia

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### Cross-correlation

**cross correlationcorrelationcorrelating**

Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either

The cross-correlation is similar in nature to the convolution of two functions.

### Deconvolution

**deconvolutedeconvoluteddeconvolution problem**

Computing the inverse of the convolution operation is known as deconvolution.

In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data.

### Convolution theorem

**convolution propertyDirect convolutionFourier convolution**

See Convolution theorem for a derivation of that property of convolution.

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.

### Finite impulse response

**FIRFIR filterFinite Impulse Response (FIR)**

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

This computation is also known as discrete convolution.

### Cauchy product

**Cauchymultiplyingproduct**

This is known as the Cauchy product of the coefficients of the sequences.

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

### Circular convolution

**Cyclic convolutioncircular (or cyclic) convolutionPeriodic convolution**

is known as a circular convolution of f and g.

The circular convolution, also known as cyclic convolution, of two aperiodic functions (i.e. Schwartz functions) occurs when one of them is convolved in the normal way with a periodic summation of the other function.

### Discrete Fourier transform

**DFTcircular convolution theoremFourier transform**

The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]].

The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.

### Overlap–save method

**overlap-saveOverlap saveOverlap-save method**

Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the Overlap–save method and Overlap–add method.

Overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x[n] and a finite impulse response (FIR) filter h[n]:

### Operation (mathematics)

**operationoperationsmathematical operation**

In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) that produces a third function expressing how the shape of one is modified by the other.

Operations on functions include composition and convolution.

### Overlap–add method

**overlap-addOverlap addoverlap-add (OLA) method**

Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the Overlap–save method and Overlap–add method.

In signal processing, the overlap–add method (OA, OLA) is an efficient way to evaluate the discrete convolution of a very long signal x[n] with a finite impulse response (FIR) filter h[n]:

### Discrete-time Fourier transform

**convolution theoremDFTDTFT § Properties**

For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution.

The significance of this result is expounded at Circular convolution and Fast convolution algorithms.

### Fourier transform

**continuous Fourier transformFourierFourier transforms**

In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created.

Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem).

### Support (mathematics)

**supportcompact supportcompactly supported**

For functions f, g supported on only

Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

### Young's convolution inequality

**Young's inequalityYoung's inequality for convolutions**

More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable

In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.

### Fast Fourier transform

**FFTFast Fourier Transform (FFT)Fast Fourier Transforms**

The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]]. Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime N, expresses a DFT of prime size N as a cyclic convolution of (composite) size N − 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).

### Laplace transform

**Laplaces-domainLaplace domain**

Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication.

### Digital signal processing

**DSPsignal processingdigital**

Digital signal processing and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.

The output of a linear digital filter to any given input may be calculated by convolving the input signal with the impulse response.

### Banach space

**Banach spacesBanachBanach-space theory**

Because the space of measures of bounded variation is a Banach space, convolution of measures can be treated with standard methods of functional analysis that may not apply for the convolution of distributions.

, where the product is the convolution of sequences.

### Periodic function

**periodicperiodperiodicity**

For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution.

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge.

### Periodic summation

**periodic sumperiodization**

The summation is called a periodic summation of the function f.

Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.

### Locally integrable function

**locally integrableintegrablelocal integrability**

More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution

Now use convolution to define the function

### Banach algebra

**Banach algebrasspectral mapping theoremalgebra norm**

is a Banach algebra under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).

### Titchmarsh convolution theorem

See also the less trivial Titchmarsh convolution theorem.

The theorem describes the properties of the support of the convolution of two functions.

### Dirac delta function

**Dirac deltadelta functionimpulse**

The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution or, at the very least (as is the case of L 1 ) admit approximations to the identity.

It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta is to time-delay f(t) by the same amount:

### Identity element

**identityneutral elementleft identity**

This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative associative algebra without identity.