Convolve

Convolve[f,g,x,y]

gives the convolution with respect to x of the expressions f and g.

Convolve[f,g,{x1,x2,},{y1,y2,}]

gives the multidimensional convolution.

Details and Options

• The convolution of two functions and is given by .
• The multidimensional convolution is given by .
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate conditions on parameters Method Automatic method to use PrincipalValue False whether to use principal value integrals

Examples

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Basic Examples(3)

Convolve a function with DiracDelta:

Convolve two unit pulses:

Convolve two exponential functions and plot the result:

Scope(5)

Univariate Convolution(3)

The convolution gives the product integral of translates:

Elementary functions:

A convolution typically smooths the function:

For this family, they all have unit area:

Multivariate Convolution(2)

The convolution gives the product integral of translates:

Convolution with multivariate delta functions acts as a point operator:

Convolution with a function of bounded support acts as a filter:

Generalizations & Extensions(1)

Multiplication by UnitStep effectively gives the convolution on a finite interval:

Options(2)

Assumptions(1)

Specify assumptions on a variable or parameter:

GenerateConditions(1)

Generate conditions for the range of a parameter:

Applications(5)

Obtain a particular solution for a linear ordinary differential equation using convolution:

Obtain the step response of a linear, time-invariant system given its impulse response h:

The step response of the system:

Convolving the PDF of UniformDistribution with itself gives a TriangularDistribution:

is the convolution of n PDFs:

ErlangDistribution[k,λ] is the convolution of k PDFs:

Properties & Relations(7)

Convolve computes an integral over the real line:

Convolution with DiracDelta gives the function itself:

Scaling:

Commutativity:

Distributivity:

The Laplace transform of a causal convolution is a product of the individual transforms:

The Fourier transform of a convolution is related to the product of the individual transforms:

Interactive Examples(1)

This demonstrates the convolution operation :

Wolfram Research (2008), Convolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Convolve.html.

Text

Wolfram Research (2008), Convolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Convolve.html.

CMS

Wolfram Language. 2008. "Convolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Convolve.html.

APA

Wolfram Language. (2008). Convolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Convolve.html

BibTeX

@misc{reference.wolfram_2021_convolve, author="Wolfram Research", title="{Convolve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Convolve.html}", note=[Accessed: 24-May-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_convolve, organization={Wolfram Research}, title={Convolve}, year={2008}, url={https://reference.wolfram.com/language/ref/Convolve.html}, note=[Accessed: 24-May-2022 ]}