Convolve
Details and Options
- Convolve is also known as Fourier convolution, acausal convolution or bilateral convolution.
- The convolution of two functions and is given by .
- The multidimensional convolution is given by .
- The following options can be given:
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Examples
open all close allBasic 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 :
UniformSumDistribution [n] is the convolution of n UniformDistribution [] PDFs:
ErlangDistribution [k,λ] is the convolution of k ExponentialDistribution [λ] 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 :
Related Guides
Related Links
History
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_2025_convolve, author="Wolfram Research", title="{Convolve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Convolve.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_convolve, organization={Wolfram Research}, title={Convolve}, year={2008}, url={https://reference.wolfram.com/language/ref/Convolve.html}, note=[Accessed: 17-November-2025]}