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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

  • 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:
  • Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditions Falsewhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PrincipalValueFalsewhether to use principal value integrals

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Convolve a function with DiracDelta:

In[1]:=1
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https://wolfram.com/xid/0bn5y66-gacf91
Out[1]=1

Convolve two unit pulses:

In[1]:=1
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https://wolfram.com/xid/0bn5y66-iu24ic
Out[1]=1

Convolve two exponential functions and plot the result:

In[1]:=1
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https://wolfram.com/xid/0bn5y66-citra4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-gfy6jn
Out[2]=2

Scope  (5)Survey of the scope of standard use cases

Univariate Convolution  (3)

The convolution gives the product integral of translates:

In[2]:=2
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https://wolfram.com/xid/0bn5y66-61a2c
In[3]:=3
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https://wolfram.com/xid/0bn5y66-sjghd
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-er2wgu
Out[4]=4
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-nv0tr
Out[4]=4

Elementary functions:

In[1]:=1
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https://wolfram.com/xid/0bn5y66-c48817
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bn5y66-cgeizj
Out[2]=2

A convolution typically smooths the function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-jcwlkp
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-p0lay6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-hm3cgk
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-d6e7pc
In[5]:=5
✖
https://wolfram.com/xid/0bn5y66-cb7izn
Out[5]=5

For this family, they all have unit area:

In[6]:=6
✖
https://wolfram.com/xid/0bn5y66-h31fav
Out[6]=6

Multivariate Convolution  (2)

The convolution gives the product integral of translates:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-e9pbr9
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-z1o8j
Out[2]=2

Convolution with multivariate delta functions acts as a point operator:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-n45qvi
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-bhlg28
Out[2]=2

Generalizations & Extensions  (1)Generalized and extended use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-buq3u4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-oxpd1a
Out[2]=2

Options  (2)Common values & functionality for each option

Assumptions  (1)

Specify assumptions on a variable or parameter:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-2wdqf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-c379s
Out[2]=2

GenerateConditions  (1)

Generate conditions for the range of a parameter:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-d48v4p
Out[1]=1

Applications  (5)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-dc8flm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-hbjfuw
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0bn5y66-bc0813
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-kfglew
Out[2]=2

The step response of the system:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-c7264b
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-vpti3
Out[4]=4

Convolving the PDF of UniformDistribution with itself gives a TriangularDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-dgbg9z
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bn5y66-bhoumj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-k5fn2r
Out[3]=3

UniformSumDistribution[n] is the convolution of n UniformDistribution[] PDFs:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-luthag
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-bgx5vy
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-bglxgq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-h9ts81
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-eukoqf
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-bbmhxe
Out[4]=4

Properties & Relations  (7)Properties of the function, and connections to other functions

Convolve computes an integral over the real line:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-mkdggp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-cebt8k
Out[2]=2

Convolution with DiracDelta gives the function itself:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-cy1n60
Out[1]=1

Scaling:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-jy2h7a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-7o0s7
Out[2]=2

Commutativity:

In[1]:=1
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https://wolfram.com/xid/0bn5y66-ed06ni
In[2]:=2
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https://wolfram.com/xid/0bn5y66-w7sp6
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-d9rc13
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-f2vetm
Out[4]=4

Distributivity:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-ccm2m9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-ig5mq
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-dni08t
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-cj7juo
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-bhkbnm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-hyhyug
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y66-btcves
In[2]:=2
✖
https://wolfram.com/xid/0bn5y66-f4yaps
In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-cqppke
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y66-ndr305
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bn5y66-bcqab5
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bn5y66-elt8x1
Out[6]=6

Interactive Examples  (1)Examples with interactive outputs

This demonstrates the convolution operation :

In[3]:=3
✖
https://wolfram.com/xid/0bn5y66-cv2dh
Out[3]=3
Wolfram Research (2008), Convolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Convolve.html.
✖
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.

✖
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.

✖
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

✖
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: 01-April-2025 ]}

✖
@misc{reference.wolfram_2025_convolve, author="Wolfram Research", title="{Convolve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Convolve.html}", note=[Accessed: 01-April-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: 01-April-2025 ]}

✖
@online{reference.wolfram_2025_convolve, organization={Wolfram Research}, title={Convolve}, year={2008}, url={https://reference.wolfram.com/language/ref/Convolve.html}, note=[Accessed: 01-April-2025 ]}