DiscreteConvolve

DiscreteConvolve[f,g,n,m]

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

DiscreteConvolve[f,g,{n1,n2,…},{m1,m2,…}]

gives the multidimensional convolution.

Details and Options

Examples

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Basic Examples  (3)

Convolve a sequence with DiscreteDelta:

Convolve two exponential sequences:

Convolve two UnitBox sequences and plot the result:

Scope  (4)

Univariate Convolution  (3)

Convolution sums the product of translates:

Convolution of elementary sequences:

Convolution of piecewise sequences:

Multivariate Convolution  (1)

Generalizations & Extensions  (1)

Multiplication by UnitStep effectively gives the convolution over 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  (2)

Obtain a particular solution for a linear difference equation:

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

The step response corresponding to this system:

Properties & Relations  (7)

DiscreteConvolve computes a sum over the set of integers:

Convolution with DiscreteDelta gives the value of a sequence at m:

Scaling:

Commutativity:

Distributivity:

The ZTransform of a causal convolution is the product of the individual transforms:

Similarly for GeneratingFunction:

The FourierSequenceTransform of a convolution is the product of the individual transforms:

Interactive Examples  (1)

This demonstrates the discrete-time convolution operation :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_discreteconvolve, author="Wolfram Research", title="{DiscreteConvolve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteConvolve.html}", note=[Accessed: 22-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_discreteconvolve, organization={Wolfram Research}, title={DiscreteConvolve}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteConvolve.html}, note=[Accessed: 22-July-2025 ]}