UnilateralDiscreteConvolve
✖
UnilateralDiscreteConvolve
gives the unilateral discrete convolution with respect to k of the expressions f and g.
gives the multidimensional unilateral discrete convolution.
Details and Options

- UnilateralDiscreteConvolve is also known as causal convolution.
- Unilateral convolution arises naturally when examining causal systems. The output of such systems at any time depends only on values of the input at the present time and in the past.
- The unilateral convolution
of two sequences
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
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Convolve a sequence with DiscreteDelta:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-h9i5g8

Convolve a pair of step sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-e3880x


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-8bmym

Convolve a linear sequence and a harmonic sequence and plot the result:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-qdv1m


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ehtu60

Unilateral convolution of multivariate sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-h5hdsd


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-w6bfu

Scope (6)Survey of the scope of standard use cases
Convolve sequences of binomial coefficients:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-mcysx

Convolution of polynomial sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cf33v2

Convolution of exponential sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-l30j4p


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nwdj0a

Convolution of trigonometric sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-hmloir


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-j5wwwc

Multivariate convolution with DiscreteDelta:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-jpsv6

Multivariate convolution of rational sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-q1awc

Applications (2)Sample problems that can be solved with this function
Obtain a particular solution for a linear ordinary difference equation using convolution:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dc8flm


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-k02cef


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-hbjfuw


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dsy21p

Find the product of two power series:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-b209po

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bfgmks

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bjklht

Verify the result using Sum:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nb7jbq

Properties & Relations (8)Properties of the function, and connections to other functions
UnilateralDiscreteConvolve computes a sum over a finite interval:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-mkdggp


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cebt8k

Convolution with DiscreteDelta gives the function itself:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cy1n60


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-jy2h7a


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ed06ni

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-w7sp6

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-d9rc13


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-f2vetm


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ccm2m9


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ig5mq

DiscreteConvolve coincides with UnilateralDiscreteConvolve for causal sequences:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-5ucg03

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-4lzpmy

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-2tmyvw


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bp7chf

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

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-1q0kv4

Verify the convolution theorem for Z transforms on the following example:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dni08t

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cj7juo

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bhkbnm


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nohtgp

The generating function of a causal convolution is a product of the individual generating functions:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bpbc

Verify the convolution theorem for generating functions on the following example:

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-juzjyv

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ij176

https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ys2pk


https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-coh8s3

Wolfram Research (2024), UnilateralDiscreteConvolve, Wolfram Language function, https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
Text
Wolfram Research (2024), UnilateralDiscreteConvolve, Wolfram Language function, https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
Wolfram Research (2024), UnilateralDiscreteConvolve, Wolfram Language function, https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
CMS
Wolfram Language. 2024. "UnilateralDiscreteConvolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
Wolfram Language. 2024. "UnilateralDiscreteConvolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
APA
Wolfram Language. (2024). UnilateralDiscreteConvolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html
Wolfram Language. (2024). UnilateralDiscreteConvolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html
BibTeX
@misc{reference.wolfram_2025_unilateraldiscreteconvolve, author="Wolfram Research", title="{UnilateralDiscreteConvolve}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html}", note=[Accessed: 19-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_unilateraldiscreteconvolve, organization={Wolfram Research}, title={UnilateralDiscreteConvolve}, year={2024}, url={https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html}, note=[Accessed: 19-June-2025
]}