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UnilateralDiscreteConvolve

New in 14.1

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gives the unilateral discrete convolution with respect to k of the expressions f and g.

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UnilateralDiscreteConvolve[f,g,{k1,…,kp},{n1,…,np}]

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$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate conditions on parameters
    MethodAutomaticmethod to use

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Convolve a sequence with DiscreteDelta:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-h9i5g8
Out[1]=1

Convolve a pair of step sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-e3880x
Out[1]=1

Plot the result:

In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-8bmym
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-qdv1m
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ehtu60
Out[2]=2

Unilateral convolution of multivariate sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-h5hdsd
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-w6bfu
Out[2]=2

Scope  (6)Survey of the scope of standard use cases

Convolve sequences of binomial coefficients:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-mcysx
Out[1]=1

Convolution of polynomial sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cf33v2
Out[1]=1

Convolution of exponential sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-l30j4p
Out[1]=1

Plot the result:

In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nwdj0a
Out[2]=2

Convolution of trigonometric sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-hmloir
Out[1]=1

Plot the result:

In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-j5wwwc
Out[2]=2

Multivariate convolution with DiscreteDelta:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-jpsv6
Out[1]=1

Multivariate convolution of rational sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-q1awc
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

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

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dc8flm
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-k02cef
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-hbjfuw
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dsy21p
Out[4]=4

Find the product of two power series:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-b209po
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bfgmks
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bjklht
Out[3]=3

Verify the result using Sum:

In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nb7jbq
Out[4]=4

Properties & Relations  (8)Properties of the function, and connections to other functions

UnilateralDiscreteConvolve computes a sum over a finite interval:

In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-mkdggp
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cebt8k
Out[4]=4

Convolution with DiscreteDelta gives the function itself:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cy1n60
Out[1]=1

Scaling:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-jy2h7a
Out[1]=1

Commutativity:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ed06ni
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-w7sp6
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-d9rc13
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-f2vetm
Out[4]=4

Distributivity:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ccm2m9
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ig5mq
Out[2]=2

DiscreteConvolve coincides with UnilateralDiscreteConvolve for causal sequences:

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-5ucg03
In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-4lzpmy
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-2tmyvw
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bp7chf
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-1q0kv4
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-dni08t
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-cj7juo
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bhkbnm
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-nohtgp
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-bpbc
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-juzjyv
In[3]:=3
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ij176
In[4]:=4
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-ys2pk
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bmvaeahaqz4ajvzfqq-coh8s3
Out[5]=5
Wolfram Research (2024), UnilateralDiscreteConvolve, Wolfram Language function, https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html.
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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: 04-June-2025 ]}

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@misc{reference.wolfram_2025_unilateraldiscreteconvolve, author="Wolfram Research", title="{UnilateralDiscreteConvolve}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html}", note=[Accessed: 04-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: 04-June-2025 ]}

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@online{reference.wolfram_2025_unilateraldiscreteconvolve, organization={Wolfram Research}, title={UnilateralDiscreteConvolve}, year={2024}, url={https://reference.wolfram.com/language/ref/UnilateralDiscreteConvolve.html}, note=[Accessed: 04-June-2025 ]}