# 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

• The 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 VerifyConvergence True whether to verify convergence

# 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:

## 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_2024_discreteconvolve, author="Wolfram Research", title="{DiscreteConvolve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteConvolve.html}", note=[Accessed: 16-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_discreteconvolve, organization={Wolfram Research}, title={DiscreteConvolve}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteConvolve.html}, note=[Accessed: 16-June-2024 ]}