# 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 :

Introduced in 2008
(7.0)