DiscreteConvolve

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:

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 :

Introduced in 2008

(7.0)

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