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Convolve

Convolve[f, g, x, y]
gives the convolution with respect to x of the expressions f and g.
Convolve[f, g, {x1, x2, ...}, {y1, y2, ...}]
gives the multidimensional convolution.
  • The convolution (f Starg) (y) of two functions f(x) and g(x) 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
PrincipalValueFalsewhether to use principal value integrals
EXAMPLESCLOSE ALL
Basic Examples   (2)
Convolve two functions:
A typical impulse response h for a system:
The step response corresponding to the same system:
Convolve two functions:
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A typical impulse response h for a system:
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The step response corresponding to the same system:
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Scope   (5)
The convolution gives the product integral of translates:
Elementary functions:
A convolution typically smooths the function:
For this family they all have unit area:
The convolution gives the product integral of translates:
Convolution with multivariate delta functions acts as a point operator:
Convolution with a function of bounded support acts as a filter:
Generalizations & Extensions   (1)
Multiplication by UnitStep effectively gives the convolution on a finite interval:
Options   (2)
Specify assumptions on a variable or parameter:
Generate conditions for the range of a parameter:
Applications   (1)
Obtain a particular solution for a linear ordinary differential equation using convolution:
Properties & Relations   (8)
Convolve computes an integral over the real line:
Convolution with DiracDelta gives the function itself:
Scaling:
Commutativity:
Distributivity:
The Laplace transform of a causal convolution is a product of the individual transforms:
The Fourier transform of a convolution is related to the product of the individual transforms:
The convolution of UnitBox with itself is UnitTriangle:
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