The convolution of two functionsMathworldPlanetmath f,g: is the function


(fg)(u) is the sum of all the terms f(x)g(y) where x+y=u. Such sums occur when investigating sums of random variablesMathworldPlanetmath, and discrete versions appear in the coefficients of products of polynomials and power series. Convolution is an important tool in data processing, in particular in digital signal and image processing. We will first define the concept in various general settings, discuss its properties and then list several convolutions of probability distributions.


If G is a locally compact (topological) Abelian groupMathworldPlanetmath ( with Haar measure μ and f and g are measurable functions on G, we define the convolution


whenever the right hand side integralDlmfPlanetmath exists (this is for instance the case if fLp(G,μ), gLq(G,μ) and 1/p+1/q=1).

The case G=n is the most important one, but G= is also useful, since it recovers the convolution of sequences which occurs when computing the coefficients of a product of polynomials or power series. The case G=n yields the so-called cyclic convolution which is often discussed in connection with the discrete Fourier transform. Based on this definition one also obtains the groupoidPlanetmathPlanetmathPlanetmathPlanetmath C*–convolution algebra (

The (Dirichlet) convolution of multiplicative functionsMathworldPlanetmath considered in number theoryMathworldPlanetmath does not quite fit the above definition, since there the functions are defined on a commutative monoid (the natural numbers under multiplication) rather than on an abelian group.

If X and Y are independentPlanetmathPlanetmath random variables with probability densities fX and fY respectively, and if X+Y has a probability density, then this density is given by the convolution fXfY. This motivates the following definition: for probability distributions P and Q on n, the convolution PQ is the probability distribution on n given by


for every Borel set A. The last equation is the result of Fubini’s theorem.

The convolution of two distributionsDlmfPlanetmathPlanetmath u and v on n is defined by


for any test function ϕ for v, assuming that ψ(t):=v(ϕ(+t)) is a suitable test function for u.


The convolution operation, when defined, is commutative, associative and distributive with respect to addition. For any f we have


where δ is the Dirac delta distribution. The Fourier transformDlmfMathworldPlanetmath F converts the convolution of two functions into their pointwise multiplication:


which provides a great simplification in the computation of convolution. Because of the availability of the Fast Fourier Transform and its inverseMathworldPlanetmathPlanetmathPlanetmath, this latter relation is often used to quickly compute discrete convolutions, and in fact the fastest known algorithms for the multiplication of numbers and polynomials are based on this idea.

Some convolutions of probability distributions

  • The convolution of two independent normal distributionsMathworldPlanetmath with zero mean and variancesMathworldPlanetmath σ12 and σ22 is a normal distribution with zero mean and variance σ2=σ12+σ22.

  • The convolution of two χ2 distributions with f1 and f2 degrees of freedom is a χ2 distribution with f1+f2 degrees of freedom.

  • The convolution of two Poisson distributionsMathworldPlanetmath with parameters λ1 and λ2 is a Poisson distribution with parameter λ=λ1+λ2.

  • The convolution of an exponential and a normal distribution is approximated by another exponential distributionMathworldPlanetmath. If the original exponential distribution has density

    f(x)=e-x/ττ(x0) or f(x)=0(x<0),

    and the normal distribution has zero mean and variance σ2, then for uσ the probability density of the sum is


    In a semi-logarithmic diagram where log(fX(x)) is plotted versus x and log(f(u)) versus u, the latter lies by the amount σ2/(2τ2) higher than the former but both are represented by parallel straight lines, the slope of which is determined by the parameter τ.

  • The convolution of a uniform and a normal distribution results in a quasi-uniform distribution smeared out at its edges. If the original distribution is uniform in the region ax<b and vanishes elsewhere and the normal distribution has zero mean and variance σ2, the probability density of the sum is




    is the distribution function of the standard normal distribution. For σ0, the function f(u) vanishes for u<a and u>b and is equal to 1/(b-a) in between. For finite σ the sharp steps at a and b are rounded off over a width of the order 2σ.


  • Adapted with permission from The Data Analysis Briefbook (

Title convolution
Canonical name Convolution
Date of creation 2013-03-22 12:32:45
Last modified on 2013-03-22 12:32:45
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 31
Author PrimeFan (13766)
Entry type Definition
Classification msc 44A35
Classification msc 94A12
Synonym convolve
Synonym fold
Synonym convolved
Synonym folded
Related topic LogarithmicConvolution
Related topic LaplaceTransformOfConvolution
Related topic GroupoidCConvolutionAlgebra