convolution

convolution

Introduction

The

convolution

of two

functions

f

,

g

:

ℝ

→

ℝ

is the function

(

f

∗

g

)

⁢

(

u

)

=

∫

-

∞

∞

f

⁢

(

x

)

⁢

g

⁢

(

u

-

x

)

⁢

𝑑

x

.

(

f

∗

g

)

⁢

(

u

)

is the sum of all the terms

f

⁢

(

x

)

⁢

g

⁢

(

y

)

where

x

+

y

=

u

. Such sums occur when investigating sums of

random variables

, 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.

Definitions

If

G

is a locally compact (topological)

Abelian group

(

http://planetmath.org/LocallyCompactGroupoids

) with Haar measure

μ

and

f

and

g

are measurable functions on

G

, we define the convolution

(

f

∗

g

)

⁢

(

u

)

:=

∫

G

f

⁢

(

x

)

⁢

g

⁢

(

u

-

x

)

⁢

𝑑

μ

⁢

(

x

)

whenever the

right hand side

integral

exists (this is for instance the case if

f

∈

L

p

⁢

(

G

,

μ

)

,

g

∈

L

q

⁢

(

G

,

μ

)

and

1

/

p

+

1

/

q

=

1

).

The case

G

=

ℝ

n

is the most important one, but

G

=

ℤ

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

=

ℤ

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

groupoid

C*–

convolution algebra

(

http://planetmath.org/GroupoidCConvolutionAlgebra

)

The (Dirichlet) convolution of

multiplicative functions

considered in

number theory

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

independent

random variables with probability densities

f

X

and

f

Y

respectively, and if

X

+

Y

has a probability density, then this

density

is given by the convolution

f

X

∗

f

Y

. This motivates the following definition: for probability distributions

P

and

Q

on

ℝ

n

, the convolution

P

∗

Q

is the probability distribution on

ℝ

n

given by

(

P

∗

Q

)

⁢

(

A

)

:=

(

P

×

Q

)

⁢

(

{

(

x

,

y

)

∣

x

+

y

∈

A

}

)

=

∫

ℝ

n

Q

⁢

(

A

-

x

)

⁢

𝑑

P

⁢

(

x

)

for every Borel set

A

. The last equation is the result of Fubini’s theorem.

The convolution of two

distributions

u

and

v

on

ℝ

n

is defined by

(

u

∗

v

)

⁢

(

ϕ

)

=

u

⁢

(

ψ

)

for any test function

ϕ

for

v

, assuming that

ψ

(

t

)

:=

v

(

ϕ

(

⋅

+

t

)

)

is a suitable test function for

u

.

Properties

The convolution operation, when defined, is

commutative

,

associative

and

distributive

with respect to

addition

. For any

f

we have

f

∗

δ

=

f

,

where

δ

is the

Dirac delta distribution

. The

Fourier transform

F

converts the convolution of two functions into their pointwise multiplication:

F

⁢

(

f

∗

g

)

=

F

⁢

(

f

)

⋅

F

⁢

(

g

)

,

which provides a great simplification in the computation of convolution.
Because of the availability of the

Fast Fourier Transform

and its

inverse

, 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 distributions

with zero mean and

variances

σ

1

2

and

σ

2

2

is a normal distribution with zero mean and variance

σ

2

=

σ

1

2

+

σ

2

2

.

•

The convolution of two

χ

2

distributions with

f

1

and

f

2

degrees of freedom

is a

χ

2

distribution with

f

1

+

f

2

degrees of freedom.

•

The convolution of two

Poisson distributions

with

parameters

λ

1

and

λ

2

is a Poisson distribution with parameter

λ

=

λ

1

+

λ

2

.

•

The convolution of an

exponential

and a normal distribution is approximated by another

exponential distribution

. If the original exponential distribution has density

f

(

x

)

=

e

-

x

/

τ

τ

(

x

≥

0

)

or

f

(

x

)

=

0

(

x

<

0

)

,

and the normal distribution has zero mean and variance

σ

2

, then for

u

≫

σ

the probability density of the sum is

f

⁢

(

u

)

≈

e

-

u

/

τ

+

σ

2

/

(

2

⁢

τ

2

)

σ

⁢

τ

⁢

2

⁢

π

In a semi-logarithmic diagram where

log

⁡

(

f

X

⁢

(

x

)

)

is plotted versus

x

and

log

⁡

(

f

⁢

(

u

)

)

versus

u

, the latter lies by the amount

σ

2

/

(

2

⁢

τ

2

)

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

a

≤

x

<

b

and vanishes elsewhere and the normal distribution has zero mean and variance

σ

2

, the probability density of the sum is

f

⁢

(

u

)

=

ψ

0

⁢

(

(

u

-

a

)

/

σ

)

-

ψ

0

⁢

(

(

u

-

b

)

/

σ

)

b

-

a

,

where

ψ

0

⁢

(

x

)

=

1

2

⁢

π

⁢

∫

-

∞

x

e

-

t

2

/

2

⁢

𝑑

t

is the

distribution function

of the

standard normal distribution

. For

σ

→

0

, 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

2

⁢

σ

.

References

•

Adapted

with permission from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

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

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