Convolution

For the usage in formal language theory, see convolution (computer science).

In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval.

Uses

Convolution and related operations are found in many applications of engineering and mathematics.

In statistics, as noted above, a weighted moving average is a convolution.
- also the probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic term for this is bokeh.
Similarly, in digital image processing, convolutional filtering plays an important role in many important algorithms in edge detection and related processes.
In acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information).
In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac delta function. See LTI system theory and digital signal processing.
In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.
In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance.

Definition

The convolution of f and g is written

f*g

. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

(f  * g )(t) = \int f(\tau) g(t - \tau)\, d\tau

The integration range depends on the domain on which the functions are defined. While the symbol $t$ is used above, it need not represent the time domain. In the case of a finite integration range, f and g are often considered to extend periodically in both directions, so that the term g(t − τ) does not imply a range violation. This use of periodic domains is sometimes called a cyclic, circular or periodic convolution. Of course, extension with zeros is also possible. Using zero-extended or infinite domains is sometimes called a linear convolution, especially in the discrete case below.

If $X$ and $Y$ are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the sum $X + Y$ is given by the convolution f $*$ g.

For discrete functions, one can use a discrete version of the convolution. It is then given by

(f  * g)(m) = \sum_n {f(n) g(m - n)} \,

When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, in this sense (using extension with zeros as mentioned above).

Generalizing the above cases, the convolution can be defined for any two integrable functions defined on a locally compact topological group (see convolutions on groups below).

A different generalization is the convolution of distributions.

Properties

The various convolution operators all satisfy the following properties:

Commutativity

f * g = g * f \,

Associativity

f  * (g  * h) = (f  * g)  * h \,

Distributivity

f  * (g + h) = (f  * g) + (f  * h) \,

Associativity with scalar multiplication

a (f  * g) = (a f)  * g = f  * (a g) \,

for any real (or complex) number

a

Differentiation rule

\mathcal{D}(f  * g) = \mathcal{D}f  * g = f  * \mathcal{D}g \,

where

\mathcal{D}f

denotes the derivative of

f

or, in the discrete case, the difference operator

\mathcal{D}f(n) = f(n+1) - f(n)

Convolution theorem

The convolution theorem states that

\mathcal{F}(f  * g) =  \mathcal{F} (f) \cdot \mathcal{F} (g)

where F(f) denotes the Fourier transform of f. Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform and Mellin transform.

Convolutions on groups

If G is a suitable group endowed with a measure m (for instance, a locally compact Hausdorff topological group with the Haar measure) and if f and g are real or complex valued m-integrable functions of G, then we can define their convolution by

(f * g)(x) = \int_G f(y)g(xy^{-1})\,dm(y) \,

In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the Peter-Weyl theorem of harmonic analysis. It is very difficult to do these calculations without more structure, and Lie groups turn out to be the setting in which these things are done.

Convolution of measures

If μ and ν are measures on the family of Borel subsets of the real line, then the convolution μ * ν is defined by

(\mu * \nu)(A) = (\mu \times \nu)(\{\, (x,y) \in \mathbb{R}^2 \,:\, x+y \in A \,\}).

In case μ and ν are absolutely continuous with respect to Lebesgue measure, so that each has a density function, then the convolution μ * ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions.

If μ and ν are probability measures, then the convolution μ * ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.

External links

Convolution, on The Data Analysis BriefBook
http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet.
http://www.jhu.edu/~signals/discreteconv2/index.html Visual convolution Java Applet for Discrete Time functions.

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