Convolution kernel

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions $f$ and $g$ that produces a third function $f*g$, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other.

Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution $f*g$ differs from cross-correlation $f \star g$ only in that either $f(x)$ or $g(x)$ is reflected about the y-axis in convolution; thus it is a cross-correlation of $g(-x)$ and $f(x)$, or $f(-x)$ and $g(x)$. For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.

Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations.

The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at .) A ''discrete convolution'' can be defined for functions on the set of integers.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Definition

The convolution of $f$ and $g$ is written $f * g$, denoting the operator with the symbol $*$. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform:

:$(f * g)(t) := \int_^\infty f(\tau) g(t - \tau) \, d\tau.$

An equivalent definition is (see commutativity):

:$(f * g)(t) := \int_^\infty f(t - \tau) g(\tau)\, d\tau.$

While the symbol $t$ is used above, it need not represent the time domain. At each $t$, the convolution formula can be described as the area under the function $f(\tau)$ weighted by the function $g(-\tau)$ shifted by the amount $t$. As $t$ changes, the weighting function $g(t-\tau)$ emphasizes different parts of the input function $f(\tau)$; If $t$ is a positive value, then $g(t-\tau)$ is equal to $g(-\tau)$ that slides or is shifted along the $\tau$-axis toward the right (toward $+\infty$) by the amount of $t$, while if $t$ is a negative value, then $g(t-\tau)$ is equal to $g(-\tau)$ that slides or is shifted toward the left (toward $-\infty$) by the amount of $, t,$.

For functions $f$, $g$ supported on only $$domain of definition'' (below).

Notation

A common engineering notational convention is:

:$f(t) * g(t) \mathrel \underbrace_,$

which has to be interpreted carefully to avoid confusion. For instance, $f(t) * g(t-t_0)$ is equivalent to $(f*g)(t-t_0)$, but $f(t-t_0) * g(t-t_0)$ is in fact equivalent to $(f * g)(t-2t_0)$.

Relations with other transforms

Given two functions $f(t)$ and $g(t)$ with Two-sided Laplace transform">bilateral Laplace transform