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March 9

Examples of convolution

I saw the wiki page, but I couldn't find any examples using actual numbers evaluating the formula. Could you give some examples of convolution, please? Mathijs Krijzer (talk) 22:41, 9 March 2013 (UTC)

Definition

The convolution of f and g is written f∗g, using an asterisk or star. 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:

$(f*g)(t)$	$\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )d\tau$
	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t-\tau )g(\tau )d\tau .$       (commutativity)

Domain of definition

The convolution of two complex-valued functions on R^d

$(f*g)(x)=\underset{{𝐑}^d}{\int }f(y)g(x-y)dy$

is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g.

Circular discrete convolution

When a function g_N is periodic, with period N, then for functions, f, such that f∗g_N exists, the convolution is also periodic and identical to:

$(f*g_N)[n]\equiv \sum _{m=0}^{N-1}(\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f[m+kN])g_N[n-m].$

Circular convolution

When a function g_T is periodic, with period T, then for functions, f, such that f∗g_T exists, the convolution is also periodic and identical to:

$(f*g_T)(t)\equiv \underset{t_0}{\overset{t_0+T}{\int }}[\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau +kT)]g_T(t-\tau )d\tau ,$

where t_o is an arbitrary choice. The summation is called a periodic summation of the function f.

Discrete convolution

For complex-valued functions f, g defined on the set Z of integers, the discrete convolution of f and g is given by:

$(f*g)[n]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}f[m]g[n-m]$

$=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}f[n-m]g[m].$       (commutativity)

When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, extended with zeros where necessary to avoid undefined terms; this is known as the Cauchy product of the coefficients of the two polynomials. Template:Collapse bottom

A convolution maps 2 functions to a third function, it does not map numbers to anything or anything to numbers, so unless you are going to point wise define a function in terms of numbers, I can't show you anything "using actual numbers". — Preceding unsigned comment added by 123.136.64.14 (talk) 05:39, 12 March 2013 (UTC)