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\r\n\r\nText for this chapter\r\n\r\nIn\u00a0\r\n\r\nText for this chapter\r\n\r\nIn\u00a0mathematics<\/a>\u00a0(in particular,\u00a0functional analysis<\/a>),\u00a0convolution<\/b>\u00a0is a\u00a0mathematical operation<\/a>\u00a0on two\u00a0functions<\/a>\u00a0(f<\/span>\u00a0and\u00a0g<\/span>) that produces a third function ({\\displaystyle f*g}<\/span><\/a> Text for this chapter<\/p>\n In\u00a0mathematics<\/a>\u00a0(in particular,\u00a0functional analysis<\/a>),\u00a0convolution<\/b>\u00a0is a\u00a0mathematical operation<\/a>\u00a0on two\u00a0functions<\/a>\u00a0(f<\/span>\u00a0and\u00a0g<\/span>) that produces a third function ({\\displaystyle f*g}<\/span> Some features of convolution are similar to\u00a0cross-correlation<\/a>: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation ({\\displaystyle f\\star g}<\/span> Convolution has applications that include\u00a0probability<\/a>,\u00a0statistics<\/a>,\u00a0acoustics<\/a>,\u00a0spectroscopy<\/a>,\u00a0signal processing<\/a>\u00a0and\u00a0image processing<\/a>,\u00a0geophysics<\/a>,\u00a0engineering<\/a>,\u00a0physics<\/a>,\u00a0computer vision<\/a>\u00a0and\u00a0differential equations<\/a>.[1]<\/a><\/sup><\/p>\n The convolution can be defined for functions on\u00a0Euclidean space<\/a>\u00a0and other\u00a0groups<\/a>.[citation needed<\/span><\/a><\/i>]<\/sup>\u00a0For example,\u00a0periodic functions<\/a>, such as the\u00a0discrete-time Fourier transform<\/a>, can be defined on a\u00a0circle<\/a>\u00a0and convolved by\u00a0periodic convolution<\/a>. (See row 18 at\u00a0DTFT \u00a7\u00a0Properties<\/a>.) A\u00a0discrete convolution<\/i>\u00a0can be defined for functions on the set of\u00a0integers<\/a>.<\/p>\n Generalizations of convolution have applications in the field of\u00a0numerical analysis<\/a>\u00a0and\u00a0numerical linear algebra<\/a>, and in the design and implementation of\u00a0finite impulse response<\/a>\u00a0filters in signal processing.[citation needed<\/span><\/a><\/i>]<\/sup><\/p>\n Computing the\u00a0inverse<\/a>\u00a0of the convolution operation is known as\u00a0deconvolution<\/a>.<\/p>\n Source: https:\/\/en.wikipedia.org\/wiki\/Convolution<\/a><\/p>\n
\n<\/p>\n<\/span>) that expresses how the shape of one is modified by the other. The term\u00a0convolution<\/i>\u00a0refers to both the result function and to the process of computing it. It is defined as the\u00a0integral<\/a>\u00a0of the product of the two functions after one is reversed and shifted. The integral is evaluated for all values of shift, producing the convolution function.<\/p>\n
<\/span>) only in that either\u00a0f<\/i>(x<\/i>)<\/span>\u00a0or\u00a0g<\/i>(x<\/i>)<\/span>\u00a0is reflected about the y-axis; thus it is a cross-correlation of\u00a0f<\/i>(x<\/i>)<\/span>\u00a0and\u00a0g<\/i>(\u2212x<\/i>)<\/span>, or\u00a0f<\/i>(\u2212x<\/i>)<\/span>\u00a0and\u00a0g<\/i>(x<\/i>)<\/span>.[A]<\/a><\/sup>\u00a0 For complex-valued functions, the cross-correlation operator is the\u00a0adjoint<\/a>\u00a0of the convolution operator.<\/p>\n