Product of convolution

In Mathematical, the produces convolution of two real functions or complex F and G generally notes “$\backslash \; ast$” and is written:

$(F\; \backslash \; ast\; G)\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (x-t)\; \backslash \; cdot\; G\; (T)\; \backslash \; cdot\; dt\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (T)\; \backslash \; cdot\; G\; (x-t)\; \backslash \; cdot\; dt$

One can regard this formula as a generalization of the Moving average idea of .

So that this definition has a direction, it is necessary to carry out certain assumptions on F and G , for example if these two functions are summable S their product of convolution is defined for almost all X and is itself summable.

Properties of the product of convolution

• the product of convolution is commutative.

$(F\; \backslash \; ast\; G)\; (X)\; \backslash \; stackrel\; \{\backslash \; mathrm\; \{def.\}\}\; \{=\}\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (x-t)\; \backslash \; cdot\; G\; (T)\; \backslash ,\; dt$

\ int_ {- \ infty} ^ {+ \ infty} F (T) \ cdot G (x-T) \, D (x-T)

\ int_ {- \ infty} ^ {+ \ infty} F (T) \ cdot G (x-T) \, dT \;

\ stackrel {\ mathrm {def.}} {=} (G \ ast F) (X)

Where T=x-t, is t=x-T.

• the product of convolution is distributive

$(F\; \backslash \; ast\; (g+h))\; (X)\; \backslash \; stackrel\; \{\backslash \; mathrm\; \{def.\}\}\; \{=\}\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (x-t)\; \backslash \; cdot\; (G\; (T)\; +h\; (T))\; \backslash ,\; dt$

\ int_ {- \ infty} ^ {+ \ infty} \ cdot G (T) + F (x-t) \ cdot H (T) \, dt

$=\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (x-t)\; \backslash \; cdot\; G\; (T)\; \backslash ,\; dt\; +\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (x-t)\; \backslash \; cdot\; H\; (T)\; \backslash ,\; dt$

\ stackrel {\ mathrm {def.}} {=} (F \ ast G) (X) + (F \ ast H) (X)

• the product of convolution corresponds to the multiplication of the transformed of Fourier of the functions

$F\; \backslash \; ast\; G\; =\; \backslash \; mathcal\; \{F\}\; ^*\; \backslash \; left\; (\backslash \; mathcal\; \{F\}\; (F)\; \backslash \; cdot\; \backslash \; mathcal\; \{F\}\; (G)\; \backslash \; right)$

where $\backslash \; mathcal\; \{F\}$ indicates the transformation of Fourier and $\backslash \; mathcal\; \{F\}\; ^*$ the opposite transformation of Fourier. The principal interest of the calculation of the product of convolution by transforms of Fourier is that these operations are less expensive in time for a computer than the direct calculation of the integral.

Use of the product of convolution

The product of convolution is used in the treatment of the signal, when filters are used (low-pass, high-pass, band pass). If there is an entering signal $S\; (T)$ and a filter element having a transfer transfer function $H\; (T)$ then the output signal $S\_s\; (T)$ will be the convolution of these two functions:

$S\_s\; (T)\; =S\; \backslash \; ast\; H$

Another use of the products of convolution is in the field of quantum mechanics, where one carries out products of convolution starting from the functions of wave will bra and ket.

In a general way, one can write the linear differential equations corresponding to many problems physique in the shape of the product of convolution of an operator by a function describing the system. One can then solve in a generic way the problem by determining the reverse of convolution of the operator (called Fonction of Green). Joseph Fourier was at the origin of this method when he sought to solve the equation of heat. Its modern formulation has to await the arrival of the theory of the distributions introduced by Laurent Schwartz.

The product of convolution spreads with many algebras of a group, for example with the algebras of a group finished. So moreover the group is abelian, then the theory of the harmonic Analyze on an abelian group finished allows to establish all the traditional results of the product of convolution.

Approaches popularized

The manner simplest to represent the product of convolution consists in considering the Fonction δ of Dirac δ _has ( X ); this function is worth 0 if X ≠ has , and its integral is worth 1. This can seem odd at first sight, one can imagine it like the limit of a succession of functions, bell-shaped curves or rectangles having all same surface 1, but increasingly fine (thus increasingly high); when the width of the curves tends towards 0, its height tends towards +∞, but surface remains equal to 1. For practical reasons, one often represents Dirac as a stick positioned in is and height 1.

Dirac: limit of a succession of functions

Because of his form, one calls also sometimes Dirac “function impulse”. The product of convolution by Dirac δ _has corresponds to a translation of the initial function of a value of has

$f\; \backslash \; ast\; \backslash \; delta\_a\; (X)\; =\; F\; (x-a)$

Produces convolution of a function by Dirac

One sees that δ₀ leaves invariant a function, it is the neutral element of the product of convolution

$f\; \backslash \; ast\; \backslash \; delta\_0\; (X)\; =\; F\; (x-0)$

If one now considers the product of convolution by a balanced Somme of two diracs (α.δ _has + β.δ _B ), one obtains the superposition of two dilated curves.

Produces convolution of a function by a balanced sum of two diracs

Let us consider a function now carries P_{a, b} ; it is a function which is worth 1 ( Ba ) between has and B , and 0 elsewhere (its integral is worth 1). This function can be seen like a succession of diracs. The convolution of F by P_{a, b} thus will be obtained while making slip F on the interval. One obtains a “widening” of F .

Produit convolution of a function by a function carries

If one considers an unspecified function now G , one can see G like a succession of diracs balanced by the value of G at the point considered. The product of convolution of F by G is thus obtained while making slip the function F and by dilating it according to the value of G .

Produces convolution of a function by an unspecified function

The product of convolution and filtering

The product of convolution is related to the concept of filtering under two conditions, namely the linearity and independence of the filter with respect to time (Système invariant). From these two conditions, the operator of convolution can be built. The convolution corresponds to the response of the filter to a given entry (noted $e\; (T)$). The filter is entirely characterized by its impulse response $h\; (T)$. Put in equation, the answer of the filter is $s\; (T)\; =\; H\; (T)\; \backslash \; ast\; E\; (T)$.

The construction of the operator of convolution is worked out in the following way. First of all, one is interested in the two conditions imposed on the filter. One notes $f\; (E)$ the filtering carried out by the filter on the entry $e$. The linearity of the filter implies that:

$f\; (\backslash \; lambda\; E)\; =\; \backslash \; lambda\; F\; (E)$

$f\; (e\_1+e\_2)\; =f\; (e\_1)\; +f\; (e\_2)$

One can note that the response of the filter to a null signal is null. The independence of time is summarized by:

$f\; (e^d)\; =\; (F\; (E))^d$

Where $e^d$ is the signal $e$ delayed $d$ quantity.

From there, one can build the response of the filter linear and independent of time to the entry $e\; (T)$. Indeed, as the filter is linear, one can break up the signal $e\; (T)$ into independent parts, using a whole of signals $e\_i$ with disjoined supports compact so that $e\; (T)\; =\; \backslash \; Sigma\_i\; e\_i\; (T)$. One injects each part of the signal in the filter then one summons the various answers. Thus filtering will give: $f\; (E)\; =\; \backslash \; Sigma\_i\; F\; (e\_i)$. This temporal decomposition of $e\; (T)$ can be carried out in a recursive way on the signals $e\_i\; (T)$. At the end, one obtains a succession of signals whose support is summarized at a point. These signals, elementary because nondecomposable temporally, correspond each one of them to the distribution of Dirac $\backslash \; delta\; (T\; \backslash \; tau)$ centered in $\backslash \; tau$ with an amplitude $e\; (\backslash \; tau)$, the impulse is written $\backslash \; delta\; (T\; \backslash \; tau)\; .e\; (\backslash \; tau)$. It is enough to summon all the impulses according to the variable $\backslash \; tau$ to obtain the signal $e\; (T)$:

$e\; (T)\; =\; \backslash \; int\; \backslash \; delta\; (T\; \backslash \; tau)\; .e\; (\backslash \; tau)\; D\; \backslash \; tau$

One applies the operation of filtering to $e\; (T)$. As the filter is linear and independent of time, we have:

$f\; (E)\; =\; \backslash \; int\; F\; (\backslash \; delta\; (T\; \backslash \; tau)\; .e\; (\backslash \; tau))\; D\; \backslash \; tau$

$=\; \backslash \; int\; E\; (\backslash \; tau)\; .f\; (\backslash \; delta\; (T\; \backslash \; tau))\; D\; \backslash \; tau\; \backslash \; \backslash$ (linearity)

$=\; \backslash \; int\; E\; (\backslash \; tau).\; (F\; \backslash \; circ\; \backslash \; delta)\; (T\; \backslash \; tau)\; D\; \backslash \; tau\; \backslash \; \backslash$ (independence of time)

The response of the filter $f$ to the impulse $\backslash \; delta\; (T)$ is named the impulse response of the filter $h\; (T)$. Finally one a:

$f\; (E)\; =\; \backslash \; int\; E\; (\backslash \; tau)\; .h\; (T\; \backslash \; tau)\; D\; \backslash \; tau$

who is only other than the product of convolution.

In conclusion: if the filter is linear and independent of time, then it is entirely characterized by its answer $h\; (T)$ and the answer of the filter to the entry $e\; (T)$ is given by the operator of convolution.

Another fundamental conclusion of the filters linear and independent of time: if one enters a signal $e\; (T)\; =e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; T\}$, the output signal will be:

$s\; (T)\; =\; \backslash \; int\; e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; \backslash \; tau\}\; H\; (T\; \backslash \; tau)\; D\; \backslash \; tau\; =\; \backslash \; int\; e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; (T\; \backslash \; tau)\}\; H\; (T)\; D\; \backslash \; tau$

$s\; (T)\; =\; e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; T\}\; \backslash \; int\; e^\; \{-\; 2\; \backslash \; pi\; \backslash \; jmath\; F\; \backslash \; tau\}\; H\; (\backslash \; tau)\; D\; \backslash \; tau$

$s\; (T)\; =\; e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; T\}\; H\; (F)$

$s\; (T)$ will be also a signal of the form $e^\; \{2\; \backslash \; pi\; \backslash \; jmath\; F\; T\}$ with the factor $H\; (F)$ near. This factor is, neither more, nor less, that the Transformée of Fourier of $h\; (T)$.

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