TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
(in particular,
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
), convolution is a
mathematical operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
on two
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
( and ) that produces a third function ($f*g$) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... of 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. Some features of convolution are similar to
cross-correlation In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ... : for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation ($f \star g$) only in that either or is reflected about the y-axis; thus it is a cross-correlation of and , or and .  For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ... ,
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ... ,
acoustics Acoustics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wo ...
,
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ... ,
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ... and
image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
,
geophysics Geophysics () is a subject of concerned with the physical processes and of the and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' sometimes refers only to solid earth applic ... ,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ... ,
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ... ,
computer vision Computer vision is an interdisciplinary scientific field that deals with how computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...
and
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
. The convolution can be defined for functions on
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
and other
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
. For example,
periodic function A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ... s, such as the
discrete-time Fourier transform In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, can be defined on a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ... and convolved by periodic convolution. (See row 18 at .) A ''discrete convolution'' can be defined for functions on the set of
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ... . Generalizations of convolution have applications in the field of
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
and
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create algorithms, computer algorithms which Algorithmic efficiency, efficiently and accurately provide approximate answers to qu ...
, and in the design and implementation of
finite impulse response In , a finite impulse response (FIR) filter is a whose (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to (IIR) filters, which may have internal feedback and may ...
filters in signal processing. Computing the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
of the convolution operation is known as
deconvolution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ... .

# Definition

The convolution of and is written , denoting the operator with the symbol . 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: :$\left(f * g\right)\left(t\right) := \int_^\infty f\left(\tau\right) g\left(t - \tau\right) \, d\tau.$ An equivalent definition is (see
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
): :$\left(f * g\right)\left(t\right) := \int_^\infty f\left(t - \tau\right) g\left(\tau\right)\, d\tau.$ While the symbol is used above, it need not represent the time domain. But in that context, the convolution formula can be described as the area under the function weighted by the function shifted by amount . As changes, the weighting function emphasizes different parts of the input function . For functions , supported on only (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: :

## __Notation_

A_common_engineering_notational_convention_is: :$_f\left(t\right)_*_g\left(t\right)_\mathrel_\underbrace_,$ which_has_to_be_interpreted_carefully_to_avoid_confusion._For_instance,__is_equivalent_to_,_but__is_in_fact_equivalent_to_.

## __Derivation_

Given_two_functions_$_f\left(t\right)_$_and_$_g\left(t\right)_$_with_bilateral_Laplace_transforms :$_F\left(s\right)_=_\int_^\infty_e^_\_f\left(u\right)_\_\textu_$ and :$_G\left(s\right)_=_\int_^\infty_e^_\_g\left(v\right)_\_\textv_$ respectively,_the_convolution_operation_$_f\left(t\right)_*_g\left(t\right)_$_can_be_defined_as_the_inverse_Laplace_transform_of_the_product_of_$_F\left(s\right)_$_and_$_G\left(s\right)_$._More_precisely, :$\begin F\left(s\right)_\cdot_G\left(s\right)_&=_\int_^\infty_e^_\_f\left(u\right)_\_\textu_\cdot_\int_^\infty_e^_\_g\left(v\right)_\_\textv_\\ &=_\int_^\infty_\int_^\infty_e^_\_f\left(u\right)_\_g\left(v\right)_\_\textu_\_\textv \end$ Let_$_v_=_t_-_u_$_such_that :$\begin F\left(s\right)_\cdot_G\left(s\right)_&=_\int_^\infty_\int_^\infty_e^_\_f\left(u\right)_\_g\left(t_-_u\right)_\_\textu_\_\textt_\\ &=_\int_^\infty_e^_\underbrace__\_\textt_\\ &=_\int_^\infty_e^_\left(f\left(t\right)_*_g\left(t\right)\right)_\_\textt \end$ Note_that_$_F\left(s\right)_\cdot_G\left(s\right)_$_is_the_bilateral_Laplace_transform_of_$_f\left(t\right)_*_g\left(t\right)_$._A_similar_derivation_can_be_done_using_the_unilateral_Laplace_transform. The_convolution_operation_also_describes_the_output_(in_terms_of_the_input)_of_an_important_class_of_operations_known_as_''linear_time-invariant''_(LTI)._See_LTI_system_theory#Overview.html" "title="#Domain of definition">domain of definition'' (below).

## Notation

A common engineering notational convention is: :$f\left(t\right) * g\left(t\right) \mathrel \underbrace_,$ which has to be interpreted carefully to avoid confusion. For instance, is equivalent to , but is in fact equivalent to .

## Derivation

Given two functions $f\left(t\right)$ and $g\left(t\right)$ with bilateral Laplace transforms :$F\left(s\right) = \int_^\infty e^ \ f\left(u\right) \ \textu$ and :$G\left(s\right) = \int_^\infty e^ \ g\left(v\right) \ \textv$ respectively, the convolution operation $f\left(t\right) * g\left(t\right)$ can be defined as the inverse Laplace transform of the product of $F\left(s\right)$ and $G\left(s\right)$. More precisely, :$\begin F\left(s\right) \cdot G\left(s\right) &= \int_^\infty e^ \ f\left(u\right) \ \textu \cdot \int_^\infty e^ \ g\left(v\right) \ \textv \\ &= \int_^\infty \int_^\infty e^ \ f\left(u\right) \ g\left(v\right) \ \textu \ \textv \end$ Let $v = t - u$ such that :$\begin F\left(s\right) \cdot G\left(s\right) &= \int_^\infty \int_^\infty e^ \ f\left(u\right) \ g\left(t - u\right) \ \textu \ \textt \\ &= \int_^\infty e^ \underbrace_ \ \textt \\ &= \int_^\infty e^ \left(f\left(t\right) * g\left(t\right)\right) \ \textt \end$ Note that $F\left(s\right) \cdot G\left(s\right)$ is the bilateral Laplace transform of $f\left(t\right) * g\left(t\right)$. A similar derivation can be done using the unilateral Laplace transform. The convolution operation also describes the output (in terms of the input) of an important class of operations known as ''linear time-invariant'' (LTI). See LTI system theory#Overview">LTI system theory In system analysis, among other fields of study, a linear time-invariant system (LTI system) is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant sy ...
for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a
transfer function In engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ... ). See
Convolution theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.

# Historical developments

One of the earliest uses of the convolution integral appeared in
D'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or com ...
's derivation of
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
in ''Recherches sur différents points importants du système du monde,'' published in 1754. Also, an expression of the type: :$\int f\left(u\right)\cdot g\left(x - u\right) \, du$ is used by Sylvestre François Lacroix on page 505 of his book entitled ''Treatise on differences and series'', which is the last of 3 volumes of the encyclopedic series: ''Traité du calcul différentiel et du calcul intégral'', Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of
Pierre Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...
,
Jean-Baptiste Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such t ...
,
Siméon Denis Poisson Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...
, and others. The term itself did not come into wide use until the 1950s or 60s. Prior to that it was sometimes known as ''Faltung'' (which means ''folding'' in
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ... ), ''composition product'', ''superposition integral'', and ''Carson's integral''. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: :$\int_0^t \varphi\left(s\right)\psi\left(t - s\right) \, ds,\quad 0 \le t < \infty,$ is a particular case of composition products considered by the Italian mathematician
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ... in 1913.

# Circular convolution

When a function is periodic, with period , then for functions, , such that exists, the convolution is also periodic and identical to: : where is an arbitrary choice. The summation is called a
periodic summation In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal pr ...
of the function . When is a periodic summation of another function, , then is known as a ''circular'' or ''cyclic'' convolution of and . And if the periodic summation above is replaced by , the operation is called a ''periodic'' convolution of and .

# Discrete convolution

For complex-valued functions defined on the set Z of integers, the ''discrete convolution'' of and is given by: : or equivalently (see
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) by: : The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the
Cauchy productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the coefficients of the sequences. Thus when has finite support in the set $\$ (representing, for instance, a
finite impulse response In , a finite impulse response (FIR) filter is a whose (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to (IIR) filters, which may have internal feedback and may ...
), a finite summation may be used: :

## Circular discrete convolution

When a function is periodic, with period , then for functions, , such that exists, the convolution is also periodic and identical to: : The summation on is called a
periodic summation In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal pr ...
of the function . If is a periodic summation of another function, , then is known as a circular convolution of and . When the non-zero durations of both and are limited to the interval ,  reduces to these common forms: The notation () for ''cyclic convolution'' denotes convolution over the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... of integers modulo . Circular convolution arises most often in the context of fast convolution with a
fast Fourier transform A fast Fourier transform (FFT) is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are u ...
(FFT) algorithm.

## Fast convolution algorithms

In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ... of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (; ). requires arithmetic operations per output value and operations for outputs. That can be significantly reduced with any of several fast algorithms.
Digital signal processing Digital signal processing (DSP) is the use of digital processing Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O( log ) complexity. The most common fast convolution algorithms use
fast Fourier transform A fast Fourier transform (FFT) is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are u ...
(FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output. Other fast convolution algorithms, such as the
Schönhage–Strassen algorithm The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971.A. Schönhage and V. Strassen,Schnelle Multiplikation großer Zahlen, ''C ...
or the Mersenne transform, use fast Fourier transforms in other rings. If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available. Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the overlap–save method and
overlap–add method In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal x with a finite impulse response (FIR) filter h where for ''m'' outside the region . The concept is to divid ...
. A hybrid convolution method that combines block and
FIR Firs (''Abies'') are a genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology Biology is the natural science that studies life and living organisms, including ...
algorithms allows for a zero input-output latency that is useful for real-time convolution computations.

# Domain of definition

The convolution of two complex-valued functions on is itself a complex-valued function on , defined by: :$\left(f * g \right)\left(x\right) = \int_ f\left(y\right)g\left(x-y\right)\,dy = \int_ f\left(x-y\right)g\left(y\right)\,dy,$ and is well-defined only if and 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 at infinity can be easily offset by sufficiently rapid decay in . The question of existence thus may involve different conditions on and :

## Compactly supported functions

If and are compactly supported
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, then their convolution exists, and is also compactly supported and continuous . More generally, if either function (say ) is compactly supported and the other is
locally integrableIn mathematics, a locally integrable function (sometimes also called locally summable function) is a function (mathematics), function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance ...
, then the convolution is well-defined and continuous. Convolution of and is also well defined when both functions are locally square integrable on and supported on an interval of the form (or both supported on ).

## Integrable functions

The convolution of and exists if and are both Lebesgue integrable functions in (), and in this case is also integrable . This is a consequence of Tonelli's theorem. This is also true for functions in , under the discrete convolution, or more generally for the convolution on any group. Likewise, if ()  and  ()  where ,  then  (),  and :$\, * g\, _p\le \, f\, _1\, g\, _p.$ In the particular case , this shows that is a
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
under the convolution (and equality of the two sides holds if and are non-negative almost everywhere). More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable spaces. Specifically, if satisfy: :$\frac+\frac=\frac+1,$ then :$\left\Vert f*g\right\Vert_r\le\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in\mathcal^p,\ g\in\mathcal^q,$ so that the convolution is a continuous bilinear mapping from to . The Young inequality for convolution is also true in other contexts (circle group, convolution on ). The preceding inequality is not sharp on the real line: when , there exists a constant such that: :$\left\Vert f*g\right\Vert_r\le B_\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in\mathcal^p,\ g\in\mathcal^q.$ The optimal value of was discovered in 1975 and independently in 1976, see Brascamp–Lieb inequality. A stronger estimate is true provided : :$\, f* g\, _r\le C_\, f\, _p\, g\, _$ where $\, g\, _$ is the weak norm. Convolution also defines a bilinear continuous map $L^\times L^\to L^$ for $1< p,q,r<\infty$, owing to the weak Young inequality: :$\, f* g\, _\le C_\, f\, _\, g\, _.$

## Functions of rapid decay

In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if ''f'' and ''g'' both decay rapidly, then ''f''∗''g'' also decays rapidly. In particular, if ''f'' and ''g'' are rapidly decreasing functions, then so is the convolution ''f''∗''g''. Combined with the fact that convolution commutes with differentiation (see #Properties), it follows that the class of
Schwartz function Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz", ...
s is closed under convolution .

## Distributions

Under some circumstances, it is possible to define the convolution of a function with a distribution, or of two distributions. If ''f'' is a compactly supported function and ''g'' is a distribution, then ''f''∗''g'' is a smooth function defined by a distributional formula analogous to :$\int_ \left(y\right)g\left(x-y\right)\,dy.$ More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law :$f* \left(g* \varphi\right) = \left(f* g\right)* \varphi$ remains valid in the case where ''f'' is a distribution, and ''g'' a compactly supported distribution .

## Measures

The convolution of any two
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s ''μ'' and ''ν'' of
bounded variation In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcu ...
is the measure $\mu*\nu$ defined by :$\int_ f\left(x\right) \, d\left(\mu*\nu\right)\left(x\right) = \int_\int_f\left(x+y\right)\,d\mu\left(x\right)\,d\nu\left(y\right).$ In particular, : $\left(\mu*\nu\right)\left(A\right) = \int_1_A\left(x+y\right)\, d\left(\mu\times\nu\right)\left(x,y\right),$ where $A\subset\mathbf R^d$ is a measurable set and $1_A$ is the
indicator function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of $A$. This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure. The convolution of measures also satisfies the following version of Young's inequality :$\, \mu* \nu\, \le \, \mu\, \, \nu\,$ where the norm is the
total variation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... of a measure. Because the space of measures of bounded variation is a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, convolution of measures can be treated with standard methods of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
that may not apply for the convolution of distributions.

# Properties

## Algebraic properties

The convolution defines a product on the
linear space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
without
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
. Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras. ;
Commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... : $f * g = g * f$ Proof: By definition: $(f * g)(t) = \int^\infty_ f(\tau)g(t - \tau)\, d\tau$ Changing the variable of integration to $u = t - \tau$ the result follows. ;
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
: $f * (g * h) = (f * g) * h$ Proof: This follows from using
Fubini's theorem In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
(i.e., double integrals can be evaluated as iterated integrals in either order). ;
Distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: $f * (g + h) = (f * g) + (f * h)$ Proof: This follows from linearity of the integral. ; Associativity with scalar multiplication: $a (f * g) = (a f) * g$ for any real (or complex) number $a$. ;
Multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
: No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution (a unitary impulse, centered at zero) or, at the very least (as is the case of ''L''1) admit approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically, $f * \delta = f$ where ''δ'' is the delta distribution. ; Inverse element: Some distributions ''S'' have an
inverse element In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
''S''−1 for the convolution which then must satisfy $S^ * S = \delta$ from which an explicit formula for ''S''−1 may be obtained.The set of invertible distributions forms an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under the convolution. ; Complex conjugation: $\overline = \overline * \overline$ ; Relationship with differentiation: $(f * g)' = f' * g = f * g'$ Proof: $\begin (f * g)' & = \frac \int^\infty_ f(\tau) g(t - \tau) \, d\tau \\$
pt & =\int^\infty_ f(\tau) \frac g(t - \tau) \, d\tau \\
pt & =\int^\infty_ f(\tau) g'(t - \tau) \, d\tau = f* g'. \end ; Relationship with integration: If $F(t) = \int^t_ f(\tau) d\tau,$ and $G(t) = \int^t_ g(\tau) \, d\tau,$ then $(F * g)(t) = (f * G)(t) = \int^t_(f * g)(\tau)\,d\tau.$

## Integration

If ''f'' and ''g'' are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals: : $\int_\left(f * g\right)\left(x\right) \, dx=\left\left(\int_f\left(x\right) \, dx\right\right) \left\left(\int_g\left(x\right) \, dx\right\right).$ This follows from
Fubini's theorem In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
. The same result holds if ''f'' and ''g'' are only assumed to be nonnegative measurable functions, by Tonelli's theorem.

## Differentiation

In the one-variable case, : $\frac\left(f * g\right) = \frac * g = f * \frac$ where ''d''/''dx'' is the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... . More generally, in the case of functions of several variables, an analogous formula holds with the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... : : $\frac\left(f * g\right) = \frac * g = f * \frac.$ A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of ''f'' and ''g'' is differentiable as many times as ''f'' and ''g'' are in total. These identities hold under the precise condition that ''f'' and ''g'' are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence of Young's convolution inequality. For instance, when ''f'' is continuously differentiable with compact support, and ''g'' is an arbitrary locally integrable function, : $\frac\left(f* g\right) = \frac * g.$ These identities also hold much more broadly in the sense of tempered distributions if one of ''f'' or ''g'' is a rapidly decreasing tempered distribution, a compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution. In the discrete case, the
difference operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
''D'' ''f''(''n'') = ''f''(''n'' + 1) − ''f''(''n'') satisfies an analogous relationship: : $D\left(f * g\right) = \left(Df\right) * g = f * \left(Dg\right).$

## Convolution theorem

The
convolution theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
states that : $\mathcal\ = k\cdot \mathcal\\cdot \mathcal\$ where $\mathcal\$ denotes the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
of $f$, and $k$ is a constant that depends on the specific
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) Normalization refers to social processes through which ideas and actions come to be seen as ' ...
of the Fourier transform. Versions of this theorem also hold for the
Laplace transform In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
,
two-sided Laplace transform In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin tr ...
,
Z-transform In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and
Mellin transformIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. On the other hand, if $\mathcal W$ is the Fourier transform matrix, then : $\mathcal W\left\left(C^x \ast C^y\right\right) = \left\left(\mathcal W C^ \bull \mathcal W C^\right\right)\left(x \otimes y\right) = \mathcal W C^x \circ \mathcal W C^y$, where $\bull$ is face-splitting product, $\otimes$ denotes
Kronecker product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, $\circ$ denotes Hadamard product (this result is an evolving of
count sketch Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by A ...
properties).

## Translational equivariance

The convolution commutes with translations, meaning that : $\tau_x \left(f * g\right) = \left(\tau_x f\right) * g = f * \left(\tau_x g\right)$ where τ''x''f is the translation of the function ''f'' by ''x'' defined by : $\left(\tau_x f\right)\left(y\right) = f\left(y - x\right).$ If ''f'' is a
Schwartz function Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz", ...
, then ''τxf'' is the convolution with a translated Dirac delta function ''τ''''x''''f'' = ''f'' ∗ ''τ''''x'' ''δ''. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution. Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds : Suppose that ''S'' is a bounded linear operator acting on functions which commutes with translations: ''S''(''τxf'') = ''τx''(''Sf'') for all ''x''. Then ''S'' is given as convolution with a function (or distribution) ''g''''S''; that is ''Sf'' = ''g''''S'' ∗ ''f''. Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of time-invariant systems, and especially LTI system theory. The representing function ''g''''S'' is the impulse response of the transformation ''S''. A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that ''S'' must be a continuous linear operator with respect to the appropriate topology. It is known, for instance, that every continuous translation invariant continuous linear operator on ''L''1 is the convolution with a finite
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. More generally, every continuous translation invariant continuous linear operator on ''L''''p'' for 1 ≤ ''p'' < ∞ is the convolution with a Distribution (mathematics)#Tempered distributions and Fourier transform, tempered distribution whose
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
is bounded. To wit, they are all given by bounded Fourier multipliers.

# Convolutions on groups

If ''G'' is a suitable group (mathematics), group endowed with a measure (mathematics), measure λ, and if ''f'' and ''g'' are real or complex valued Lebesgue integral, integrable functions on ''G'', then we can define their convolution by :$\left(f * g\right)\left(x\right) = \int_G f\left(y\right) g\left\left(y^x\right\right)\,d\lambda\left(y\right).$ It is not commutative in general. In typical cases of interest ''G'' is a locally compact Hausdorff space, Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless ''G'' is unimodular group, unimodular, the convolution defined in this way is not the same as $\int f\left(xy^\right)g(y) \, d\lambda(y)$. The preference of one over the other is made so that convolution with a fixed function ''g'' commutes with left translation in the group: :$L_h\left(f* g\right) = \left(L_hf\right)* g.$ Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former. On locally compact
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, a version of the
convolution theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The circle group T with the Lebesgue measure is an immediate example. For a fixed ''g'' in ''L''1(T), we have the following familiar operator acting on the Hilbert space ''L''2(T): :$T \left(x\right) = \frac \int_ \left(y\right) g\left( x - y\right) \, dy.$ The operator ''T'' is compact operator on Hilbert space, compact. A direct calculation shows that its adjoint ''T* '' is convolution with :$\bar\left(-y\right).$ By the commutativity property cited above, ''T'' is normal operator, normal: ''T''* ''T'' = ''TT''* . Also, ''T'' commutes with the translation operators. Consider the family ''S'' of operators consisting of all such convolutions and the translation operators. Then ''S'' is a commuting family of normal operators. According to compact operator on Hilbert space, spectral theory, there exists an orthonormal basis that simultaneously diagonalizes ''S''. This characterizes convolutions on the circle. Specifically, we have :$h_k \left(x\right) = e^, \quad k \in \mathbb,\;$ which are precisely the Character (mathematics), characters of T. Each convolution is a compact multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above. A discrete example is a finite
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... of order ''n''. Convolution operators are here represented by circulant matrices, and can be diagonalized by the discrete Fourier transform. A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in ''L''2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.

# Convolution of measures

Let ''G'' be a (multiplicatively written) topological group. If μ and ν are finite
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s on ''G'', then their convolution ''μ''∗''ν'' is defined as the pushforward measure of the Group action (mathematics), group action and can be written as :$\left(\mu * \nu\right)\left(E\right) = \iint 1_E\left(xy\right) \,d\mu\left(x\right) \,d\nu\left(y\right)$ for each measurable subset ''E'' of ''G''. The convolution is also a finite measure, whose
total variation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... satisfies :$\, \mu * \nu\, \le \left\, \mu\right\, \left\, \nu\right\, .$ In the case when ''G'' is locally compact with (left-)Haar measure λ, and μ and ν are absolute continuity, absolutely continuous with respect to a λ, Radon–Nikodym theorem, 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 on the topological group then the convolution ''μ''∗''ν'' is the probability distribution of the sum ''X'' + ''Y'' of two statistical independence, independent random variables ''X'' and ''Y'' whose respective distributions are μ and ν.

# Bialgebras

Let (''X'', Δ, ∇, ''ε'', ''η'') be a bialgebra with comultiplication Δ, multiplication ∇, unit η, and counit ''ε''. The convolution is a product defined on the endomorphism algebra End(''X'') as follows. Let ''φ'', ''ψ'' ∈ End(''X''), that is, ''φ'', ''ψ'': ''X'' → ''X'' are functions that respect all algebraic structure of ''X'', then the convolution ''φ''∗''ψ'' is defined as the composition :$X \mathrel X \otimes X \mathrel X \otimes X \mathrel X.$ The convolution appears notably in the definition of Hopf algebras . A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism ''S'' such that :$S * \operatorname_X = \operatorname_X * S = \eta\circ\varepsilon.$

# Applications Convolution and related operations are found in many applications in science, engineering and mathematics. * In
image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
*: In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes. *: In optics, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is bokeh. *: In image processing applications such as adding blurring. * In digital data processing *: In analytical chemistry, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve signal-to-noise ratio with minimal distortion of the spectra *: In
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ... , a weighted moving average is a convolution. * In
acoustics Acoustics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wo ...
, reverberation is the convolution of the original sound with echo (phenomenon), echoes from objects surrounding the sound source. *: In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal. *: In electronic music convolution is the imposition of a Spectrum, spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other. * In electrical engineering, the convolution of one function (the Signal (electrical engineering), input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred. * In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ... , wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ... line broadening due to the Doppler effect on its own gives a Normal distribution, Gaussian spectral line shape and collision broadening alone gives a Cauchy distribution, Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function. *: In Time-resolved spectroscopy#Time-resolved fluorescence spectroscopy, 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 computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost. * In probability theory, the probability distribution of the sum of two independent (probability), independent random variables is the convolution of their individual distributions. *: In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian. * In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm. * In structural reliability, the reliability index can be defined based on the convolution theorem. *: The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the joint distribution function. In fact, the joint distribution function can be obtained using the convolution theory. * Convolutional neural networks apply multiple cascaded ''convolution'' kernels with applications in machine vision and artificial intelligence. Though these are actually cross-correlations rather than convolutions in most cases. * In Smoothed-particle hydrodynamics, simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle $i$, some physical quantity $A_i$ is calculated as a convolution of $A_j$ with a weighting function, where $j$ denotes the neighbors of particle $i$: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.

* Analog signal processing * Circulant matrix * Convolution for optical broad-beam responses in scattering media * Convolution power * Deconvolution * Dirichlet convolution * Generalized signal averaging * Jan Mikusinski * List of convolutions of probability distributions * LTI system theory#Impulse response and convolution * Multidimensional discrete convolution * Scaled correlation * Titchmarsh convolution theorem * Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where each row is a shifted copy of the convolution kernel)

# References

* . * * * Dominguez-Torres, Alejandro (Nov 2, 2010). "Origin and history of convolution". 41 pgs. http://www.slideshare.net/Alexdfar/origin-adn-history-of-convolution. Cranfield, Bedford MK43 OAL, UK. Retrieved Mar 13, 2013. * * * . * . * . * . * . * * * . * * . * . * . * . * * * .

Earliest Uses: The entry on Convolution has some historical information.

o

* 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 * https://get-the-solution.net/projects/discret-convolution discret-convolution online calculator *https://lpsa.swarthmore.edu/Convolution/CI.html Convolution demo and visualization in javascript *https://phiresky.github.io/convolution-demo/ Another convolution demo in javascript
Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution.
by Alan Peters * * https://archive.org/details/Lectures_on_Image_Processing
Convolution Kernel Mask Operation Interactive tutorial

at MathWorld
Freeverb3 Impulse Response Processor
Opensource zero latency impulse response processor with VST plugins * Stanford University CS 17

showing how spatial convolution works.
A video lecture on the subject of convolution
given by Salman Khan (educator), Salman Khan
Example of FFT convolution for pattern-recognition (image processing)Intuitive Guide to Convolution
A blogpost about an intuitive interpretation of convolution. {{Differentiable computing Functional analysis Image processing Fourier analysis Bilinear operators Feature detection (computer vision)