In

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 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 ($f\; \backslash star\; g$) 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 ofintegral 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 ...

:
:$(f\; *\; g)(t)\; :=\; \backslash int\_^\backslash infty\; f(\backslash tau)\; g(t\; -\; \backslash tau)\; \backslash ,\; d\backslash tau.$
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 ...

):
:$(f\; *\; g)(t)\; :=\; \backslash int\_^\backslash infty\; f(t\; -\; \backslash tau)\; g(\backslash tau)\backslash ,\; d\backslash tau.$
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:
:$(f\; *\; g\; )(t)\; =\; \backslash int\_^\; f(\backslash tau)\; g(t\; -\; \backslash tau)\backslash ,\; d\backslash tau\; \backslash quad\; \backslash \; \backslash text\; f,\; g\; :;\; href="/html/ALL/s/,\_\backslash infty)\_\backslash to\_\backslash mathbb.$__Notation_

A_common_engineering_notational_convention_is: :$\_f(t)\_*\_g(t)\_\backslash mathrel\_\backslash underbrace\_,$ which_has_to_be_interpreted_carefully_to_avoid_confusion._For_instance,__is_equivalent_to_,_but__is_in_fact_equivalent_to_.__Derivation_

Given_two_functions_$\_f(t)\_$_and_$\_g(t)\_$_with_bilateral_Laplace_transforms :$\_F(s)\_=\_\backslash int\_^\backslash infty\_e^\_\backslash \_f(u)\_\backslash \_\backslash textu\_$ and :$\_G(s)\_=\_\backslash int\_^\backslash infty\_e^\_\backslash \_g(v)\_\backslash \_\backslash textv\_$ respectively,_the_convolution_operation_$\_f(t)\_*\_g(t)\_$_can_be_defined_as_the_inverse_Laplace_transform_of_the_product_of_$\_F(s)\_$_and_$\_G(s)\_$._More_precisely, :$\backslash begin\; F(s)\_\backslash cdot\_G(s)\_\&=\_\backslash int\_^\backslash infty\_e^\_\backslash \_f(u)\_\backslash \_\backslash textu\_\backslash cdot\_\backslash int\_^\backslash infty\_e^\_\backslash \_g(v)\_\backslash \_\backslash textv\_\backslash \backslash \; \&=\_\backslash int\_^\backslash infty\_\backslash int\_^\backslash infty\_e^\_\backslash \_f(u)\_\backslash \_g(v)\_\backslash \_\backslash textu\_\backslash \_\backslash textv\; \backslash end$ Let_$\_v\_=\_t\_-\_u\_$_such_that :$\backslash begin\; F(s)\_\backslash cdot\_G(s)\_\&=\_\backslash int\_^\backslash infty\_\backslash int\_^\backslash infty\_e^\_\backslash \_f(u)\_\backslash \_g(t\_-\_u)\_\backslash \_\backslash textu\_\backslash \_\backslash textt\_\backslash \backslash \; \&=\_\backslash int\_^\backslash infty\_e^\_\backslash underbrace\_\_\backslash \_\backslash textt\_\backslash \backslash \; \&=\_\backslash int\_^\backslash infty\_e^\_(f(t)\_*\_g(t))\_\backslash \_\backslash textt\; \backslash end$ Note_that_$\_F(s)\_\backslash cdot\_G(s)\_$_is_the_bilateral_Laplace_transform_of_$\_f(t)\_*\_g(t)\_$._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: :$f(t)\; *\; g(t)\; \backslash mathrel\; \backslash underbrace\_,$ which has to be interpreted carefully to avoid confusion. For instance, is equivalent to , but is in fact equivalent to .Derivation

Given two functions $f(t)$ and $g(t)$ with bilateral Laplace transforms :$F(s)\; =\; \backslash int\_^\backslash infty\; e^\; \backslash \; f(u)\; \backslash \; \backslash textu$ and :$G(s)\; =\; \backslash int\_^\backslash infty\; e^\; \backslash \; g(v)\; \backslash \; \backslash textv$ respectively, the convolution operation $f(t)\; *\; g(t)$ can be defined as the inverse Laplace transform of the product of $F(s)$ and $G(s)$. More precisely, :$\backslash begin\; F(s)\; \backslash cdot\; G(s)\; \&=\; \backslash int\_^\backslash infty\; e^\; \backslash \; f(u)\; \backslash \; \backslash textu\; \backslash cdot\; \backslash int\_^\backslash infty\; e^\; \backslash \; g(v)\; \backslash \; \backslash textv\; \backslash \backslash \; \&=\; \backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; e^\; \backslash \; f(u)\; \backslash \; g(v)\; \backslash \; \backslash textu\; \backslash \; \backslash textv\; \backslash end$ Let $v\; =\; t\; -\; u$ such that :$\backslash begin\; F(s)\; \backslash cdot\; G(s)\; \&=\; \backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; e^\; \backslash \; f(u)\; \backslash \; g(t\; -\; u)\; \backslash \; \backslash textu\; \backslash \; \backslash textt\; \backslash \backslash \; \&=\; \backslash int\_^\backslash infty\; e^\; \backslash underbrace\_\; \backslash \; \backslash textt\; \backslash \backslash \; \&=\; \backslash int\_^\backslash infty\; e^\; (f(t)\; *\; g(t))\; \backslash \; \backslash textt\; \backslash end$ Note that $F(s)\; \backslash cdot\; G(s)$ is the bilateral Laplace transform of $f(t)\; *\; g(t)$. 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 ...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.
Visual explanation

Historical developments

One of the earliest uses of the convolution integral appeared inD'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:
:$\backslash int\; f(u)\backslash cdot\; g(x\; -\; u)\; \backslash ,\; du$
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:
:$\backslash int\_0^t\; \backslash varphi(s)\backslash psi(t\; -\; s)\; \backslash ,\; ds,\backslash quad\; 0\; \backslash le\; t\; <\; \backslash infty,$
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: :$(f\; *\; g\_T)(t)\; \backslash equiv\; \backslash int\_^\; \backslash left;\; href="/html/ALL/s/sum\_^\backslash infty\_f(\backslash tau\_+\_kT)\backslash right.html"\; ;"title="sum\_^\backslash infty\; f(\backslash tau\; +\; kT)\backslash right">sum\_^\backslash infty\; f(\backslash tau\; +\; kT)\backslash right$ where is an arbitrary choice. The summation is called aperiodic 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: :$(f\; *\; g);\; href="/html/ALL/s/.html"\; ;"title="">$ or equivalently (seecommutativity
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:
:$(f\; *\; g);\; href="/html/ALL/s/.html"\; ;"title="">$
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 $\backslash $ (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:
:$(f*\; g);\; href="/html/ALL/s/.html"\; ;"title="">$
Circular discrete convolution

When a function is periodic, with period , then for functions, , such that exists, the convolution is also periodic and identical to: :$(f\; *\; g\_N);\; href="/html/ALL/s/.html"\; ;"title="">$ The summation on is called aperiodic 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 inmultiplication
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: :$(f\; *\; g\; )(x)\; =\; \backslash int\_\; f(y)g(x-y)\backslash ,dy\; =\; \backslash int\_\; f(x-y)g(y)\backslash ,dy,$ 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 supportedcontinuous 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 :$\backslash ,\; *\; g\backslash ,\; \_p\backslash le\; \backslash ,\; f\backslash ,\; \_1\backslash ,\; g\backslash ,\; \_p.$ In the particular case , this shows that is aBanach 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:
:$\backslash frac+\backslash frac=\backslash frac+1,$
then
:$\backslash left\backslash Vert\; f*g\backslash right\backslash Vert\_r\backslash le\backslash left\backslash Vert\; f\backslash right\backslash Vert\_p\backslash left\backslash Vert\; g\backslash right\backslash Vert\_q,\backslash quad\; f\backslash in\backslash mathcal^p,\backslash \; g\backslash in\backslash mathcal^q,$
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:
:$\backslash left\backslash Vert\; f*g\backslash right\backslash Vert\_r\backslash le\; B\_\backslash left\backslash Vert\; f\backslash right\backslash Vert\_p\backslash left\backslash Vert\; g\backslash right\backslash Vert\_q,\backslash quad\; f\backslash in\backslash mathcal^p,\backslash \; g\backslash in\backslash mathcal^q.$
The optimal value of was discovered in 1975 and independently in 1976, see Brascamp–Lieb inequality.
A stronger estimate is true provided :
:$\backslash ,\; f*\; g\backslash ,\; \_r\backslash le\; C\_\backslash ,\; f\backslash ,\; \_p\backslash ,\; g\backslash ,\; \_$
where $\backslash ,\; g\backslash ,\; \_$ is the weak norm. Convolution also defines a bilinear continuous map $L^\backslash times\; L^\backslash to\; L^$ for $1<\; p,q,r<\backslash infty$, owing to the weak Young inequality:
:$\backslash ,\; f*\; g\backslash ,\; \_\backslash le\; C\_\backslash ,\; f\backslash ,\; \_\backslash ,\; g\backslash ,\; \_.$
at genezanazwisk.pl (the webpage cites the sources)

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 ofSchwartz 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 :$\backslash int\_\; (y)g(x-y)\backslash ,dy.$ More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law :$f*\; (g*\; \backslash varphi)\; =\; (f*\; g)*\; \backslash varphi$ remains valid in the case where ''f'' is a distribution, and ''g'' a compactly supported distribution .Measures

The convolution of any twoBorel 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 $\backslash mu*\backslash nu$ defined by
:$\backslash int\_\; f(x)\; \backslash ,\; d(\backslash mu*\backslash nu)(x)\; =\; \backslash int\_\backslash int\_f(x+y)\backslash ,d\backslash mu(x)\backslash ,d\backslash nu(y).$
In particular,
: $(\backslash mu*\backslash nu)(A)\; =\; \backslash int\_1\_A(x+y)\backslash ,\; d(\backslash mu\backslash times\backslash nu)(x,y),$
where $A\backslash subset\backslash mathbf\; R^d$ 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 Ltotal 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 thelinear 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: $$(f\; *\; g)(t)\; =\; \backslash int^\backslash infty\_\; f(\backslash tau)g(t\; -\; \backslash tau)\backslash ,\; d\backslash tau$$ Changing the variable of integration to $u\; =\; t\; -\; \backslash tau$ 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''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''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: $\backslash overline\; =\; \backslash overline\; *\; \backslash overline$
; Relationship with differentiation: $$(f\; *\; g)\text{'}\; =\; f\text{'}\; *\; g\; =\; f\; *\; g\text{'}$$ Proof: $$\backslash begin\; (f\; *\; g)\text{'}\; \&\; =\; \backslash frac\; \backslash int^\backslash infty\_\; f(\backslash tau)\; g(t\; -\; \backslash tau)\; \backslash ,\; d\backslash tau\; \backslash \backslash $$ & =\int^\infty_ f(\tau) \frac g(t - \tau) \, d\tau \\ & =\int^\infty_ f(\tau) g'(t - \tau) \, d\tau = f* g'.
\end
; Relationship with integration: If $F(t)\; =\; \backslash int^t\_\; f(\backslash tau)\; d\backslash tau,$ and $G(t)\; =\; \backslash int^t\_\; g(\backslash tau)\; \backslash ,\; d\backslash tau,$ then $$(F\; *\; g)(t)\; =\; (f\; *\; G)(t)\; =\; \backslash int^t\_(f\; *\; g)(\backslash tau)\backslash ,d\backslash tau.$$
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: : $\backslash int\_(f\; *\; g)(x)\; \backslash ,\; dx=\backslash left(\backslash int\_f(x)\; \backslash ,\; dx\backslash right)\; \backslash left(\backslash int\_g(x)\; \backslash ,\; dx\backslash right).$ This follows fromFubini'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, : $\backslash frac(f\; *\; g)\; =\; \backslash frac\; *\; g\; =\; f\; *\; \backslash frac$ where ''d''/''dx'' is thederivative
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 ...

:
: $\backslash frac(f\; *\; g)\; =\; \backslash frac\; *\; g\; =\; f\; *\; \backslash frac.$
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 (Ldifference 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:
: $D(f\; *\; g)\; =\; (Df)\; *\; g\; =\; f\; *\; (Dg).$
Convolution theorem

Theconvolution 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
: $\backslash mathcal\backslash \; =\; k\backslash cdot\; \backslash mathcal\backslash \backslash cdot\; \backslash mathcal\backslash $
where $\backslash mathcal\backslash $ 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 $\backslash mathcal\; W$ is the Fourier transform matrix, then
: $\backslash mathcal\; W\backslash left(C^x\; \backslash ast\; C^y\backslash right)\; =\; \backslash left(\backslash mathcal\; W\; C^\; \backslash bull\; \backslash mathcal\; W\; C^\backslash right)(x\; \backslash otimes\; y)\; =\; \backslash mathcal\; W\; C^x\; \backslash circ\; \backslash mathcal\; W\; C^y$,
where $\backslash bull$ is face-splitting product, $\backslash otimes$ 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 ...

, $\backslash circ$ 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 : $\backslash tau\_x\; (f\; *\; g)\; =\; (\backslash tau\_x\; f)\; *\; g\; =\; f\; *\; (\backslash tau\_x\; g)$ where τSchwartz 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 ''τBorel 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''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 :$(f\; *\; g)(x)\; =\; \backslash int\_G\; f(y)\; g\backslash left(y^x\backslash right)\backslash ,d\backslash lambda(y).$ 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 $\backslash int\; f\backslash left(xy^\backslash right)g(y)\; \backslash ,\; d\backslash lambda(y)$. The preference of one over the other is made so that convolution with a fixed function ''g'' commutes with left translation in the group: :$L\_h(f*\; g)\; =\; (L\_hf)*\; g.$ 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 compactabelian 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''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''Convolution of measures

Let ''G'' be a (multiplicatively written) topological group. If μ and ν are finiteBorel 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
:$(\backslash mu\; *\; \backslash nu)(E)\; =\; \backslash iint\; 1\_E(xy)\; \backslash ,d\backslash mu(x)\; \backslash ,d\backslash nu(y)$
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
:$\backslash ,\; \backslash mu\; *\; \backslash nu\backslash ,\; \backslash le\; \backslash left\backslash ,\; \backslash mu\backslash right\backslash ,\; \backslash left\backslash ,\; \backslash nu\backslash right\backslash ,\; .$
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 :$X\; \backslash mathrel\; X\; \backslash otimes\; X\; \backslash mathrel\; X\; \backslash otimes\; X\; \backslash mathrel\; X.$ 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 :$S\; *\; \backslash operatorname\_X\; =\; \backslash operatorname\_X\; *\; S\; =\; \backslash eta\backslash circ\backslash varepsilon.$Applications

Convolution and related operations are found in many applications in science, engineering and mathematics. * Inimage 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.
See also

* 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)Notes

References

Further reading

* . * * * 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. * * * . * . * . * . * . * * * . * * . * . * . * . * * * .External links

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)