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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(in particular,
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
), convolution is a
mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most c ...
on two functions ( 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, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
). 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, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; 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 concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
,
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acousticia ...
,
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
,
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
,
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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
and
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. The convolution can be defined for functions on
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and other
groups A group is a number of persons 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 ide ...
(as
algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
). For example,
periodic function A periodic function is a 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 throughout science to des ...
s, such as the
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
, can be defined on a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
and convolved by
periodic convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
. (See row 18 at .) A ''discrete convolution'' can be defined for functions on the set of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. 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 manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
and
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
, and in the design and implementation of
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
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 a ...
of the convolution operation is known as
deconvolution In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
.


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 reflected about the y-axis and shifted. As such, it is a particular kind of
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
: :(f * g)(t) := \int_^\infty f(\tau) g(t - \tau) \, d\tau. An equivalent definition is (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
): :(f * g)(t) := \int_^\infty f(t - \tau) g(\tau)\, d\tau. While the symbol is used above, it need not represent the time domain. At each ''t'', the convolution formula can be described as the area under the function weighted by the function shifted by the amount . As changes, the weighting function emphasizes different parts of the input function ; If is a positive value, then is equal to that slides or is shifted along the \tau-axis toward the right (toward ) by the amount of , while if is a negative value, then is equal to that slides or is shifted toward the left (toward ) by the amount of . For functions , supported on only (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: :(f * g )(t) = \int_^ f(\tau) g(t - \tau)\, d\tau \quad \ \text f, g : domain_of_definition''_(below).


__Notation_

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


__Relations_with_other_transforms_

Given_two_functions__f(t)__and__g(t)__with_Two-sided_Laplace_transform.html" ;"title="#Domain of definition">domain of definition'' (below).


Notation

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


Relations with other transforms

Given two functions f(t) and g(t) with Two-sided Laplace transform">bilateral Laplace transforms (two-sided Laplace transform) : F(s) = \int_^\infty e^ \ f(u) \ \textu and : G(s) = \int_^\infty e^ \ g(v) \ \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, : \begin F(s) \cdot G(s) &= \int_^\infty e^ \ f(u) \ \textu \cdot \int_^\infty e^ \ g(v) \ \textv \\ &= \int_^\infty \int_^\infty e^ \ f(u) \ g(v) \ \textu \ \textv \end Let t = u + v such that : \begin F(s) \cdot G(s) &= \int_^\infty \int_^\infty e^ \ f(u) \ g(t - u) \ \textu \ \textt \\ &= \int_^\infty e^ \underbrace_ \ \textt \\ &= \int_^\infty e^ (f(t) * g(t)) \ \textt \end Note that F(s) \cdot G(s) is the bilateral Laplace transform of f(t) * g(t) . A similar derivation can be done using the unilateral Laplace transform (one-sided 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 LTI can refer to: * '' LTI – Lingua Tertii Imperii'', a book by Victor Klemperer * Language Technologies Institute, a division of Carnegie Mellon University * Linear time-invariant system, an engineering theory that investigates the response o ...
for a derivation of convolution as the result of LTI constraints. In terms of the
Fourier transforms A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
). See
Convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
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 in
D'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopé ...
'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 th ...
in ''Recherches sur différents points importants du système du monde,'' published in 1754. Also, an expression of the type: :\int f(u)\cdot g(x - u) \, du is used by
Sylvestre François Lacroix Sylvestre François Lacroix (28 April 176524 May 1843) was a French mathematician. Life He was born in Paris, and was raised in a poor family who still managed to obtain a good education for their son. Lacroix's path to mathematics started wi ...
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,
Jean-Baptiste Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and har ...
,
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
, 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: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
), ''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(s)\psi(t - s) \, 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 and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...
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) \equiv \int_^ \left sum_^\infty f(\tau + kT)\rightg_T(t - \tau)\, d\tau, where is an arbitrary choice. The summation is called a
periodic summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
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) = \sum_^\infty f g - m or equivalently (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
) by: :(f * g) = \sum_^\infty f -mg 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, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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 product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...
of the coefficients of the sequences. Thus when has finite support in the set \ (representing, for instance, a
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
), a finite summation may be used: :(f* g) \sum_^M f -m


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) \equiv \sum_^ \left(\sum_^\infty + kNright) g_N - m The summation on is called a
periodic summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
of the function . If is a periodic summation of another function, , then is known as a
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
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, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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 that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in ...
(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 Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
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, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
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 that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in ...
(FFT) algorithms via the circular convolution theorem. Specifically, the
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
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, ...
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. A hybrid convolution method that combines block and FIR 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) = \int_ f(y)g(x-y)\,dy = \int_ f(x-y)g(y)\,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, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
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 integrable, 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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
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 In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
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_ (y)g(x-y)\,dy. More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law :f* (g* \varphi) = (f* g)* \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, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
s ''μ'' and ''ν'' of bounded variation is the measure \mu*\nu defined by :\int_ f(x) \, d(\mu*\nu)(x) = \int_\int_f(x+y)\,d\mu(x)\,d\nu(y). In particular, : (\mu*\nu)(A) = \int_1_A(x+y)\, d(\mu\times\nu)(x,y), where A\subset\mathbf R^d is a measurable set and 1_A is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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 of a measure. Because the space of measures of bounded variation is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, convolution of measures can be treated with standard methods of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
that may not apply for the convolution of distributions.


Properties


Algebraic properties

The convolution defines a product on the linear space 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, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
without
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
. Other linear spaces of functions, such as the space of continuous functions of compact support, are
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under the convolution, and so also form commutative associative algebras. ;
Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: 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 valid rule of replacement ...
: f * (g * h) = (f * g) * h Proof: This follows from using
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
(i.e., double integrals can be evaluated as iterated integrals in either order). ;
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
: 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, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
: 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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
(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 mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
''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, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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_(f * g)(x) \, dx=\left(\int_f(x) \, dx\right) \left(\int_g(x) \, dx\right). This follows from
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
. 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(f * g) = \frac * g = f * \frac where ''d''/''dx'' is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. More generally, in the case of functions of several variables, an analogous formula holds with the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
: : \frac(f * g) = \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 In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. Statement Euclidean Space In real analysis, the following result is called Young's convolution ...
. For instance, when ''f'' is continuously differentiable with compact support, and ''g'' is an arbitrary locally integrable function, : \frac(f* g) = \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 ''D'' ''f''(''n'') = ''f''(''n'' + 1) − ''f''(''n'') satisfies an analogous relationship: : D(f * g) = (Df) * g = f * (Dg).


Convolution theorem

The
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
states that : \mathcal\ = k\cdot \mathcal\\cdot \mathcal\ where \mathcal\ denotes the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of f, and k is a constant that depends on the specific normalization of the Fourier transform. Versions of this theorem also hold for the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
, two-sided Laplace transform,
Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
and
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used ...
. On the other hand, if \mathcal W is the Fourier transform matrix, then : \mathcal W\left(C^x \ast C^y\right) = \left(\mathcal W C^ \bull \mathcal W C^\right)(x \otimes y) = \mathcal W C^x \circ \mathcal W C^y, where \bull is face-splitting product, \otimes denotes
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
, \circ denotes Hadamard product (this result is an evolving of count sketch properties).


Translational equivariance

The convolution commutes with translations, meaning that : \tau_x (f * g) = (\tau_x f) * g = f * (\tau_x g) where τ''x''f is the translation of the function ''f'' by ''x'' defined by : (\tau_x f)(y) = f(y - x). If ''f'' is a
Schwartz function In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
, 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
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 LTI can refer to: * '' LTI – Lingua Tertii Imperii'', a book by Victor Klemperer * Language Technologies Institute, a division of Carnegie Mellon University * Linear time-invariant system, an engineering theory that investigates the response o ...
. The representing function ''g''''S'' is the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
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 In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear op ...
with respect to the appropriate
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
. More generally, every continuous translation invariant continuous linear operator on ''L''''p'' for 1 ≤ ''p'' < ∞ is the convolution with a
tempered distribution Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives ...
whose
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is bounded. To wit, they are all given by bounded Fourier multipliers.


Convolutions on groups

If ''G'' is a suitable
group A group is a number of persons 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 ide ...
endowed with a measure λ, and if ''f'' and ''g'' are real or complex valued
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
functions on ''G'', then we can define their convolution by :(f * g)(x) = \int_G f(y) g\left(y^x\right)\,d\lambda(y). It is not commutative in general. In typical cases of interest ''G'' is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Hausdorff
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
and λ is a (left-)
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thou ...
. In that case, unless ''G'' is 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(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 compact
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, a version of the
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''L''2(T): :T (x) = \frac \int_ (y) g( x - y) \, dy. The operator ''T'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
. A direct calculation shows that its adjoint ''T* '' is convolution with :\bar(-y). By the commutativity property cited above, ''T'' is 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
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
, there exists an orthonormal basis that simultaneously diagonalizes ''S''. This characterizes convolutions on the circle. Specifically, we have :h_k (x) = e^, \quad k \in \mathbb,\; which are precisely the
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
s 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, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n''. Convolution operators are here represented by circulant matrices, and can be diagonalized by the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s form an orthonormal basis in ''L''2 by the
Peter–Weyl theorem In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
, and an analog of the convolution theorem continues to hold, along with many other aspects of
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
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, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
s on ''G'', then their convolution ''μ''∗''ν'' is defined as the pushforward measure of the
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
and can be written as :(\mu * \nu)(E) = \iint 1_E(x+y) \,d\mu(x) \,d\nu(y) for each measurable subset ''E'' of ''G''. The convolution is also a finite measure, whose total variation satisfies :\, \mu * \nu\, \le \left\, \mu\right\, \left\, \nu\right\, . In the case when ''G'' is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
with (left-)
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thou ...
λ, and μ and ν are
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
with respect to a λ, 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 measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s on the topological group then the convolution ''μ''∗''ν'' is the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of the sum ''X'' + ''Y'' of two
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s ''X'' and ''Y'' whose respective distributions are μ and ν.


Infimal convolution

In
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...
, the infimal convolution of proper (not identically +\infty)
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
s f_1,\dots,f_m on \mathbb R^n is defined by: (f_1*\cdots*f_m)(x)=\inf_x \. It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the
Legendre transform In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
: \varphi^*(x) = \sup_y ( x\cdot y - \varphi(y)). We have: (f_1*\cdots *f_m)^*(x) = f_1^*(x) + \cdots + f_m^*(x).


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 In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gro ...
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 algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Swe ...
s . 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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
** In
digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
convolutional filtering plays an important role in many important
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s in
edge detection Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuitie ...
and related processes (see Kernel (image processing)) ** In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is
bokeh In photography, bokeh ( or ; ) is the aesthetic quality of the blur produced in out-of-focus parts of an image. Bokeh has also been defined as "the way the lens renders out-of-focus points of light". Differences in lens aberrations and ...
. ** In image processing applications such as adding blurring. * In digital data processing ** In
analytical chemistry Analytical chemistry studies and uses instruments and methods to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separati ...
, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in de ...
with minimal distortion of the spectra ** In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a weighted
moving average In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
is a convolution. * In
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acousticia ...
,
reverberation Reverberation (also known as reverb), in acoustics, is a persistence of sound, after a sound is produced. Reverberation is created when a sound or signal is reflected causing numerous reflections to build up and then decay as the sound is abso ...
is the convolution of the original sound with
echoes Echoes may refer to: * Echo (phenomenon) Film and television * ''Echoes'' (2014 film), an American supernatural horror film * ''Echoes'' (miniseries), a 2022 Netflix original drama series * "Echoes" (''Fear Itself''), an episode of ''Fear Itse ...
from objects surrounding the sound source. ** In digital signal processing, convolution is used to map the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
of a real room on a digital audio signal. ** In
electronic music Electronic music is a genre of music that employs electronic musical instruments, digital instruments, or circuitry-based music technology in its creation. It includes both music made using electronic and electromechanical means ( electro ...
convolution is the imposition of a
spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars J ...
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 Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a
linear time-invariant system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
(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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, wherever there is a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction ...
with a "
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
", a convolution operation makes an appearance. For instance, in
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter ...
line broadening due to the Doppler effect on its own gives a
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
spectral line shape and collision broadening alone gives a Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function. ** In 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 Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
, 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 Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of the sum of two
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s is the convolution of their individual distributions. ** In
kernel density estimation In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on '' kernels'' as ...
, 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 In deep learning, a convolutional neural network (CNN, or ConvNet) is a class of artificial neural network (ANN), most commonly applied to analyze visual imagery. CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Networ ...
apply multiple cascaded ''convolution'' kernels with applications in
machine vision Machine vision (MV) is the technology and methods used to provide imaging-based automatic inspection and analysis for such applications as automatic inspection, process control, and robot guidance, usually in industry. Machine vision refers to ...
and
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
. Though these are actually cross-correlations rather than convolutions in most cases. * In
Smoothed-particle hydrodynamics Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysica ...
, 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 In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplit ...
* Convolution for optical broad-beam responses in scattering media *
Convolution power In mathematics, the convolution power is the ''n''-fold iteration of the convolution with itself. Thus if x is a function on Euclidean space R''d'' and n is a natural number, then the convolution power is defined by : x^ = \underbrace_n,\quad x^ ...
*
Deconvolution In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
* Dirichlet convolution *
Generalized signal averaging Within signal processing, in many cases only one image with noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vib ...
*
Jan Mikusinski Jan, JaN or JAN may refer to: Acronyms * Jackson, Mississippi (Amtrak station), US, Amtrak station code JAN * Jackson-Evers International Airport, Mississippi, US, IATA code * Jabhat al-Nusra (JaN), a Syrian militant group * Japanese Article Numb ...
*
List of convolutions of probability distributions In probability theory, the probability distribution of the sum of two or more independent (probability), independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass ...
* LTI system theory#Impulse response and convolution * Multidimensional discrete convolution *
Scaled correlation In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), sca ...
*
Titchmarsh convolution theorem The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926. Titchmarsh convolution theorem If \varphi(t)\, and \psi(t) are integrable functio ...
*
Toeplitz matrix In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b ...
(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 ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

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 Abdul Rashid Salim Salman Khan (; 27 December 1965) is an Indian actor, film producer, and television personality who works in Hindi films. In a film career spanning over thirty years, Khan has received numerous awards, including two Nation ...

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 maps Feature detection (computer vision)