In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions and, in particular, functional analysis Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral, convolution is a mathematical operation An operator is a mapping from one vector space or module to another. Operators are of critical importance to both linear algebra and functional analysis. Important properties that various operators may exhibit include linearity, continuity, and boundedness on two functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or inner-product. It is commonly used to search a long duration signal for a shorter, known feature. It also has applications in pattern recognition, single particle. It has applications that include statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, computer vision Computer vision is the science and technology of machines that see. As a scientific discipline, computer vision is concerned with the theory behind artificial systems that extract information from images. The image data can take many forms, such as video sequences, views from multiple cameras, or multi-dimensional data from a medical scanner, image In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as photographs or frames of video; the output of image processing can be either an image or a set of characteristics or parameters related to the image. Most image-processing techniques involve treating the image and signal processing Signal processing is an area of electrical engineering, systems engineering, and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals. Signals of interest can include sound, images, time-varying measurement values and sensor data, for example, electrical engineering Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical power supply. It now covers a range of subtopics, and differential equations A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

The convolution can be defined for functions on groups In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from other than Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity. In particular, the circular convolution can be defined for periodic functions In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity (that is, functions on the circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius), and the discrete convolution can be defined for functions on the set of integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6. These generalizations of the convolution have applications in the field of numerical analysis Numerical analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics (as distinguished from discrete mathematics) and numerical linear algebra Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, data mining,, and in the design and implementation of finite impulse response A finite impulse response filter is a type of a digital filter. The impulse response, the filter's response to a Kronecker delta input, is finite because it settles to zero in a finite number of sample intervals. This is in contrast to infinite impulse response (IIR) filters, which have internal feedback and may continue to respond indefinitely filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing. Because these techniques are in turn widely used in many scientific and engineering disciplines, deconvolution finds many.

Contents 1 History 2 Definition 2.1 Circular convolution 3 Discrete convolution 3.1 Circular discrete convolution 3.2 Fast convolution algorithms 4 Domain of definition 4.1 Compactly supported functions 4.2 Integrable functions 4.3 Functions of rapid decay 4.4 Distributions 4.5 Measures 5 Properties 5.1 Algebraic properties 5.2 Integration 5.3 Differentiation 5.4 Convolution theorem 5.5 Translation invariance 6 Convolutions on groups 7 Convolution of measures 8 Bialgebras 9 Applications 10 See also 11 Notes 12 References 13 External links

History

The operation

is a particular case of composition products considered by the Italian mathematician Vito Volterra Vito Volterra was an Italian mathematician and physicist, best known for his contributions to mathematical biology.^[1]

The convolution was originally known under the name "Faltung" (which means folding in German German (Deutsch, [ˈdɔʏtʃ] ) is a West Germanic language, thus related to and classified alongside English and Dutch. It is one of the world's major languages and the most widely spoken first language in the European Union. Globally, German is spoken by approximately 120 million native speakers and also by about 80 million non-native speakers), introduced by a German mathematician Gustav Doetsch.^[2]

This section requires expansion.

Definition

The convolution of ƒ and g is written ƒ∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform The input of this transform is a function f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator:

	(commutativity)

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function ƒ(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.

More generally, if f and g are complex-valued functions on R^d, then their convolution may be defined as the integral:

Visual explanation of convolution.

1.Express each function in terms of a dummy variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder , or a wildcard character that stands for an unspecified symbol τ. 2.Reflect one of the functions: g(τ)→g( − τ). 3.Add a time-offset, t, which allows g(t − τ) to slide along the τ-axis. 4.Start t at -∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function f(τ), where the weighting function is g( − τ). The resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of.

Circular convolution

Main article: Circular convolution

When a function g_T is periodic, with period T, then for functions, ƒ, such that ƒ∗g_T exists, the convolution is also periodic and identical to:

where t_o is an arbitrary choice. The summation is called a periodic extension of the function ƒ.

If g_T is a periodic extension of another function, g, then ƒ∗g_T is known as a circular, cyclic, or periodic convolution of ƒ and g.

Discrete convolution

For complex-valued functions ƒ, g defined on the set Z of integers, the discrete convolution of ƒ and g is given by:

(commutativity)

When multiplying two polynomials In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x, the coefficients of the product are given by the convolution of the original coefficient sequences In mathematics, a sequence is an ordered list of objects . Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence. A sequence is a, extended with zeros where necessary to avoid undefined terms; this is known as the Cauchy product of the coefficients of the two polynomials.

Circular discrete convolution

When a function g_N is periodic, with period N, then for functions, ƒ, such that ƒ∗g_N exists, the convolution is also periodic and identical to:

The summation on k is called a periodic extension of the function ƒ.

If g_N is a periodic extension of another function, g, then ƒ∗g_N is known as a circular convolution of ƒ and g.

When the non-zero durations of both ƒ and g are limited to the interval [0, N-1], ƒ∗g_N reduces to these common forms:

(Eq.1)

The notation for cyclic convolution denotes convolution over the cyclic group In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive) of integers modulo N In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus. Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

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 is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division) of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C; von zur Gathen & Gerhard 2003, §8.2).

Eq.1 requires N arithmetic operations per output value and N² operations for N outputs. That can be significantly reduced with any of several fast algorithms. Digital signal processing Digital signal processing is concerned with the representation of signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing. DSP includes subfields like: audio and speech signal processing, sonar and radar signal processing, sensor and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.

The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem In mathematics, the discrete Fourier transform is a specific kind of Fourier transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function (which is often a function in the time domain). But the DFT requires an input function that is. 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, use fast Fourier transforms in other rings In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under.

Domain of definition

The convolution of two complex-valued functions on R^d

is well-defined only if ƒ and g 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 g at infinity can be easily offset by sufficiently rapid decay in ƒ. The question of existence thus may involve different conditions on ƒ and g.

Compactly supported functions

If ƒ and g are compactly supported In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set. This concept is used very widely in mathematical analysis. In the form of functions with support that is bounded, it also plays a major part in various types of mathematical duality theories continuous functions In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is, then their convolution exists, and is also compactly supported and continuous (Hörmander). More generally, if either function (say ƒ) is compactly supported and the other is locally integrable, then the convolution ƒ∗g is well-defined and continuous.

Integrable functions

The convolution of ƒ and g exists if ƒ and g are both Lebesgue integrable functions In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. This article focuses on the more general (in L¹(R^d) In mathematics, the Lp spaces are function spaces defined using natural generalizations of p-norms for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to Bourbaki (1987) they were first introduced by Riesz (1910). They form an important class of examples of Banach spaces), and in this case ƒ∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem In mathematical analysis, Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to change the order of integration. Likewise, if ƒ ∈ L¹(R^d) and g ∈ L^p(R^d) where 1 ≤ p ≤ ∞, then ƒ∗g ∈ L^p(R^d) and

In the particular case p= 1, this shows that L¹ is a Banach algebra under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).

More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable L^p spaces. Specifically, if 1 ≤ p,q,r ≤ ∞ satisfy

then

so that the convolution is a continuous bilinear mapping from L^p×L^q to L^r.

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 ƒ and g both decay rapidly, then ƒ∗g also decays rapidly. In particular, if ƒ and g are rapidly decreasing functions, then so is the convolution ƒ∗g. Combined with the fact that convolution commutes with differentiation (see Properties), it follows that the class of Schwartz functions is closed under convolution.

Distributions

Main article: Distribution (mathematics) In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivative does not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used to formulate generalized solutions of partial

Under some circumstances, it is possible to define the convolution of a function with a distribution, or of two distributions. If ƒ is a compactly supported function and g is a distribution, then ƒ∗g is a smooth function defined by a distributional formula analogous to

More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law

remains valid in the case where ƒ is a distribution, and g a compactly supported distribution (Hörmander 1983, §4.2).

Measures

The convolution of any two Borel measures In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a μ and ν of bounded variation is the measure λ defined by

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

where the norm is the total variation of a measure. Because the space of measures of bounded variation is a Banach space In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them. Many of the infinite-dimensional function spaces studied in analysis are examples of Banach spaces, convolution of measures can be treated with standard methods of functional analysis Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral that may not apply for the convolution of distributions.

Properties

Algebraic properties

See also: Convolution algebra

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 algebra without identity In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts (Strichartz 1994, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are closed In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not under the convolution, and so also form commutative algebras.

Commutativity In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the

Associativity In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such

Distributivity In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:

Associativity with scalar multiplication

for any real (or complex) number .

Multiplicative identity

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 or, at the very least (as is the case of L¹) admit approximations to the identity^{[disambiguation needed]}. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,

where δ is the delta distribution.

Inverse element

Some distributions have an inverse element In abstract algebra, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic for the convolution, S⁽⁻¹⁾, which is defined by

The set of invertible distributions forms an abelian group 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 their order . Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel under the convolution.

Complex conjugation

Integration

If ƒ and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:

This follows from Fubini's theorem In mathematical analysis, Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to change the order of integration. The same result holds if ƒ and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem.

Differentiation

In the one-variable case,

where d/dx is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of ƒ and g is differentiable as many times as ƒ and g are together.

These identities hold under the precise condition that ƒ and g are absolutely integrable and at least one of them has an absolutely integrable (L¹) weak derivative, as a consequence of Young's inequality. For instance, when ƒ is continuously differentiable with compact support, and g is an arbitrary locally integrable function,

These identities also hold much more broadly in the sense of tempered distributions if one of ƒ or g is a compactly supported 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 ƒ(n) = ƒ(n + 1) − ƒ(n) satisfies an analogous relationship:

Convolution theorem

The convolution theorem states that

where denotes the Fourier transform of f, and k is a constant that depends on the specific normalization of the Fourier transform (see “Properties of the fourier transform”). Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

See also the less trivial Titchmarsh convolution theorem.

Translation invariance

The convolution commutes with translations, meaning that

where τ_xƒ is the translation of the function ƒ by x defined by

If ƒ is a Schwartz function, then τ_xƒ is the convolution with a translated Dirac delta function τ_xƒ = ƒ∗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 linear operator acting on functions which commutes with translations: S(τ_xƒ) = τ_x(Sƒ) for all x. Then S is given as convolution with a function (or distribution) g_S; that is Sƒ = g_S∗ƒ.

Thus any translation invariant operation can be represented as a convolution. Convolutions play an important role in the study of time-invariant systems, and especially LTI system theory. The representing function g_S is the impulse response of the transformation S.

A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a continuous linear operator with respect to the appropriate topology. It is known, for instance, that every continuous translation invariant continuous linear operator on L¹ is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear operator on L^p for 1 ≤ p < ∞ is the convolution with a tempered distribution whose Fourier transform is bounded. To wit, they are all given by bounded Fourier multipliers.

Convolutions on groups

If G is a suitable group endowed with a measure λ, and if f and g are real or complex valued integrable functions on G, then we can define their convolution by

In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless G is unimodular, the convolution defined in this way is not the same as . The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:

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 groups, a version of the convolution theorem 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 L¹(T), we have the following familiar operator acting on the Hilbert space L²(T):

The operator T is compact. A direct calculation shows that its adjoint T* is convolution with

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, there exists an orthonormal basis {h_k} that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have

which are precisely the characters of T. Each convolution is a compact multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above.

A discrete example is a finite cyclic group 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 L² by the Peter-Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.

Convolution of measures

Let G be a topological group. If μ and ν are finite Borel measures on a group G, then their convolution μ∗ν is defined by

for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies

In the case when G is locally compact with (left-)Haar measure λ, and μ and ν are absolutely continuous 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 measures, then the convolution μ∗ν is the probability distribution of the sum X + Y of two 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

The convolution appears notably in the definition of Hopf algebras (Kassel 1995, §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that

Applications

Convolution and related operations are found in many applications of engineering and mathematics.

• In electrical engineering, the convolution of one function (the 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 digital signal processing and image processing applications, the entire input function is often available for computing every sample of the output function. In that case, the constraint that each output is the effect of only prior inputs can be relaxed.
  • Convolution amplifies or attenuates each frequency component of the input independently of the other components.
• In statistics, as noted above, a weighted moving average is a convolution.
• In probability theory, the probability distribution of the sum of two independent random variables is the convolution of their individual distributions.
• In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic term for this is bokeh.
• Similarly, in digital image processing, convolutional filtering plays an important role in many important algorithms in edge detection and related processes.
• In linear acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
• In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information).
• 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 radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm.
• In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance.
• In geographic information systems, the result of a kernel estimate of the intensity function of the point pattern is the convolution of the isotropic Gaussian kernel of a standard deviation with point masses at each of the data points. (Diggle 1995) cited by documentation of the "Kernel Smoothed Intensity of Point Pattern" of the SDA4PP QGIS plugin.

See also

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

Notes

1. ^ According to [Lothar von Wolfersdorf (2000), "Einige Klassen quadratischer Integralgleichungen", Sitzungsberichte der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-naturwissenschaftliche Klasse, Band 128, Heft 2, 6–7], the source is Volterra, Vito (1913), "Leçons sur les fonctions de linges". Gauthier-Villars, Paris 1913.
2. ^ Gustav Doetsch, "Die Integrodifferentialgleichungen vom Faltungstypus". Mathematische Annalen 89 (1923), 192–207

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• Rudin, Walter (1962), Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York–London, MR 0152834, ISBN 047152364X .
• Sobolev, V.I. (2001), "Convolution of functions", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c026430.htm .
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 0-691-08078-X .
• Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, ISBN 0849382734 .
• Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), New York, N.Y.: Chelsea Pub. Co. (published 1986), ISBN 978-0828403245 .
• Uludag, A. M. (1998), "On possible deterioration of smoothness under the operation of convolution", J. Math. Anal. Appl. 227 no. 2, 335--358
• Treves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press, ISBN 0486453529 .
• von zur Gathen, J.; Gerhard, J. (2003), Modern Computer Algebra, Cambridge University Press, ISBN 0-521-82646-2 .
• Diggle, P. J. (1995), "A kernel method for smoothing point process data", Journal of the Royal Statistical Society, Series C) 34 (1985) 138–147

External links

Look up convolution in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: Convolution

• Earliest Uses: The entry on Convolution has some historical information.
• http://www.nitte.ac.in/downloads/Conv-LTI.pdf
• Convolution, on The Data Analysis BriefBook
• 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.
• 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.
  • http://www.vuse.vanderbilt.edu/~rap2/EECE253/EECE253_01_Intro.pdf
• A collection of 2D convolution kernels
• Convolution Kernel Mask Operation Interactive tutorial
• Convolution at MathWorld
• Freeverb3 Impulse Response Processor: Opensource zero latency impulse response processor with VST plugins

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