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Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. Also, the symmetry of f is the reason
{\displaystyle f*g}
{\displaystyle f\star g}
are identical in this example. Comparison convolution correlation.svg
Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. Also, the symmetry of f is the reason and are identical in this example.

In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function 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. Convolution is similar to cross-correlation. For real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f (x) or g(x) is reflected about the y-axis; thus it is a cross-correlation of f (x) and g(−x), or f (−x) and g(x). [note 1]   For continuous functions, the cross-correlation operator is the adjoint of the convolution operator.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Functional analysis branch of mathematical analysis concerned with infinite-dimensional topological vector spaces, often spaces of functions

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 and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Operation (mathematics) mathematical procedure which produces a result from operands; calculation from zero or more input values (called operands) to an output value. The number of operands is the arity of the operation

In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations, such as addition and multiplication, and unary operations, such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is supposed to be finite. However, infinitary operations are sometimes considered, in which context the "usual" operations of finite arity are called finitary operations.


Convolution has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.[ citation needed ]

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

Statistics study of the collection, organization, analysis, interpretation, and presentation of data

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

Computer vision is an interdisciplinary scientific field that deals with how computers can be made to gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to automate tasks that the human visual system can do.

The convolution can be defined for functions on Euclidean space, and other groups. [ citation needed ] For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 13 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers.

Euclidean space Generalization of Euclidean geometry to higher dimensions

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

Group (mathematics) set with an invertible, associative internal operation admitting a neutral element

In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.[ citation needed ]

Numerical analysis study of algorithms that use numerical approximation for the problems of mathematical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis, and a type of linear algebra. Because computers use floating-point arithmetic, they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision.

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

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 applications.


The convolution of f and g is written fg, 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:

An asterisk (*); from Late Latin asteriscus, from Ancient Greek ἀστερίσκος, asteriskos, "little star", is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star.

In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain. The solution is then mapped back to the original domain using the inverse of the integral transform.

An equivalent definition is (see 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 f (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.

For functions f, g supported on only [0, ∞) (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:

For the multi-dimensional formulation of convolution, see domain of definition (below).


A common engineering convention is: [1]

which has to be interpreted carefully to avoid confusion.  For instance, f (t)∗g(tt0) is equivalent to (fg)(tt0),  but f (tt0)∗g(tt0) is in fact equivalent to (fg)(t − 2t0). [2]


Convolution describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem 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

Visual explanations of convolution
  1. Express each function in terms of a dummy variable
  2. Reflect one of the functions:
  3. Add a time-offset, t, which allows to slide along the -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-sum of function where the weighting function is
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). Formally:
In this example, the red-colored "pulse", is an even function so convolution is equivalent to correlation. A snapshot of this "movie" shows functions and (in blue) for some value of parameter which is arbitrarily defined as the distance from the axis to the center of the red pulse. The amount of yellow is the area of the product computed by the convolution/correlation integral. The movie is created by continuously changing and recomputing the integral. The result (shown in black) is a function of but is plotted on the same axis as for convenience and comparison.
Convolution of box signal with itself2.gif
In this depiction, could represent the response of an RC circuit to a narrow pulse that occurs at In other words, if the result of convolution is just But when is the wider pulse (in red), the response is a "smeared" version of It begins at because we defined as the distance from the axis to the center of the wide pulse (instead of the leading edge).
Convolution of spiky function with box2.gif

Historical developments

One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. [3]

Also, an expression of the type:

is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. [4] Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, 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), composition product, superposition integral, and Carson's integral. [5] Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. [6] [7]

The operation:

is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. [8]

Circular convolution

When a function gT is periodic, with period T, then for functions, f, such that fgT exists, the convolution is also periodic and identical to:

where to is an arbitrary choice. The summation is called a periodic summation of the function f.

When gT is a periodic summation of another function, g, then fgT is known as a circular or cyclic convolution of f and g.

And if the periodic summation above is replaced by fT, the operation is called a periodic convolution of fT and gT.

Discrete convolution

For complex-valued functions f, g defined on the set Z of integers, the discrete convolution of f and g is given by: [9]

or equivalently (see commutativity) by:

The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, 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 of the coefficients of the sequences.

Thus when g has finite support in the set (representing, for instance, a finite impulse response), a finite summation may be used: [10]

Circular discrete convolution

When a function gN is periodic, with period N, then for functions, f, such that fgN exists, the convolution is also periodic and identical to:

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

If gN is a periodic summation of another function, g, then fgN is known as a circular convolution of f and g.

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






The notation (fNg) for cyclic convolution denotes convolution over the cyclic group of integers modulo N.

Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (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 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 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. 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 or the Mersenne transform, [11] 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. [12] 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. [13] 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. [14]

Domain of definition

The convolution of two complex-valued functions on Rd is itself a complex-valued function on Rd, defined by:

is well-defined only if f 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 f. The question of existence thus may involve different conditions on f and g:

Compactly supported functions

If f and g are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous ( Hörmander 1983 , Chapter 1). More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution fg is well-defined and continuous.

Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [a, +∞) (or both supported on [−∞, a]).

Integrable functions

The convolution of f and g exists if f and g are both Lebesgue integrable functions in L1(Rd), and in this case fg is also integrable ( Stein & Weiss 1971 , Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L1, under the discrete convolution, or more generally for the convolution on any group.

Likewise, if fL1(Rd)  and  gLp(Rd)  where 1 ≤ p ≤ ∞,  then  fgLp(Rd),  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 Lp spaces. Specifically, if 1 ≤ p, q, r ≤ ∞ satisfy:


so that the convolution is a continuous bilinear mapping from Lp×Lq to Lr. The Young inequality for convolution is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 < p, q, r < ∞, there exists a constant Bp,q < 1 such that:

The optimal value of Bp,q was discovered in 1975. [15]

A stronger estimate is true provided 1 < p, q, r < ∞ :

where is the weak Lq norm. Convolution also defines a bilinear continuous map for , owing to the weak Young inequality: [16]

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 fg also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the convolution fg. Combined with the fact that convolution commutes with differentiation (see Properties), it follows that the class of Schwartz functions is closed under convolution ( Stein & Weiss 1971 , Theorem 3.3).


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 fg 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 f is a distribution, and g a compactly supported distribution ( Hörmander 1983 , §4.2).


The convolution of any two Borel measures μ and ν of bounded variation is the measure λ defined by ( Rudin 1962 )

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, convolution of measures can be treated with standard methods of functional analysis that may not apply for the convolution of distributions.


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 without identity ( Strichartz 1994 , §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras.


Proof: By definition

Changing the variable of integration to and the result follows.


Proof: This follows from using Fubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).


Proof: This follows from linearity of the integral.

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. 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 for the convolution, S(1), which is defined by

The set of invertible distributions forms an abelian group under the convolution.

Complex conjugation
Relationship with differentiation


Relationship with integration
If and then


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:

This follows from Fubini's theorem. The same result holds if f and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem.


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 f and g is differentiable as many times as f and g are in total.

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

These identities also hold much more broadly in the sense of tempered distributions if one of f 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 Df(n) = f(n + 1) f(n) satisfies an analogous relationship:

Convolution theorem

The convolution theorem states that

where denotes the Fourier transform of , and is a constant that depends on the specific normalization 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.

Translational equivariance

The convolution commutes with translations, meaning that

where τxf is the translation of the function f by x defined by

If f is a Schwartz function, then τxf is the convolution with a translated Dirac delta function τxf = fτxδ. 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

Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of time-invariant systems, and especially LTI system theory. The representing function gS 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 Lp 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

It is not commutative in general. 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 {hk} 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 (multiplicatively written) topological group. If μ and ν are finite Borel measures on G, then their convolution μ∗ν is defined as the pushforward measure of the group action and can be written as

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 on the topological group (R,+), 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.


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 : XX 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


Gaussian blur can be used to obtain a smooth grayscale digital image of a halftone print. Halftone, Gaussian Blur.jpg
Gaussian blur can be used to obtain a smooth grayscale digital image of a halftone print.

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

In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes.
In optics, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is bokeh.
In image processing applications such as adding blurring.
In analytical chemistry, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve signal-to-noise ratio with minimal distortion of the spectra.
In statistics, a weighted moving average is a convolution.
In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal.
In electronic music convolution is the imposition of a spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other. [17]
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, 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 kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian. ( Diggle 1995 ).
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. ( Ghasemi-Nowak 2017 ).

See also


  1. Reasons for the reflection include:

Related Research Articles


Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

Normal distribution probability distribution

In probability theory, the normaldistribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

Dirac delta function pseudo-function δ such that an integral of δ(x-c)f(x) always takes the value of f(c)

In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

Fourier transform mathematical transform that expresses a mathematical function of time as a function of frequency

The Fourier transform (FT) decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest, one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time. In image processing the notion of a time domain is replaced by that of a spatial domain where the intensity of a signal is identified by its spatial position rather than at any point in time.

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain. Versions of the convolution theorem are true for various Fourier-related transforms. Let and be two functions with convolution .

Spectral density Relative importance of certain frequencies in a composite signal

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.

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 operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.

Pontryagin duality theorem

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as , the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function :

Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant system to an arbitrary input signal. Trajectories of these systems are commonly measured and tracked as they move through time, but in applications like image processing and field theory, the LTI systems also have trajectories in spatial dimensions. Thus, these systems are also called linear translation-invariant to give the theory the most general reach. In the case of generic discrete-time systems, linear shift-invariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.

In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S1 = G / AN and G / A = S1 × S1 \ diag S1. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III0 is an ergodic flow on a measure space.

In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.

The circular convolution, also known as cyclic convolution, of two aperiodic functions occurs when one of them is convolved in the normal way with a periodic summation of the other function. That situation arises in the context of the circular convolution theorem. The identical operation can also be expressed in terms of the periodic summations of both functions, if the infinite integration interval is reduced to just one period. That situation arises in the context of the discrete-time Fourier transform (DTFT) and is also called periodic convolution. In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences.

Wigner distribution function

The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis.

A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed cross-terms.

Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time; this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.

In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.

In mathematics, lifting theory was first introduced by John von Neumann in his (1931) pioneering paper, followed later by Dorothy Maharam’s (1958) paper, and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper. Lifting theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph, now a standard reference in the field. Lifting theory continued to develop after 1969, yielding significant new results and applications.


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  3. Dominguez-Torres, p 2
  4. Dominguez-Torres, p 4
  5. R. N. Bracewell (2005), "Early work on imaging theory in radio astronomy", in W. T. Sullivan, The Early Years of Radio Astronomy: Reflections Fifty Years After Jansky's Discovery, Cambridge University Press, p. 172, ISBN   978-0-521-61602-7
  6. John Hilton Grace and Alfred Young (1903), The algebra of invariants, Cambridge University Press, p. 40
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  8. According to [Lothar von Wolfersdorf (2000), "Einige Klassen quadratischer Integralgleichungen", Sitzungsberichte der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-naturwissenschaftliche Klasse, volume 128, number 2, 6–7], the source is Volterra, Vito (1913), "Leçons sur les fonctions de linges". Gauthier-Villars, Paris 1913.
  9. Damelin & Miller 2011 , p. 232
  10. Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1989). Numerical Recipes in Pascal. Cambridge University Press. p. 450. ISBN   0-521-37516-9.
  11. Rader, C.M. (December 1972). "Discrete Convolutions via Mersenne Transforms". IEEE Transactions on Computers. 21 (12): 1269–1273. doi:10.1109/T-C.1972.223497 . Retrieved 17 May 2013.
  12. Madisetti, Vijay K. (1999). "Fast Convolution and Filtering" in the "Digital Signal Processing Handbook" (PDF). CRC Press LLC. p. Section 8. ISBN   9781420045635.
  13. Juang, B.H. "Lecture 21: Block Convolution" (PDF). EECS at the Georgia Institute of Technology. Retrieved 17 May 2013.
  14. Gardner, William G. (November 1994). "Efficient Convolution without Input/Output Delay" (PDF). Audio Engineering Society Convention 97. Paper 3897. Retrieved 17 May 2013.
  15. Beckner, William (1975), "Inequalities in Fourier analysis", Ann. of Math. (2) 102: 159182. Independently, Brascamp, Herm J. and Lieb, Elliott H. (1976), "Best constants in Young's inequality, its converse, and its generalization to more than three functions", Advances in Math. 20: 151173. See Brascamp–Lieb inequality
  16. Reed & Simon 1975 , IX.4
  17. Zölzer, Udo, ed. (2002). DAFX:Digital Audio Effects, p.48–49. ISBN   0471490784.

Further reading