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In
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 defined o ...
), 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 (
) 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 wit ...
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). The integral is evaluated for all values of shift, producing the convolution function.
Some features of convolution are similar to
cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (
) differs from cross-correlation (
) 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
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
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 speakin ...
,
statistics,
acoustics,
spectroscopy,
signal processing and
image processing,
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 r ...
,
computer vision and
differential equations.
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 Euclidean ...
and other
groups (as
algebraic structures). For example,
periodic functions, such as the
discrete-time Fourier transform, 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 con ...
and convolved by
periodic convolution. (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, 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 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 ...
:
:
An equivalent definition is (see
commutativity):
:
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
-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:
:
For the multi-dimensional formulation of convolution, see ''#Domain of definition">domain of definition
In mathematics, a partial function from a Set (mathematics), set to a set is a function from a subset of (possibly itself) to . The subset , that is, the Domain of a function, domain of viewed as a function, is called the domain of defini ...
'' (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 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 on page 505 of his book entitled ''Treatise on differences and series'', which is the last of 3 volumes of the encyclopedic series: ''Traité du calcul différentiel et du calcul intégral'', Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of
Pierre Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
,
Jean-Baptiste Joseph Fourier,
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), ''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) 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 example ...
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 (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 (; ).
requires arithmetic operations per output value and operations for 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( log ) complexity.
The most common fast convolution algorithms use
fast Fourier transform (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 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 In signal processing, ''overlap–save'' is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x /math> and a finite impulse response (FIR) filter h /math>:
where for ''m'' outside the regio ...
and
overlap–add method
In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal x /math> with a finite impulse response (FIR) filter h /math>:
where for ''m'' outside the region .
This article uses c ...
.
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
continuous functions, 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 In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
, 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 functions 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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
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 measures ''μ'' and ''ν'' of
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
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 of
A.
This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L
1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.
The convolution of measures also satisfies the following version of Young's inequality
:
\, \mu* \nu\, \le \, \mu\, \, \nu\,
where the norm is the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
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
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 defined o ...
that may not apply for the convolution of distributions.
Properties
Algebraic properties
The convolution defines a product on the
linear space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 . Other linear spaces of functions, such as the space of continuous functions of compact support, are
closed under the convolution, and so also form commutative associative algebras.
;
Commutativity:
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: No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a
delta distribution (a unitary impulse, centered at zero) or, at the very least (as is the case of ''L''
1) admit
approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,
f * \delta = f where ''δ'' is the delta distribution.
; Inverse element: Some distributions ''S'' have an
inverse element
In 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:
:
\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 (L
1) weak derivative, as a consequence of
Young's convolution inequality. For instance, when ''f'' is continuously differentiable with compact support, and ''g'' is an arbitrary locally integrable function,
:
\frac(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
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
''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 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,
two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin t ...
,
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,
\circ denotes
Hadamard product (this result is an evolving of
count sketch
Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms.
It was invented by
Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by ...
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, then ''τ
xf'' is the convolution with a translated Dirac delta function ''τ''
''x''''f'' = ''f'' ∗ ''τ''
''x'' ''δ''. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.
Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds
: Suppose that ''S'' is a bounded
linear operator acting on functions which commutes with translations: ''S''(''τ
xf'') = ''τ
x''(''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 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. 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
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 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 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 T with the Lebesgue measure is an immediate example. For a fixed ''g'' in ''L''
1(T), we have the following familiar operator acting on the
Hilbert space ''L''
2(T):
:
T (x) = \frac \int_ (y) g( x - y) \, dy.
The operator ''T'' is
compact. A direct calculation shows that its adjoint ''T* '' is convolution with
:
\bar(-y).
By the commutativity property cited above, ''T'' is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
: ''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 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
characters of T. Each convolution is a compact
multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above.
A discrete example is a finite
cyclic group
In group theory, 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.
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 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 In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
Definition
Given meas ...
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
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
satisfies
:
\, \mu * \nu\, \le \left\, \mu\right\, \left\, \nu\right\, .
In the case when ''G'' is
locally compact with (left-)
Haar measure λ, and μ and ν are
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 measures on the topological group then the convolution ''μ''∗''ν'' is the
probability distribution 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
* Independ ...
random variables ''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 functions
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:
\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 algebras . A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism ''S'' such that
:
S * \operatorname_X = \operatorname_X * S = \eta\circ\varepsilon.
Applications
Convolution and related operations are found in many applications in science, engineering and mathematics.
* In
image processing
** 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 image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an image.
Details
The general ...
)
** 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 with minimal distortion of the spectra
** In
statistics, 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,
reverberation is the convolution of the original sound with
echoes from objects surrounding the sound source.
** In digital signal processing, convolution is used to map the
impulse response of a real room on a digital audio signal.
** In
electronic music
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 ( electroa ...
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.
* 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
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 r ...
, wherever there is a
linear system with a "
superposition principle", a convolution operation makes an appearance. For instance, in
spectroscopy line broadening due to the Doppler effect on its own gives a
Gaussian spectral line shape Spectral line shape describes the form of a feature, observed in spectroscopy, corresponding to an energy change in an atom, molecule or ion. This shape is also referred to as the spectral line profile. Ideal line shapes include Lorentzian, Gaussi ...
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
The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy distribution, Cauchy-Lorentz distribution and a Normal distribution, Gaussian distribution. It is often used in analyzing data from spe ...
.
** 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 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
* Independ ...
random variables 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 r ...
. Though these are actually cross-correlations rather than convolutions in most cases.
* In
Smoothed-particle hydrodynamics, simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle
i, some physical quantity
A_i is calculated as a convolution of
A_j with a weighting function, where
j denotes the neighbors of particle
i: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.
See also
*
Analog signal processing Analog signal processing is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete digital signal processing where the signal processing is carried out by a digital process). "Analog" indi ...
*
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
*
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
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
*
Generalized signal averaging
*
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
*
Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where each row is a shifted copy of the convolution kernel)
Notes
References
Further reading
* .
*
*
* Dominguez-Torres, Alejandro (Nov 2, 2010). "Origin and history of convolution". 41 pgs. http://www.slideshare.net/Alexdfar/origin-adn-history-of-convolution. Cranfield, Bedford MK43 OAL, UK. Retrieved Mar 13, 2013.
*
*
* .
* .
* .
* .
* .
*
*
* .
*
* .
* .
* .
* .
*
*
* .
External links
Earliest Uses: The entry on Convolution has some historical information. o
* http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet
* http://www.jhu.edu/~signals/discreteconv2/index.html Visual convolution Java Applet for discrete-time functions
* https://get-the-solution.net/projects/discret-convolution discret-convolution online calculator
*https://lpsa.swarthmore.edu/Convolution/CI.html Convolution demo and visualization in javascript
*https://phiresky.github.io/convolution-demo/ Another convolution demo in javascript
Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution. by Alan Peters
* * https://archive.org/details/Lectures_on_Image_Processing
Convolution Kernel Mask Operation Interactive tutorialat
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 convolutiongiven by
Salman Khan
Example of FFT convolution for pattern-recognition (image processing)Intuitive Guide to ConvolutionA blogpost about an intuitive interpretation of convolution.
{{Differentiable computing
Functional analysis
Image processing
Fourier analysis
Bilinear maps
Feature detection (computer vision)