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In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval. Euclid, detail from The School of Athens by Raphael. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
Partial plot of a function f. ...
The term moving average is used in different contexts. ...
In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...
In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...
Uses
Convolution and related operations are found in many applications of engineering and mathematics. - In statistics, as noted above, a weighted moving average is a convolution.
- also the probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
- In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic term for this is bokeh.
- In acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
- In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information).
- In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac delta function. See LTI system theory and digital signal processing.
- In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is sum of exponential decays from each delta pulse.
- In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance.
A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ...
The term moving average is used in different contexts. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
Table of Opticks, 1728 Cyclopaedia Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ...
In geometry, two sets have the same shape if one can be transformed to another by a combination of translations, rotations and uniform scalings. ...
The astounding bokeh of a Helios-40 lens A photograph of jonquil flowers with background bokeh Compare a photograph of jonquil flowers with low background bokeh Bokeh (from the Japanese boke ã¼ã‘, blur) is a photographic term describing the subjective aesthetic qualities of out-of-focus areas in an...
Acoustics is a branch of physics and is the study of sound, mechanical waves in gases, liquids, and solids. ...
When sound is produced in an enclosed space multiple reflections build up and blend together creating reverberation or reverb. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ...
In mathematics, an invariant is something that does not change under a set of transformations. ...
A linear system is a model of a system based on some kind of linear operator. ...
The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...
In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
Fluorescence spectroscopy or fluorometry is a type of electromagnetic spectroscopy used for analyzing fluorescent spectra. ...
Physics is the science of Nature. ...
A linear system is a model of a system based on some kind of linear operator. ...
In physics, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...
Definition The convolution of f and g is written f * g. 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: In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ...
 The integration range depends on the domain on which the functions are defined. While the symbol t is used above, it needs not represent the time domain. In the case of a finite integration range, f and g are often considered to extend periodically in both directions, so that the term g(t − τ) does not imply a range violation. This use of periodic domains is sometimes called a cyclic, circular or periodic convolution. Of course, extension with zeros is also possible. Using zero-extended or infinite domains is sometimes called a linear convolution, especially in the discrete case below. In mathematics, the domain of a function is the set of all input values to the function. ...
In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...
It has been suggested that this article or section be merged into convolution. ...
If X and Y are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the sum X + Y is given by the convolution f * g. A random variable is a term used in mathematics and statistics. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
For discrete functions, one can use a discrete version of the convolution. It is then given by
 When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, in this sense (using extension with zeros as mentioned above). In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
Generalizing the above cases, the convolution can be defined for any two integrable functions defined on a locally compact topological group. A different generalization is the convolution of distributions. Integrability is a mathematical concept used in different areas. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
Properties The various convolution operators all satisfy the following properties:
 In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
 In mathematics, associativity is a property that a binary operation can have. ...
 In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
 for any real (or complex) number a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...
 where  denotes the derivative of f or, in the discrete case, the difference operator  . In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...
In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ...
Convolution theorem The convolution theorem states that In mathematics, the convolution theorem states that the Fourier transform of a convolution is the point-wise product of Fourier transforms. ...
 where F(f) denotes the Fourier transform of f. Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform and Mellin transform. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...
In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. ...
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ...
Convolutions on groups If G is a suitable group endowed with a measure m (for instance, a locally compact Hausdorff topological group with the Haar measure) and if f and g are real or complex valued m-integrable functions of G, then we can define their convolution by In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, a measure is a function that assigns a number, e. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...
In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...
 In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the Peter-Weyl theorem of Harmonic analysis. It is very difficult to do these calculations without more structure, and Lie groups turn out to be the setting in which these things are done. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
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. ...
Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...
In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...
See also Deconvolution is a process used to reverse the effects of convolution on recorded data. ...
External links - Convolution on PlanetMath
- Convolution, on The Data Analysis BriefBook
- http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet.
- http://www.jhu.edu/~signals/discreteconv2/index.html Visual convolution Java Applet for Discrete Time functions.
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