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The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). [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 18 at DTFT § Properties.)
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete 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 discrete-time Fourier transform (DTFT).
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding ...