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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).
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in L 2 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.
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.
1.1.2.1 Using the convolution theorem. 1.1.3 Geometric proof. 2 Correlated random variables. Toggle Correlated random variables subsection. 2.1 Proof. 3 References. 4 ...
The Poisson summation formula is a particular case of the convolution theorem on tempered distributions. If one of the two factors is the Dirac comb, one obtains periodic summation on one side and sampling on the other side of the equation.
Then many of the values of the circular convolution are identical to values of x∗h, which is actually the desired result when the h sequence is a finite impulse response (FIR) filter. Furthermore, the circular convolution is very efficient to compute, using a fast Fourier transform (FFT) algorithm and the circular convolution theorem.
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.
in which form it is clearly recognizable as an umbral variant of the binomial theorem (for more on umbral variants of the binomial theorem, see binomial type). The Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that