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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 mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (and ) that produces a third function (). The term convolution refers to both the resulting function and to the process of computing it.
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.
As seen above, the discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. It has been shown [9] [10] that any linear transform that turns convolution into pointwise product is the DFT up to a permutation of coefficients ...
In mathematics, the discrete-time Fourier transform (DTFT) ... An important special case is the circular convolution of sequences s and y defined by ...
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .
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).
O&S (1999) uses both Periodic convolution and Circular convolution for a summation over N terms of the product of two sequences. The only distinction is whether: (1) the sequences are N-periodic (infinitely long), or (2) they are just one period of each sequence, with one of the sequences addressed by modulo N indexing for the n-sample offsets.