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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 particular, the DTFT of the product of two discrete sequences ...
The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution 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 ...
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.
If instead one uses functions on the circle (periodic functions), integration kernels are then biperiodic functions; convolution by functions on the circle yields circular convolution. If one uses functions on the cyclic group of order n (C n or Z/nZ), one obtains n × n matrices as integration kernels; convolution corresponds to circulant ...
10 Notes. 11 Page citations. 12 References. 13 Further reading. ... An important special case is the circular convolution of sequences s and y defined by ...
The cyclotomic fast Fourier transform is a type of fast Fourier transform algorithm over finite fields. [1] This algorithm first decomposes a DFT into several circular convolutions, and then derives the DFT results from the circular convolution results.
where:. DFT N and IDFT N refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and; L is customarily chosen such that N = L+M-1 is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.
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.