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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 use of zero-padding for the convolution in Bluestein's algorithm deserves some additional comment. Suppose we zero-pad to a length M ≥ 2N–1. This means that a n is extended to an array A n of length M, where A n = a n for 0 ≤ n < N and A n = 0 otherwise—the usual meaning of "zero-padding".
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.
The following is a pseudocode of the algorithm: (Overlap-add algorithm for linear convolution) h = FIR_filter M = length(h) Nx = length(x) N = 8 × 2^ceiling( log2(M) ) (8 times the smallest power of two bigger than filter length M.
Rader's algorithm (1968), [1] named for Charles M. Rader of MIT Lincoln Laboratory, is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution).
Therefore, the case < is often referred to as zero-padding. Spectral leakage, which increases as L {\displaystyle L} decreases, is detrimental to certain important performance metrics, such as resolution of multiple frequency components and the amount of noise measured by each DTFT sample.
English: Circular convolution can be expedited by the FFT algorithm, so it is often used with an FIR filter to efficiently compute linear convolutions. These graphs illustrate how that is possible. These graphs illustrate how that is possible.
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.