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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 premise behind the circular convolution approach on multidimensional signals is to develop a relation between the Convolution theorem and the Discrete Fourier transform (DFT) that can be used to calculate the convolution between two finite-extent, discrete-valued signals.
Instead, one can compute a length-(N–1) cyclic convolution exactly by zero-padding it to a length of at least 2(N–1)–1, say to a power of two, which can then be evaluated in O(N log N) time without the recursive application of Rader's algorithm. This algorithm, then, requires O(N) additions plus O(N log N) time for the
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.
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.
The procedure is sometimes referred to as zero-padding, which is a particular implementation used in conjunction with the fast Fourier transform (FFT) algorithm. The inefficiency of performing multiplications and additions with zero-valued "samples" is more than offset by the inherent efficiency of the FFT.
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".
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 ...