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In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in spectroscopy line broadening due to the Doppler effect on its own gives a Gaussian spectral line shape and collision broadening alone gives a Lorentzian line shape.
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 ...
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 leading and trailing edge-effects of circular convolution are overlapped and added, [C] and subsequently discarded. [D]
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.
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).
Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function (,, …,); this is a discrete circular convolution. The formula for the convolution of the functions ():= () is
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.
There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups.The set of complex-valued functions on a finite group, , together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of over the complex numbers, [].