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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 ).
Fourier inversion theorem; Plancherel's theorem; Convolution; Convolution theorem; Positive-definite function; Poisson summation formula; Paley-Wiener theorem; Sobolev space; Time–frequency representation; Quantum Fourier transform
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in L 2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.
There are two common methods used to implement discrete convolution: the definition of convolution and fast Fourier transformation (FFT and IFFT) according to the convolution theorem. To calculate the optical broad-beam response, the impulse response of a pencil beam is convolved with the beam function.
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.
When that interval is [,], the applicable reconstruction formula is the Whittaker–Shannon interpolation formula. This is a cornerstone in the foundation of digital signal processing . Another reason to be interested in S 1 T ( f ) {\displaystyle S_{\tfrac {1}{T}}(f)} is that it often provides insight into the amount of aliasing caused by 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.
Most of the important attributes of the complex DFT, including the inverse transform, the convolution theorem, and most fast Fourier transform (FFT) algorithms, depend only on the property that the kernel of the transform is a principal root of unity. These properties also hold, with identical proofs, over arbitrary rings.