Other Fast Algorithms for the FFT - Bellanger, Maurice (Digital Signal Processing, 2024)
Algorithms for the fast calculation of a discrete Fourier transform (DFT) are based on factorization of the matrix of the transform. In order to use these fast algorithms and thus to exploit to the full both the characteristics of the signals to be processed and the various technological possibilities, one must use a suitable mathematical tool – the Kronecker product of matrices. The Kronecker product is a tensor operation which is a generalization of the multiplication of a matrix by a scalar. The Kronecker product can be combined with conventional matrix products. Factorization as Kronecker products forms the basis of algorithms which have various properties. It also applies to partial transforms, which are of great practical importance. In image compression, lapped transforms introduce smoothing and attenuate the so‐called blocking effects. Winograd algorithms generally introduce a reduction in the amount of computation. This reduction can be quite significant when compared to fast Fourier transform algorithms.