Fast Algorithms of Fourier and Hartleytransform and Their Implementation In

This paper is mainly intended as survey article on construction of fast algorithms for the computation of the discrete Fourier transform (DFT) and discrete Hartley transform (DHT), all in relation to discrete linear and cyclic convolution which are fundamental operations in many data processing tasks. The exposition prefers purely algebraic approach to explain the basic ideas in concise but clear manner. The beneets of author's new algebraic setting of generalized Kronecker product are demonstrated in deriving fast algorithms of Cooley-Tukey type for the computation of multidimensional fast Fourier and Hartley transform. These algorithms have been implemented as FORTRAN MEX-les in MATLAB which makes it easy to use them and evaluate their performance. Compared with other commonly used procedures the results of performance tests exhibit equal or better numerical stability and for most larger transform lengths time eeciency superior to that of the comparative procedures. The results concerning the new algorithm for fast Hartley transform are stated without proofs and will be published elsewhere in more detail. 1. Introduction Discrete Fourier transform (DFT), its alternative discrete Hartley transform (DHT) and discrete convolution (DC) play fundamental role in many elds of mathematics and applied sciences, the traditional one being digital signal processing. This is evidenced by a large number of monographs devoted to this topic DC stands behind linear techniques for data processing which are widely used under synonyms moving average method or linear digital ltration. The basic idea is to achieve a desired modiication of the data sequence simply by replacing each entry with a weighted average of values in its neighbourhood. The weights remain xed and \move" along the data sequence. DC is closely connected with DFT, actually either of DC or DFT may be computed via the latter operation. DFT and DHT themselves are a useful tool for nding an approximate Fourier expansion of the data sequence and play a fundamental role of their own in the spectral representation of signals. Thus the problem of fast computation is mainly reduced to the problem of nding fast algorithm for