algorithm machine calculation complex fourier von cooley james john tukey (1 Ergebnisse)

Contained in Mathematics of Computation, Vol. 19, No. 90, pp. 297-301. 8vo. pp. 177-364. Original printed wrappers bound in. Black cloth, printed paper spine label. Institutional exlib stamp on printed wrapper cover, ownership marks on printed covers. Fine. FIRST EDITION. This important work on the fast Fourier transform (FFT) algorithm, which is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers. "By far the most common FFT is the Cooley-Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size N = N1N2 into many smaller DFTs of sizes N1 and N2, along with O(N) multiplications by complex roots of unity, traditionally called twiddle factors (after Gentleman and Sande, 1966). This method (and the general idea of an FFT) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). The most well-known use of the Cooley-Tukey algorithm is to divide the transform into two pieces of size N / 2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey). These are called the radix-2 and mixed-radix cases, respectively (and other variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT." [Wikip.]. Norman, Origins of Cyberspace 548.