Computational frameworks for the fast fourier transform by Charles Van Loan

By Charles Van Loan

The main entire therapy of FFTs to this point. Van mortgage captures the interaction among arithmetic and the layout of potent numerical algorithms--a severe connection as extra complex machines turn into to be had. A stylized Matlab notation, that's frequent to these engaged in high-performance computing, is used. the short Fourier rework (FFT) family members of algorithms has revolutionized many components of medical computation. The FFT is without doubt one of the most generally used algorithms in technological know-how and engineering, with functions in nearly each self-discipline. This quantity is key for pros drawn to linear algebra in addition to these operating with numerical tools. The FFT is usually a superb motor vehicle for educating key points of clinical computing.

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C. Singleton (1968a). "On Computing the Fast Fourier Transform," Comm. ACM 10, 647-654. C. Temes (1977). "Worst Case Error Analysis for the Fast Fourier Transform," IEEE J. Electron. Circuits Syitema 1, 110-115. T. Thong and B. Liu (1977). "Accumulation of Roundoff Errors in Floating Point FFT," IEEE Trana. Circuita and Syatemt CAS-24, 132-143. In many VLSI settings it is necessary to work in fixed-point arithmetic. R. Knight and R. Kaiser (1979). "A Simple Fixed Point Error Bound for the Fast Fourier Transform," IEEE Trana.

S. W. Parks (1985). DFT/FFT and Convolution Algorithms, John Wiley fc Sons, New York. G. J. Lambiotte (1979). "Computing the Fast Fourier Transform on a Vector Computer," Math. Comp. 33, 977-992. H. Stone (1971). "Parallel Processing with the Perfect Shuffle," IEEE Trans. Comput. C-20, 153161. Twelve different bit-reversal algorithms are compared in P. Rosel (1989). "Timing of Some Bit Reversal Algorithms," Signal Processing 18, 425-433. 44 CHAPTER 1. C. Elster (1989). "Fast Bit-Reversal Algorithms," ICASSP '89 Proceeding*, 1099-1102.

2): These can be thought of as interpolation formulae, since A is "in between" A — B and A + B. To derive the method, it is convenient to define Ck = cos(27rk/L) and sk = sm(27rk/L) for k = 0:1+ - 1. 3) that for integers p and j. Suppose L — 64 and that for p — 0, 1, 2, 4, 8, 16, and 32 we have computed (c p,sp). 4. WEIGHT AND BUTTERFLY COMPUTATIONS 27 In general, the weight vector u;32(0:31) is filled in through a sequence of steps which we depict in Fig. 1. The numbers indicate the indices of the cosine/sine pairs that FIG.

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