Some applications of generalized FFT's

Abstract

Generalized FFTs are eecient algorithms for computing a Fourier transform of a function deened on nite group, or a bandlimited function de-ned on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now \classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey 21], it may be viewed as an algorithm which eeciently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make eecient data analysis of one sort or another. To cite but a few examples, Gauss was interested in eeciently interpolating the orbits of asteroids 43]; Danielson and Lanczos were concerned with x-ray diiraction 23]; Yates 103] and Good 47] needed the algorithm for statistics; Cooley and Tukey were interested in eecient time series analysis and digital signal processing 21]. For thorough historical overviews see 19, 20, 50]. Recently, there has developed a growing literature related to the construction of algorithms which generalize the FFT from the point of view of the theory of group representations (see e.g., 5, 17, 18, 29, 82]). These sorts of generalizations are \natural" as mathematical constructs, but in point of fact, they too have been motivated by applications. For example, the seemingly earliest construction of \nonabelian" FFTs (due to Willsky) was motivated by the search for new eecient lters 102]. Later constructions have been motivated by applications such as eecient data analysis (cf. 26]) and circuit design (cf. 6]), just to name a few examples. The purpose of this paper is to survey some of the applications of generalized FFTs and thereby (hopefully!) motivate further work in this direction. 1 2 DANIEL N. ROCKMORE One early version of the FFT is due to the statistician Yates. He was interested in the eecient analysis of data from factorial designs. Section 2 reviews this algorithm and then explains in some detail its generalization in the form of eecient computation of spectral analysis for data on a nite group or its quotient. This is illustrated by a brief discussion of one of the more successful applications to date …