Lesser Known FFT Algorithms

Abstract

The introduction of the Cooley-Tukey Fast Fourier transform (C-T FFT) algorithm in 1968 was a critical step in advancing the widespread use of digital com- puters in scientific and technological applications. Initial efforts focused on realizing the potential of the immense reduction in arithmetic complexity afforded by the FFT for computing the finite Fourier transform and convolution. On existing serial butter- fly architectures, this limited implementations of the FFT to transform sizes a power of two. The original C-T FFT algorithms and its many extensions rested mainly on addi- tive structures of the data indexing set and divide-and-conquer strategies for reducing complexity. The relative cost of multiplications as compared with additions on these early machines motivated the study of new algorithms for reducing multiplications usually at the cost of increasing additions. These new algorithms were based on mul- tiplicative structures of the data indexing set. They resulted in greater flexibility as compared to earlier efforts by allowing for fast small prime size computations which could be embedded in a wide collection of large transform sizes. During the 1980s, the increasing importance of RISC and parallel computers re- moved some of the initial motivation for these multiplicative FFTs. Many RISC ar- chitectures featured a hardwired multiply and accumulate which permitted multiplica- tions to be nested in additions. The goal was to arrange the computations so that most multiplications were followed by an addition. Parallel architectures placed the major algorithmic burden on controlling the data flow. The exotic data flow of the multi- plicative FFTs were often incompatible or at least required an immense coding effort for efficient parallel computation. Only the need for flexible transform sizes justified continued efforts. The recent importance of FPGAs and reconfigurable hardware renews the need to reduce multiplication counts. In this talk, we will review some of the work on multiplicative approaches introduced mainly at IBM in the 1970s and 1980s and place these results within the realm of harmonic analysis.