FFT Algorithms

The DFT usually arises as an approximation to the continuous Fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete Fourier transform is a matrix-vector multiplication W~g, and would take O(N) operations for N data-points. The general principle of the Fast Fourier Transform algorithms is to use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, typically reducing the operation count to O(N ∑ fi) if N can be factorized into smaller integers, N = f1f2 . . . fn. This chapter explains the algorithms used in the GSL FFT routines and provides some information on how to extend them. To learn more about the FFT you should read the review article Fast Fourier Transforms: A Tutorial Review and A State of the Art by Duhamel and Vetterli [1]. There are several introductory books on the FFT with example programs, such as The Fast Fourier Transform by Brigham [2] and DFT/FFT and Convolution Algorithms by Burrus and Parks [3]. In 1979 the IEEE published a compendium of carefully-reviewed Fortran FFT programs in Programs for Digital Signal Processing [4] which is a useful reference for implementations of many different FFT algorithms. If you are interested in using DSPs then the Handbook of Real-Time Fast Fourier Transforms [5] provides detailed information on the algorithms and hardware needed to design, build and test DSP applications. Many FFT algorithms rely on results from number theory. These results are covered in the books Fast transforms: algorithms, analyses, applications, by Elliott and Rao [6], Fast Algorithms for Digital Signal Processing by Blahut [7] and Number Theory in Digital Signal Processing by McClellan and Rader [8]. There is also an annotated bibliography of papers on the FFT and related topics by Burrus [9].