Computational Frameworks for the Fast Fourier Transform

The front matter includes the Kronecker product properties, title, series, copyright, dedication, TOC, preface, and preliminary remarks.

A fast Fourier transform (FFT) is a quick method for forming the matrix-vector product Fnx, where Fn is the discrete Fourier transform (DFT) matrix. Our examination of this area begins in the simplest setting: the case when n=2t. This permits the orderly repetition of the central divide-and-conquer process that underlies all FFT work. Our approach is based upon the factorization of Fn into the product of t=log2 n sparse matrix factors. Different factorizations correspond to different FFT frameworks. Within each framework different implementations are possible.

To navigate this hierarchy of ideas, we rely heavily upon block matrix notation, which we detail in §1.1. This “language” revolves around the Kronecker product and is used in §1.2 to establish the “radix-2 splitting,” a factorization that indicates how a 2m-point DFT can be rapidly determined from a pair of m-point DFTs. The repeated application of this process leads to our first complete FFT procedure, the famous Cooley—Tukey algorithm in §1.3.

The frameworks of Chapter 1 revolve around the efficient synthesis of an evenorder DFT from two half-length DFTs. However, if n is not a power of two, then the radix-2 divide-and-conquer process breaks down. The mission of this section is to show that it is possible to formulate FFT frameworks for general n, as long as n is highly composite. In some important cases, the resulting algorithms involve fewer flops and better memory traffic patterns than their radix-2 counterparts.

The first three sections cover the underlying mathematics and generalize the factorization notions developed in Chapter 1. As a case study, we detail the central calculations associated with radix-4 and radix-8 frameworks in §2.4. We conclude in §2.5 with a discussion of the split-radix algorithm, an interesting reduced-arithmetic rearrangement of the radix-2 Cooley—Tukey algorithm.

Suppose X∈ℂn1×n2. The computations X←Fn1X and X←XFn2 are referred to as multiple DFT problems. Applying an FFT algorithm to the columns or rows of a matrix is an important problem in many applications. However, there is more to the problem than just the repeated application of a single-vector technique, as we show in §3.1.

Surprisingly, we have not yet exhausted the set of all possible factorizations of the DFT matrix Fn. In fact, a case can be made that we are but halfway along in our survey, for until now we have neglected the prime factor FFTs. This framework is based upon a number-theoretic splitting of the DFT matrix and a large fraction of the FFT literature is devoted to its implementation and various properties.

Elsewhere in this final chapter we examine two leading application areas that rely heavily upon FFTs: convolution and the numerical solution of the Poisson equation. Methods for one- and two-dimensional convolution are given in §4.2. The duality between fast convolution and FFTs is established and some new FFT frameworks emerge. Fast methods for the product of a Toeplitz matrix and a vector are also discussed.

A real-data FFT framework is derived in §4.3 and comparisons are made with the rival Hartley transform. By exploiting structure, it is possible to halve the amount of required arithmetic. Fast algorithms for various sine and cosine transforms are given in §4.4 and then used in §4.5 to solve the Poisson equation problem. The trigonometric transforms that we develop have many applications and are a fitting tribute to the FFT and its wide applicability.

The back matter includes bibliography and index.