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# 1. Introduction

This manual documents version 2.1.5 of FFTW, the Fastest Fourier Transform in the West. FFTW is a comprehensive collection of fast C routines for computing the discrete Fourier transform (DFT) in one or more dimensions, of both real and complex data, and of arbitrary input size. FFTW also includes parallel transforms for both shared- and distributed-memory systems. We assume herein that the reader is already familiar with the properties and uses of the DFT that are relevant to her application. Otherwise, see e.g. The Fast Fourier Transform by E. O. Brigham (Prentice-Hall, Englewood Cliffs, NJ, 1974). Our web page also has links to FFT-related information online.

FFTW is usually faster (and sometimes much faster) than all other freely-available Fourier transform programs found on the Net. For transforms whose size is a power of two, it compares favorably with the FFT codes in Sun’s Performance Library and IBM’s ESSL library, which are targeted at specific machines. Moreover, FFTW’s performance is portable. Indeed, FFTW is unique in that it automatically adapts itself to your machine, your cache, the size of your memory, the number of registers, and all the other factors that normally make it impossible to optimize a program for more than one machine. An extensive comparison of FFTW’s performance with that of other Fourier transform codes has been made. The results are available on the Web at the benchFFT home page.

In order to use FFTW effectively, you need to understand one basic concept of FFTW’s internal structure. FFTW does not used a fixed algorithm for computing the transform, but it can adapt the DFT algorithm to details of the underlying hardware in order to achieve best performance. Hence, the computation of the transform is split into two phases. First, FFTW’s planner is called, which “learns” the fastest way to compute the transform on your machine. The planner produces a data structure called a plan that contains this information. Subsequently, the plan is passed to FFTW’s executor, along with an array of input data. The executor computes the actual transform, as dictated by the plan. The plan can be reused as many times as needed. In typical high-performance applications, many transforms of the same size are computed, and consequently a relatively-expensive initialization of this sort is acceptable. On the other hand, if you need a single transform of a given size, the one-time cost of the planner becomes significant. For this case, FFTW provides fast planners based on heuristics or on previously computed plans.

The pattern of planning/execution applies to all four operation modes of FFTW, that is, I) one-dimensional complex transforms (FFTW), II) multi-dimensional complex transforms (FFTWND), III) one-dimensional transforms of real data (RFFTW), IV) multi-dimensional transforms of real data (RFFTWND). Each mode comes with its own planner and executor.

Besides the automatic performance adaptation performed by the planner, it is also possible for advanced users to customize FFTW for their special needs. As distributed, FFTW works most efficiently for arrays whose size can be factored into small primes (2, 3, 5, and 7), and uses a slower general-purpose routine for other factors. FFTW, however, comes with a code generator that can produce fast C programs for any particular array size you may care about. For example, if you need transforms of size 513&nbsp;=&nbsp;19*3<sup>3</sup>, you can customize FFTW to support the factor 19 efficiently.

FFTW can exploit multiple processors if you have them. FFTW comes with a shared-memory implementation on top of POSIX (and similar) threads, as well as a distributed-memory implementation based on MPI. We also provide an experimental parallel implementation written in Cilk, the superior programming tool of choice for discriminating hackers (Olin Shivers). (See the Cilk home page.)

For more information regarding FFTW, see the paper, “The Fastest Fourier Transform in the West,” by M. Frigo and S. G. Johnson, which is the technical report MIT-LCS-TR-728 (Sep. ’97). See also, “FFTW: An Adaptive Software Architecture for the FFT,” by M. Frigo and S. G. Johnson, which appeared in the 23rd International Conference on Acoustics, Speech, and Signal Processing (Proc. ICASSP 1998 3, p. 1381). The code generator is described in the paper “A Fast Fourier Transform Compiler”, by M. Frigo, to appear in the Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), Atlanta, Georgia, May 1999. These papers, along with the latest version of FFTW, the FAQ, benchmarks, and other links, are available at the FFTW home page. The current version of FFTW incorporates many good ideas from the past thirty years of FFT literature. In one way or another, FFTW uses the Cooley-Tukey algorithm, the Prime Factor algorithm, Rader’s algorithm for prime sizes, and the split-radix algorithm (with a variation due to Dan Bernstein). Our code generator also produces new algorithms that we do not yet completely understand. The reader is referred to the cited papers for the appropriate references.

The rest of this manual is organized as follows. We first discuss the sequential (one-processor) implementation. We start by describing the basic features of FFTW in Tutorial. This discussion includes the storage scheme of multi-dimensional arrays (Multi-dimensional Array Format) and FFTW’s mechanisms for storing plans on disk (Words of Wisdom). Next, FFTW Reference provides comprehensive documentation of all FFTW’s features. Parallel transforms are discussed in their own chapter Parallel FFTW. Fortran programmers can also use FFTW, as described in Calling FFTW from Fortran. Installation and Customization explains how to install FFTW in your computer system and how to adapt FFTW to your needs. License and copyright information is given in License and Copyright. Finally, we thank all the people who helped us in Acknowledgments.

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