Introduction

Although there are a wide range of fast Fourier transform (FFT) algorithms, involving a wealth of mathematics from number theory to polynomial algebras, the vast majority of FFT implementations in practice employ some variation on the Cooley-Tukey algorithm [9]. The Cooley-Tukey algorithm can be derived in two or three lines of elementary algebra. It can be implemented almost as easily, especially if only power-of-two sizes are desired; numerous popular textbooks list short FFT subroutines for power-of-two sizes, written in the language du jour. The implementation of the Cooley-Tukey algorithm, at least, would therefore seem to be a long-solved problem. In this chapter, however, we will argue that matters are not as straightforward as they might appear.

For many years, the primary route to improving upon the Cooley-Tukey FFT seemed to be reductions in the count of arithmetic operations, which often dominated the execution time prior to the ubiquity of fast floating-point hardware (at least on non-embedded processors). Therefore, great effort was expended towards finding new algorithms with reduced arithmetic counts [13], from Winograd's method to achieve Θ(n)Θ(n) multiplications1 (at the cost of many more additions) [54], [22], [12], [13] to the split-radix variant on Cooley-Tukey that long achieved the lowest known total count of additions and multiplications for power-of-two sizes [56], [10], [53], [32], [13] (but was recently improved upon [27], [31]). The question of the minimum possible arithmetic count continues to be of fundamental theoretical interest—it is not even known whether better than Θ(nlogn)Θ(nlogn) complexity is possible, since Ω(nlogn)Ω(nlogn) lower bounds on the count of additions have only been proven subject to restrictive assumptions about the algorithms [33], [36], [37]. Nevertheless, the difference in the number of arithmetic operations, for power-of-two sizes nn, between the 1965 radix-2 Cooley-Tukey algorithm (∼5nlog2n∼5nlog2n [9]) and the currently lowest-known arithmetic count (∼349nlog2n∼349nlog2n [27], [31]) remains only about 25%.

Figure 1: The ratio of speed (1/time) between a highly optimized FFT (FFTW 3.1.2 [15], [16]) and a typical textbook radix-2 implementation (Numerical Recipes in C [38]) on a 3 GHz Intel Core Duo with the Intel C compiler 9.1.043, for single-precision complex-data DFTs of size nn, plotted versus log2nlog2n. Top line (squares) shows FFTW with SSE SIMD instructions enabled, which perform multiple arithmetic operations at once (see section ); bottom line (circles) shows FFTW with SSE disabled, which thus requires a similar number of arithmetic instructions to the textbook code. (This is not intended as a criticism of Numerical Recipes—simple radix-2 implementations are reasonable for pedagogy—but it illustrates the radical differences between straightforward and optimized implementations of FFT algorithms, even with similar arithmetic costs.) For n≳219n≳219, the ratio increases because the textbook code becomes much slower (this happens when the DFT size exceeds the level-2 cache).

And yet there is a vast gap between this basic mathematical theory and the actual practice—highly optimized FFT packages are often an order of magnitude faster than the textbook subroutines, and the internal structure to achieve this performance is radically different from the typical textbook presentation of the “same” Cooley-Tukey algorithm. For example, Figure 1 plots the ratio of benchmark speeds between a highly optimized FFT [15], [16] and a typical textbook radix-2 implementation [38], and the former is faster by a factor of 5–40 (with a larger ratio as nn grows). Here, we will consider some of the reasons for this discrepancy, and some techniques that can be used to address the difficulties faced by a practical high-performance FFT implementation.2

In particular, in this chapter we will discuss some of the lessons learned and the strategies adopted in the FFTW library. FFTW [15], [16] is a widely used free-software library that computes the discrete Fourier transform (DFT) and its various special cases. Its performance is competitive even with manufacturer-optimized programs [16], and this performance is portable thanks the structure of the algorithms employed, self-optimization techniques, and highly optimized kernels (FFTW's codelets) generated by a special-purpose compiler.

This chapter is structured as follows. First "Review of the Cooley-Tukey FFT", we briefly review the basic ideas behind the Cooley-Tukey algorithm and define some common terminology, especially focusing on the many degrees of freedom that the abstract algorithm allows to implementations. Next, in "Goals and Background of the FFTW Project", we provide some context for FFTW's development and stress that performance, while it receives the most publicity, is not necessarily the most important consideration in the implementation of a library of this sort. Third, in "FFTs and the Memory Hierarchy", we consider a basic theoretical model of the computer memory hierarchy and its impact on FFT algorithm choices: quite general considerations push implementations towards large radices and explicitly recursive structure. Unfortunately, general considerations are not sufficient in themselves, so we will explain in "Adaptive Composition of FFT Algorithms" how FFTW self-optimizes for particular machines by selecting its algorithm at runtime from a composition of simple algorithmic steps. Furthermore, "Generating Small FFT Kernels" describes the utility and the principles of automatic code generation used to produce the highly optimized building blocks of this composition, FFTW's codelets. Finally, we will briefly consider an important non-performance issue, in "Numerical Accuracy in FFTs".

Review of the Cooley-Tukey FFT

The (forward, one-dimensional) discrete Fourier transform (DFT) of an array XX of nn complex numbers is the array YY given by

where 0≤k<n0≤k<n and ωn=exp(-2πi/n)ωn=exp(-2πi/n) is a primitive root of unity. Implemented directly, Equation 1 would require Θ(n2)Θ(n2) operations; fast Fourier transforms are O(nlogn)O(nlogn) algorithms to compute the same result. The most important FFT (and the one primarily used in FFTW) is known as the “Cooley-Tukey” algorithm, after the two authors who rediscovered and popularized it in 1965 [9], although it had been previously known as early as 1805 by Gauss as well as by later re-inventors [24]. The basic idea behind this FFT is that a DFT of a composite size n=n1n2n=n1n2 can be re-expressed in terms of smaller DFTs of sizes n1n1 and n2n2—essentially, as a two-dimensional DFT of size n1×n2n1×n2 where the output is transposed. The choices of factorizations of nn, combined with the many different ways to implement the data re-orderings of the transpositions, have led to numerous implementation strategies for the Cooley-Tukey FFT, with many variants distinguished by their own names [13], [52]. FFTW implements a space of many such variants, as described in "Adaptive Composition of FFT Algorithms", but here we derive the basic algorithm, identify its key features, and outline some important historical variations and their relation to FFTW.

The Cooley-Tukey algorithm can be derived as follows. If nn can be factored into n=n1n2n=n1n2, Equation 1 can be rewritten by letting ℓ=ℓ1n2+ℓ2ℓ=ℓ1n2+ℓ2 and k=k1+k2n1k=k1+k2n1. We then have:

where k1,2=0,...,n1,2-1k1,2=0,...,n1,2-1. Thus, the algorithm computes n2n2 DFTs of size n1n1 (the inner sum), multiplies the result by the so-called [21] twiddle factors ωnℓ2k1ωnℓ2k1, and finally computes n1n1 DFTs of size n2n2 (the outer sum). This decomposition is then continued recursively. The literature uses the term radix to describe an n1n1 or n2n2 that is bounded (often constant); the small DFT of the radix is traditionally called a butterfly.

Many well-known variations are distinguished by the radix alone. A decimation in time (DIT) algorithm uses n2n2 as the radix, while a decimation in frequency (DIF) algorithm uses n1n1 as the radix. If multiple radices are used, e.g. for nn composite but not a prime power, the algorithm is called mixed radix. A peculiar blending of radix 2 and 4 is called split radix, which was proposed to minimize the count of arithmetic operations [56], [10], [53], [32], [13] although it has been superseded in this regard [27], [31]. FFTW implements both DIT and DIF, is mixed-radix with radices that are adapted to the hardware, and often uses much larger radices (e.g. radix 32) than were once common. On the other end of the scale, a “radix” of roughly nn has been called a four-step FFT algorithm (or six-step, depending on how many transposes one performs) [2]; see "FFTs and the Memory Hierarchy" for some theoretical and practical discussion of this algorithm.

A key difficulty in implementing the Cooley-Tukey FFT is that the n1n1 dimension corresponds to discontiguous inputs ℓ1ℓ1 in XX but contiguous outputs k1k1 in YY, and vice-versa for n2n2. This is a matrix transpose for a single decomposition stage, and the composition of all such transpositions is a (mixed-base) digit-reversal permutation (or bit-reversal, for radix 2). The resulting necessity of discontiguous memory access and data re-ordering hinders efficient use of hierarchical memory architectures (e.g., caches), so that the optimal execution order of an FFT for given hardware is non-obvious, and various approaches have been proposed.

Figure 2: Schematic of traditional breadth-first (left) vs. recursive depth-first (right) ordering for radix-2 FFT of size 8: the computations for each nested box are completed before doing anything else in the surrounding box. Breadth-first computation performs all butterflies of a given size at once, while depth-first computation completes one subtransform entirely before moving on to the next (as in the algorithm below).

One ordering distinction is between recursion and iteration. As expressed above, the Cooley-Tukey algorithm could be thought of as defining a tree of smaller and smaller DFTs, as depicted in Figure 2; for example, a textbook radix-2 algorithm would divide size nn into two transforms of size n/2n/2, which are divided into four transforms of size n/4n/4, and so on until a base case is reached (in principle, size 1). This might naturally suggest a recursive implementation in which the tree is traversed “depth-first” as in Figure 2(right) and the algorithm of Code 1—one size n/2n/2 transform is solved completely before processing the other one, and so on. However, most traditional FFT implementations are non-recursive (with rare exceptions [46]) and traverse the tree “breadth-first” [52] as in Figure 2(left)—in the radix-2 example, they would perform nn (trivial) size-1 transforms, then n/2n/2 combinations into size-2 transforms, then n/4n/4 combinations into size-4 transforms, and so on, thus making log2nlog2n passes over the whole array. In contrast, as we discuss in "Discussion", FFTW employs an explicitly recursive strategy that encompasses both depth-first and breadth-first styles, favoring the former since it has some theoretical and practical advantages as discussed in "FFTs and the Memory Hierarchy".

Y[0,...,n-1]←recfft 2(n,X,ι)Y[0,...,n-1]←recfft 2(n,X,ι):
IF n=1 THEN
     Y[0]←X[0]Y[0]←X[0]
ELSE
     Y[0,...,n/2-1]←recfft2(n/2,X,2ι)Y[0,...,n/2-1]←recfft2(n/2,X,2ι)
     Y[n/2,...,n-1]←recfft2(n/2,X+ι,2ι)Y[n/2,...,n-1]←recfft2(n/2,X+ι,2ι)
     FOR k1=0k1=0 TO (n/2)-1(n/2)-1 DO
          t←Y[k1]t←Y[k1]
          Y[k1]←t+ωnk1Y[k1+n/2]Y[k1]←t+ωnk1Y[k1+n/2]
          Y[k1+n/2]←t-ωnk1Y[k1+n/2]Y[k1+n/2]←t-ωnk1Y[k1+n/2]
     END FOR
END IF

A second ordering distinction lies in how the digit-reversal is performed. The classic approach is a single, separate digit-reversal pass following or preceding the arithmetic computations; this approach is so common and so deeply embedded into FFT lore that many practitioners find it difficult to imagine an FFT without an explicit bit-reversal stage. Although this pass requires only O(n)O(n) time [29], it can still be non-negligible, especially if the data is out-of-cache; moreover, it neglects the possibility that data reordering during the transform may improve memory locality. Perhaps the oldest alternative is the Stockham auto-sort FFT [47], [52], which transforms back and forth between two arrays with each butterfly, transposing one digit each time, and was popular to improve contiguity of access for vector computers [48]. Alternatively, an explicitly recursive style, as in FFTW, performs the digit-reversal implicitly at the “leaves” of its computation when operating out-of-place (see section "Discussion"). A simple example of this style, which computes in-order output using an out-of-place radix-2 FFT without explicit bit-reversal, is shown in the algorithm of Code 1 [corresponding to Figure 2(right)]. To operate in-place with O(1)O(1) scratch storage, one can interleave small matrix transpositions with the butterflies [26], [49], [40], [23], and a related strategy in FFTW [16] is briefly described by "Discussion".

Finally, we should mention that there are many FFTs entirely distinct from Cooley-Tukey. Three notable such algorithms are the prime-factor algorithm for gcd(n1,n2)=1gcd(n1,n2)=1 [35], along with Rader's [41] and Bluestein's [4], [43], [35] algorithms for prime nn. FFTW implements the first two in its codelet generator for hard-coded nn "Generating Small FFT Kernels" and the latter two for general prime nn (sections "Plans for prime sizes" and "Goals and Background of the FFTW Project"). There is also the Winograd FFT [54], [22], [12], [13], which minimizes the number of multiplications at the expense of a large number of additions; this trade-off is not beneficial on current processors that have specialized hardware multipliers.

Goals and Background of the FFTW Project

The FFTW project, begun in 1997 as a side project of the authors Frigo and Johnson as graduate students at MIT, has gone through several major revisions, and as of 2008 consists of more than 40,000 lines of code. It is difficult to measure the popularity of a free-software package, but (as of 2008) FFTW has been cited in over 500 academic papers, is used in hundreds of shipping free and proprietary software packages, and the authors have received over 10,000 emails from users of the software. Most of this chapter focuses on performance of FFT implementations, but FFTW would probably not be where it is today if that were the only consideration in its design. One of the key factors in FFTW's success seems to have been its flexibility in addition to its performance. In fact, FFTW is probably the most flexible DFT library available:

Our design philosophy has been to first define the most general reasonable functionality, and then to obtain the highest possible performance without sacrificing this generality. In this section, we offer a few thoughts about why such flexibility has proved important, and how it came about that FFTW was designed in this way.

FFTW's generality is partly a consequence of the fact the FFTW project was started in response to the needs of a real application for one of the authors (a spectral solver for Maxwell's equations [28]), which from the beginning had to run on heterogeneous hardware. Our initial application required multi-dimensional DFTs of three-component vector fields (magnetic fields in electromagnetism), and so right away this meant: (i) multi-dimensional FFTs; (ii) user-accessible loops of FFTs of discontiguous data; (iii) efficient support for non-power-of-two sizes (the factor of eight difference between n×n×nn×n×n and 2n×2n×2n2n×2n×2n was too much to tolerate); and (iv) saving a factor of two for the common real-input case was desirable. That is, the initial requirements already encompassed most of the features above, and nothing about this application is particularly unusual.

Even for one-dimensional DFTs, there is a common misperception that one should always choose power-of-two sizes if one cares about efficiency. Thanks to FFTW's code generator (described in "Generating Small FFT Kernels"), we could afford to devote equal optimization effort to any nn with small factors (2, 3, 5, and 7 are good), instead of mostly optimizing powers of two like many high-performance FFTs. As a result, to pick a typical example on the 3 GHz Core Duo processor of Figure 1, n=3600=24·32·52n=3600=24·32·52 and n=3840=28·3·5n=3840=28·3·5 both execute faster than n=4096=212n=4096=212. (And if there are factors one particularly cares about, one can generate code for them too.)

One initially missing feature was efficient support for large prime sizes; the conventional wisdom was that large-prime algorithms were mainly of academic interest, since in real applications (including ours) one has enough freedom to choose a highly composite transform size. However, the prime-size algorithms are fascinating, so we implemented Rader's O(nlogn)O(nlogn) prime-nn algorithm [41] purely for fun, including it in FFTW 2.0 (released in 1998) as a bonus feature. The response was astonishingly positive—even though users are (probably) never forced by their application to compute a prime-size DFT, it is rather inconvenient to always worry that collecting an unlucky number of data points will slow down one's analysis by a factor of a million. The prime-size algorithms are certainly slower than algorithms for nearby composite sizes, but in interactive data-analysis situations the difference between 1 ms and 10 ms means little, while educating users to avoid large prime factors is hard.

Another form of flexibility that deserves comment has to do with a purely technical aspect of computer software. FFTW's implementation involves some unusual language choices internally (the FFT-kernel generator, described in "Generating Small FFT Kernels", is written in Objective Caml, a functional language especially suited for compiler-like programs), but its user-callable interface is purely in C with lowest-common-denominator datatypes (arrays of floating-point values). The advantage of this is that FFTW can be (and has been) called from almost any other programming language, from Java to Perl to Fortran 77. Similar lowest-common-denominator interfaces are apparent in many other popular numerical libraries, such as LAPACK [1]. Language preferences arouse strong feelings, but this technical constraint means that modern programming dialects are best hidden from view for a numerical library.

Ultimately, very few scientific-computing applications should have performance as their top priority. Flexibility is often far more important, because one wants to be limited only by one's imagination, rather than by one's software, in the kinds of problems that can be studied.

FFTs and the Memory Hierarchy

There are many complexities of computer architectures that impact the optimization of FFT implementations, but one of the most pervasive is the memory hierarchy. On any modern general-purpose computer, memory is arranged into a hierarchy of storage devices with increasing size and decreasing speed: the fastest and smallest memory being the CPU registers, then two or three levels of cache, then the main-memory RAM, then external storage such as hard disks.3 Most of these levels are managed automatically by the hardware to hold the most-recently-used data from the next level in the hierarchy.4 There are many complications, however, such as limited cache associativity (which means that certain locations in memory cannot be cached simultaneously) and cache lines (which optimize the cache for contiguous memory access), which are reviewed in numerous textbooks on computer architectures. In this section, we focus on the simplest abstract principles of memory hierarchies in order to grasp their fundamental impact on FFTs.

Because access to memory is in many cases the slowest part of the computer, especially compared to arithmetic, one wishes to load as much data as possible in to the faster levels of the hierarchy, and then perform as much computation as possible before going back to the slower memory devices. This is called temporal locality: if a given datum is used more than once, we arrange the computation so that these usages occur as close together as possible in time.

Adaptive Composition of FFT Algorithms

As alluded to several times already, FFTW implements a wide variety of FFT algorithms (mostly rearrangements of Cooley-Tukey) and selects the “best” algorithm for a given nn automatically. In this section, we describe how such self-optimization is implemented, and especially how FFTW's algorithms are structured as a composition of algorithmic fragments. These techniques in FFTW are described in greater detail elsewhere [16], so here we will focus only on the essential ideas and the motivations behind them.

An FFT algorithm in FFTW is a composition of algorithmic steps called a plan. The algorithmic steps each solve a certain class of problems (either solving the problem directly or recursively breaking it into sub-problems of the same type). The choice of plan for a given problem is determined by a planner that selects a composition of steps, either by runtime measurements to pick the fastest algorithm, or by heuristics, or by loading a pre-computed plan. These three pieces: problems, algorithmic steps, and the planner, are discussed in the following subsections.

The space of plans in FFTW

Here, we describe a subset of the possible plans considered by FFTW; while not exhaustive [16], this subset is enough to illustrate the basic structure of FFTW and the necessity of including the vector loop(s) in the problem definition to enable several interesting algorithms. The plans that we now describe usually perform some simple “atomic” operation, and it may not be apparent how these operations fit together to actually compute DFTs, or why certain operations are useful at all. We shall discuss those matters in "Discussion".

Roughly speaking, to solve a general DFT problem, one must perform three tasks. First, one must reduce a problem of arbitrary vector rank to a set of loops nested around a problem of vector rank 0, i.e., a single (possibly multi-dimensional) DFT. Second, one must reduce the multi-dimensional DFT to a sequence of of rank-1 problems, i.e., one-dimensional DFTs; for simplicity, however, we do not consider multi-dimensional DFTs below. Third, one must solve the rank-1, vector rank-0 problem by means of some DFT algorithm such as Cooley-Tukey. These three steps need not be executed in the stated order, however, and in fact, almost every permutation and interleaving of these three steps leads to a correct DFT plan. The choice of the set of plans explored by the planner is critical for the usability of the FFTW system: the set must be large enough to contain the fastest possible plans, but it must be small enough to keep the planning time acceptable.

Rank-0 plans

The rank-0 problem dft(,V,I,O)dft(,V,I,O) denotes a permutation of the input array into the output array. FFTW does not solve arbitrary rank-0 problems, only the following two special cases that arise in practice.

• When V=1V=1 and I≠OI≠O, FFTW produces a plan that copies the input array into the output array. Depending on the strides, the plan consists of a loop or, possibly, of a call to the ANSI C function memcpy, which is specialized to copy contiguous regions of memory.
• When V=2V=2, I=OI=O, and the strides denote a matrix-transposition problem, FFTW creates a plan that transposes the array in-place. FFTW implements the square transposition dft(,(n,ι,o),(n,o,ι),I,O)dft(,(n,ι,o),(n,o,ι),I,O) by means of the cache-oblivious algorithm from [19], which is fast and, in theory, uses the cache optimally regardless of the cache size (using principles similar to those described in the section "FFTs and the Memory Hierarchy"). A generalization of this idea is employed for non-square transpositions with a large common factor or a small difference between the dimensions, adapting algorithms from [11].

Rank-1 plans

Rank-1 DFT problems denote ordinary one-dimensional Fourier transforms. FFTW deals with most rank-1 problems as follows.

Direct plans

When the DFT rank-1 problem is “small enough” (usually, n≤64n≤64), FFTW produces a direct plan that solves the problem directly. These plans operate by calling a fragment of C code (a codelet) specialized to solve problems of one particular size, whose generation is described in "Generating Small FFT Kernels". More precisely, the codelets compute a loop (V≤1V≤1) of small DFTs.

Cooley-Tukey plans

For problems of the form dft((n,ι,o),V,I,O)dft((n,ι,o),V,I,O) where n=rmn=rm, FFTW generates a plan that implements a radix-rr Cooley-Tukey algorithm "Review of the Cooley-Tukey FFT". Both decimation-in-time and decimation-in-frequency plans are supported, with both small fixed radices (usually, r≤64r≤64) produced by the codelet generator "Generating Small FFT Kernels" and also arbitrary radices (e.g. radix-nn).

The most common case is a decimation in time (DIT) plan, corresponding to a radix r=n2r=n2 (and thus m=n1m=n1) in the notation of "Review of the Cooley-Tukey FFT": it first solves dft((m,r·ι,o),V∪(r,ι,m·o),I,O)dft((m,r·ι,o),V∪(r,ι,m·o),I,O), then multiplies the output array OO by the twiddle factors, and finally solves dft((r,m·o,m·o),V∪(m,o,o),O,O)dft((r,m·o,m·o),V∪(m,o,o),O,O). For performance, the last two steps are not planned independently, but are fused together in a single “twiddle” codelet—a fragment of C code that multiplies its input by the twiddle factors and performs a DFT of size rr, operating in-place on OO.

Plans for higher vector ranks

These plans extract a vector loop to reduce a DFT problem to a problem of lower vector rank, which is then solved recursively. Any of the vector loops of VV could be extracted in this way, leading to a number of possible plans corresponding to different loop orderings.

Formally, to solve dft(N,V,I,O)dft(N,V,I,O), where V=(n,ι,o)∪V1V=(n,ι,o)∪V1, FFTW generates a loop that, for all kk such that 0≤k<n0≤k<n, invokes a plan for dft(N,V1,I+k·ι,O+k·o)dft(N,V1,I+k·ι,O+k·o).

Indirect plans

Indirect plans transform a DFT problem that requires some data shuffling (or discontiguous operation) into a problem that requires no shuffling plus a rank-0 problem that performs the shuffling.

Formally, to solve dft(N,V,I,O)dft(N,V,I,O) where N>0N>0, FFTW generates a plan that first solves dft(,N∪V,I,O)dft(,N∪V,I,O), and then solves dft(copy-o(N),copy-o(V),O,O)dft(copy-o(N),copy-o(V),O,O). Here we define copy-o(t)copy-o(t) to be the I/O tensor (n,o,o)∣(n,ι,o)∈t(n,o,o)∣(n,ι,o)∈t: that is, it replaces the input strides with the output strides. Thus, an indirect plan first rearranges/copies the data to the output, then solves the problem in place.

Plans for prime sizes

As discussed in "Goals and Background of the FFTW Project", it turns out to be surprisingly useful to be able to handle large prime nn (or large prime factors). Rader plans implement the algorithm from [41] to compute one-dimensional DFTs of prime size in Θ(nlogn)Θ(nlogn) time. Bluestein plans implement Bluestein's “chirp-z” algorithm, which can also handle prime nn in Θ(nlogn)Θ(nlogn) time [4], [43], [35]. Generic plans implement a naive Θ(n2)Θ(n2) algorithm (useful for n≲100n≲100).

Discussion

Although it may not be immediately apparent, the combination of the recursive rules in "The space of plans in FFTW" can produce a number of useful algorithms. To illustrate these compositions, we discuss three particular issues: depth- vs. breadth-first, loop reordering, and in-place transforms.

Figure 3: Two possible decompositions for a size-30 DFT, both for the arbitrary choice of DIT radices 3 then 2 then 5, and prime-size codelets. Items grouped by a "{{" result from the plan for a single sub-problem. In the depth-first case, the vector rank was reduced to zero as per "Plans for higher vector ranks" before decomposing sub-problems, and vice-versa in the breadth-first case.

As discussed previously in sections "Review of the Cooley-Tukey FFT" and "Understanding FFTs with an ideal cache", the same Cooley-Tukey decomposition can be executed in either traditional breadth-first order or in recursive depth-first order, where the latter has some theoretical cache advantages. FFTW is explicitly recursive, and thus it can naturally employ a depth-first order. Because its sub-problems contain a vector loop that can be executed in a variety of orders, however, FFTW can also employ breadth-first traversal. In particular, a 1d algorithm resembling the traditional breadth-first Cooley-Tukey would result from applying "Cooley-Tukey plans" to completely factorize the problem size before applying the loop rule "Plans for higher vector ranks" to reduce the vector ranks, whereas depth-first traversal would result from applying the loop rule before factorizing each subtransform. These two possibilities are illustrated by an example in Figure 3.

Another example of the effect of loop reordering is a style of plan that we sometimes call vector recursion (unrelated to “vector-radix” FFTs [13]). The basic idea is that, if one has a loop (vector-rank 1) of transforms, where the vector stride is smaller than the transform size, it is advantageous to push the loop towards the leaves of the transform decomposition, while otherwise maintaining recursive depth-first ordering, rather than looping “outside” the transform; i.e., apply the usual FFT to “vectors” rather than numbers. Limited forms of this idea have appeared for computing multiple FFTs on vector processors (where the loop in question maps directly to a hardware vector) [48]. For example, Cooley-Tukey produces a unit input-stride vector loop at the top-level DIT decomposition, but with a large output stride; this difference in strides makes it non-obvious whether vector recursion is advantageous for the sub-problem, but for large transforms we often observe the planner to choose this possibility.

In-place 1d transforms (with no separate bit reversal pass) can be obtained as follows by a combination DIT and DIF plans "Cooley-Tukey plans" with transposes "Rank-0 plans". First, the transform is decomposed via a radix-pp DIT plan into a vector of pp transforms of size qmqm, then these are decomposed in turn by a radix-qq DIF plan into a vector (rank 2) of p×qp×q transforms of size mm. These transforms of size mm have input and output at different places/strides in the original array, and so cannot be solved independently. Instead, an indirect plan "Indirect plans" is used to express the sub-problem as pqpq in-place transforms of size mm, followed or preceded by an m×p×qm×p×q rank-0 transform. The latter sub-problem is easily seen to be mm in-place p×qp×q transposes (ideally square, i.e. p=qp=q). Related strategies for in-place transforms based on small transposes were described in [26], [49], [40], [23]; alternating DIT/DIF, without concern for in-place operation, was also considered in [34], [44].

Generating Small FFT Kernels

The base cases of FFTW's recursive plans are its codelets, and these form a critical component of FFTW's performance. They consist of long blocks of highly optimized, straight-line code, implementing many special cases of the DFT that give the planner a large space of plans in which to optimize. Not only was it impractical to write numerous codelets by hand, but we also needed to rewrite them many times in order to explore different algorithms and optimizations. Thus, we designed a special-purpose “FFT compiler” called genfft that produces the codelets automatically from an abstract description. genfft is summarized in this section and described in more detail by [20].

A typical codelet in FFTW computes a DFT of a small, fixed size nn (usually, n≤64n≤64), possibly with the input or output multiplied by twiddle factors "Cooley-Tukey plans". Several other kinds of codelets can be produced by genfft , but we will focus here on this common case.

In principle, all codelets implement some combination of the Cooley-Tukey algorithm from Equation 2 and/or some other DFT algorithm expressed by a similarly compact formula. However, a high-performance implementation of the DFT must address many more concerns than Equation 2 alone suggests. For example, Equation 2 contains multiplications by 1 that are more efficient to omit. Equation 2 entails a run-time factorization of nn, which can be precomputed if nn is known in advance. Equation 2 operates on complex numbers, but breaking the complex-number abstraction into real and imaginary components turns out to expose certain non-obvious optimizations. Additionally, to exploit the long pipelines in current processors, the recursion implicit in Equation 2 should be unrolled and re-ordered to a significant degree. Many further optimizations are possible if the complex input is known in advance to be purely real (or imaginary). Our design goal for genfft was to keep the expression of the DFT algorithm independent of such concerns. This separation allowed us to experiment with various DFT algorithms and implementation strategies independently and without (much) tedious rewriting.

genfft is structured as a compiler whose input consists of the kind and size of the desired codelet, and whose output is C code. genfft operates in four phases: creation, simplification, scheduling, and unparsing.

In the creation phase, genfft produces a representation of the codelet in the form of a directed acyclic graph (dag). The dag is produced according to well-known DFT algorithms: Cooley-Tukey Equation 2, prime-factor [35], split-radix [56], [10], [53], [32], [13], and Rader [41]. Each algorithm is expressed in a straightforward math-like notation, using complex numbers, with no attempt at optimization. Unlike a normal FFT implementation, however, the algorithms here are evaluated symbolically and the resulting symbolic expression is represented as a dag, and in particular it can be viewed as a linear network [8] (in which the edges represent multiplication by constants and the vertices represent additions of the incoming edges).

In the simplification phase, genfft applies local rewriting rules to each node of the dag in order to simplify it. This phase performs algebraic transformations (such as eliminating multiplications by 1) and common-subexpression elimination. Although such transformations can be performed by a conventional compiler to some degree, they can be carried out here to a greater extent because genfft can exploit the specific problem domain. For example, two equivalent subexpressions can always be detected, even if the subexpressions are written in algebraically different forms, because all subexpressions compute linear functions. Also, genfft can exploit the property that network transposition (reversing the direction of every edge) computes the transposed linear operation [8], in order to transpose the network, simplify, and then transpose back—this turns out to expose additional common subexpressions [20]. In total, these simplifications are sufficiently powerful to derive DFT algorithms specialized for real and/or symmetric data automatically from the complex algorithms. For example, it is known that when the input of a DFT is real (and the output is hence conjugate-symmetric), one can save a little over a factor of two in arithmetic cost by specializing FFT algorithms for this case—with genfft , this specialization can be done entirely automatically, pruning the redundant operations from the dag, to match the lowest known operation count for a real-input FFT starting only from the complex-data algorithm [20], [27]. We take advantage of this property to help us implement real-data DFTs [20], [16], to exploit machine-specific “SIMD” instructions "SIMD instructions" [16], and to generate codelets for the discrete cosine (DCT) and sine (DST) transforms [20], [27]. Furthermore, by experimentation we have discovered additional simplifications that improve the speed of the generated code. One interesting example is the elimination of negative constants [20]: multiplicative constants in FFT algorithms often come in positive/negative pairs, but every C compiler we are aware of will generate separate load instructions for positive and negative versions of the same constants.11 We thus obtained a 10–15% speedup by making all constants positive, which involves propagating minus signs to change additions into subtractions or vice versa elsewhere in the dag (a daunting task if it had to be done manually for tens of thousands of lines of code).

In the scheduling phase, genfft produces a topological sort of the dag (a schedule). The goal of this phase is to find a schedule such that a C compiler can subsequently perform a good register allocation. The scheduling algorithm used by genfft offers certain theoretical guarantees because it has its foundations in the theory of cache-oblivious algorithms [19] (here, the registers are viewed as a form of cache), as described in "Memory strategies in FFTW". As a practical matter, one consequence of this scheduler is that FFTW's machine-independent codelets are no slower than machine-specific codelets generated by SPIRAL [55].

In the stock genfft implementation, the schedule is finally unparsed to C. A variation from [18] implements the rest of a compiler back end and outputs assembly code.

Numerical Accuracy in FFTs

An important consideration in the implementation of any practical numerical algorithm is numerical accuracy: how quickly do floating-point roundoff errors accumulate in the course of the computation? Fortunately, FFT algorithms for the most part have remarkably good accuracy characteristics. In particular, for a DFT of length nn computed by a Cooley-Tukey algorithm with finite-precision floating-point arithmetic, the worst-case error growth is O(logn)O(logn) [21], [50] and the mean error growth for random inputs is only O(logn)O(logn) [45], [50]. This is so good that, in practical applications, a properly implemented FFT will rarely be a significant contributor to the numerical error.

The amazingly small roundoff errors of FFT algorithms are sometimes explained incorrectly as simply a consequence of the reduced number of operations: since there are fewer operations compared to a naive O( n2 )O( n2 ) algorithm, the argument goes, there is less accumulation of roundoff error. The real reason, however, is more subtle than that, and has to do with the ordering of the operations rather than their number. For example, consider the computation of only the output Y[0]Y[0] in the radix-2 algorithm of Code 1, ignoring all of the other outputs of the FFT. Y[0]Y[0] is the sum of all of the inputs, requiring n-1n-1 additions. The FFT does not change this requirement, it merely changes the order of the additions so as to re-use some of them for other outputs. In particular, this radix-2 DIT FFT computes Y[0]Y[0] as follows: it first sums the even-indexed inputs, then sums the odd-indexed inputs, then adds the two sums; the even- and odd-indexed inputs are summed recursively by the same procedure. This process is sometimes called cascade summation, and even though it still requires n-1n-1 total additions to compute Y[0]Y[0] by itself, its roundoff error grows much more slowly than simply adding X[0]X[0], X[1]X[1], X[2]X[2] and so on in sequence. Specifically, the roundoff error when adding up nn floating-point numbers in sequence grows as O(n)O(n) in the worst case, or as O(n)O(n) on average for random inputs (where the errors grow according to a random walk), but simply reordering these n-1 additions into a cascade summation yields O(logn)O(logn) worst-case and O(logn)O(logn) average-case error growth [57].

However, these encouraging error-growth rates only apply if the trigonometric “twiddle” factors in the FFT algorithm are computed very accurately. Many FFT implementations, including FFTW and common manufacturer-optimized libraries, therefore use precomputed tables of twiddle factors calculated by means of standard library functions (which compute trigonometric constants to roughly machine precision). The other common method to compute twiddle factors is to use a trigonometric recurrence formula—this saves memory (and cache), but almost all recurrences have errors that grow as O(n)O(n), O(n)O(n), or even O(n2)O(n2) [51], which lead to corresponding errors in the FFT. For example, one simple recurrence is ei(k+1)θ=eikθeiθei(k+1)θ=eikθeiθ, multiplying repeatedly by eiθeiθ to obtain a sequence of equally spaced angles, but the errors when using this process grow as O(n)O(n) [51]. A common improved recurrence is ei(k+1)θ=eikθ+eikθ(eiθ-1)ei(k+1)θ=eikθ+eikθ(eiθ-1), where the small quantity12 eiθ-1=cos(θ)-1+isin(θ)eiθ-1=cos(θ)-1+isin(θ) is computed using cos(θ)-1=-2sin2(θ/2)cos(θ)-1=-2sin2(θ/2) [46]; unfortunately, the error using this method still grows as O(n)O(n) [51], far worse than logarithmic.

There are, in fact, trigonometric recurrences with the same logarithmic error growth as the FFT, but these seem more difficult to implement efficiently; they require that a table of Θ(logn)Θ(logn) values be stored and updated as the recurrence progresses [5], [51]. Instead, in order to gain at least some of the benefits of a trigonometric recurrence (reduced memory pressure at the expense of more arithmetic), FFTW includes several ways to compute a much smaller twiddle table, from which the desired entries can be computed accurately on the fly using a bounded number (usually <3<3) of complex multiplications. For example, instead of a twiddle table with nn entries ωnkωnk, FFTW can use two tables with Θ(n)Θ(n) entries each, so that ωnkωnk is computed by multiplying an entry in one table (indexed with the low-order bits of kk) by an entry in the other table (indexed with the high-order bits of kk).

There are a few non-Cooley-Tukey algorithms that are known to have worse error characteristics, such as the “real-factor” algorithm [42], [13], but these are rarely used in practice (and are not used at all in FFTW). On the other hand, some commonly used algorithms for type-I and type-IV discrete cosine transforms [48], [38], [6] have errors that we observed to grow as nn even for accurate trigonometric constants (although we are not aware of any theoretical error analysis of these algorithms), and thus we were forced to use alternative algorithms [16].

To measure the accuracy of FFTW, we compare against a slow FFT implemented in arbitrary-precision arithmetic, while to verify the correctness we have found the O(nlogn)O(nlogn) self-test algorithm of [14] very useful.

Concluding Remarks

It is unlikely that many readers of this chapter will ever have to implement their own fast Fourier transform software, except as a learning exercise. The computation of the DFT, much like basic linear algebra or integration of ordinary differential equations, is so central to numerical computing and so well-established that robust, flexible, highly optimized libraries are widely available, for the most part as free/open-source software. And yet there are many other problems for which the algorithms are not so finalized, or for which algorithms are published but the implementations are unavailable or of poor quality. Whatever new problems one comes across, there is a good chance that the chasm between theory and efficient implementation will be just as large as it is for FFTs, unless computers become much simpler in the future. For readers who encounter such a problem, we hope that these lessons from FFTW will be useful:

We should also mention one final lesson that we haven't discussed in this chapter: you can't optimize in a vacuum, or you end up congratulating yourself for making a slow program slightly faster. We started the FFTW project after downloading a dozen FFT implementations, benchmarking them on a few machines, and noting how the winners varied between machines and between transform sizes. Throughout FFTW's development, we continued to benefit from repeated benchmarks against the dozens of high-quality FFT programs available online, without which we would have thought FFTW was “complete” long ago.

Acknowledgements

SGJ was supported in part by the Materials Research Science and Engineering Center program of the National Science Foundation under award DMR-9400334; MF was supported in part by the Defense Advanced Research Projects Agency (DARPA) under contract No. NBCH30390004. We are also grateful to Sidney Burrus for the opportunity to contribute this chapter, and for his continual encouragement—dating back to his first kind words in 1997 for the initial FFT efforts of two graduate students venturing outside their fields.