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This chapter describes SFFT: a high-performance FFT library for SIMD microprocessors that is, in many cases, faster than the state of the art FFT libraries reviewed in Existing libraries .
Implementation details described some simple implementations of the FFT and concluded with an analysis of the performance bottlenecks. The implementations presented in this chapter are designed to improve spatial locality, and utilize larger straight line blocks of code at the leaves, corresponding to sub-transforms of sizes 8 through to 64, in order to reduce latency and stack overheads.
In distinct contrast to the simple FFT programs of Chapter 3 , this chapter employs meta-programming. Rather than describe FFT programs, we describe programs that statically elaborate the FFT into a DAG of nodes representing the computation, apply some optimizing transformations to the graph, and then generate code. Many other auto-vectorization techniques, such as those employed by SPIRAL, operate at the instruction level [link] , but the techniques presented in this chapter vectorize blocks of computation at the algorithm level of abstraction, thus enabling some of the algorithms structure to be utilized.
Three types of implementation are described in this chapter, and the performance of each depends on the parameters of the transform to be computed and the characteristics of the underlying machine.For a given machine and FFT to be computed (which has parameters such as length and precision), the fastest configuration is selected from among a small set of up to eight possible FFT configurations – a much smaller space compared to FFTW's exhaustive search of all possible FFTs. The fastest configuration is easily selected by timing each of the possible options, but it is shown in Results and discussion that it is also possible to use machine learning to build a classifier that will predict the fastest based on attributes such as the size of the cache.
SFFT comprises three types of conjugate-pair implementation, which are:
Statically elaborating a DAG that represents a depth-first recursive FFT is much like computing a depth-first recursive FFT: instead of performing computation at the leaves of the recursion and where smaller DFTs are combined into one, a node representing the computation is appended to the end of a list, and the list of nodes, i.e., a topological ordering of the DAG, is later translated into a program that can be compiled and executed.
Emitting code with a vector length of 1 (i.e., scalar code or vector code where only one complex element fits in a vector register) is relatively simple and is described in "Vector length 1" . For vector lengths above 1, vectorizing the topological ordering of nodes poses some subtle challenges, and these details are described in "Other vector lengths" . The fully hard-coded FFTs described in this section are generally only practical for smaller sizes of transforms, typically where $N\le 128$ , however these techniques are expanded in later sections to scale the performance to larger sizes.
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