Introduction

In 1990, it was estimated that Cray Research's installed base of approximately 200 machines spent 40% of all CPU cycles computing the fast Fourier transform (FFT)  [link] . With each machine worth about USD$25 million, the performance of the FFT was of prime importance.

Today, use of the FFT is even more pervasive, and it is counted among the 10 algorithms that have had the greatest influence on the development andpractice of science and engineering in the 20 ^th century  [link] . Huge numbers of mobile smartphones, tablets and PCs  [link] , [link] , most of which are equipped with single-instruction, multiple-data (SIMD)  [link] , [link] microprocessors, compute the FFT on a large scale for a plethora of sound, video and imageprocessing applications. In the space of a few years, mobile applications have become a part of many people's everyday lives  [link] .

This thesis shows that the key to optimizing the performance of the split-radix FFT algorithms on SIMD microprocessors is latency and spatiallocality optimizations, and in some cases, a variant of the split-radix FFT called the conjugate-pairalgorithm  [link] , [link] , [link] , [link] . It is also shown that extensive machine specific calibration may besuperfluous.

Hypotheses

FFTW  [link] , [link] , [link] , SPIRAL  [link] , [link] , [link] and UHFFT  [link] , [link] , [link] , [link] , [link] are state of the art FFT libraries that employ automatic empirical optimization. SPIRAL automatically performs machine-specific optimizations atcompile time, and FFTW and UHFFT automatically adapt to a machine at run-time. Aside from the use of automatic optimization, a common denominator among theselibraries is the use of large straight line blocks of code and optimized memory locality.

The hypotheses outlined below test whether good heuristics and model-based optimization can be used in the place of automatic empirical optimization.

Hypothesis 1: accessing memory in sequential “streams” is critical for best performance

Large FFT exhibit poor temporal locality, and when computing these transforms on microprocessor based systems that feature a cache, bestperformance is typically achieved when “streaming” sequential data through the CPU. Hypothesis 1 is tested in Implementation Details with replicated coefficient lookup tables that trade-off increased memory size forbetter spatial locality, and in Streaming FFT by topologically sorting a directed acyclic graph (DAG) of sub-transforms to again improve spatial locality.

Hypothesis 2: the conjugate-pair algorithm is faster than the ordinary split-radix algorithm

Hypothesis 2 is based on the idea that memory bandwidth is a bottleneck, and on the fact that the conjugate-pair algorithm requires only half the number oftwiddle factor loads. This hypothesis is tested in Split-radix vs. conjugate-pair , where a highly optimized implementation of the conjugate-pair algorithm is benchmarked against anequally highly optimized implementation of the ordinary split-radix algorithm.

Hypothesis 3: the performance of an fft can be predicted based on characteristics of the underlying machine and the compiler

Exploratory experiments suggest that good results can be obtained without empirical techniques, and that certain parameters can be predicted based on thecharacteristics of the underlying machine and the compiler used. Hypothesis 3 is tested in Results and Discussion by building a model that predicts performance, and by benchmarking FFTW against an implementation that does notrequire extensive calibration, on 18 different machines.