Engineering a Fast Fourier Transform

Computing the discrete Fourier transform is one of the most important in applied computer science, with applications in fields as diverse as seismology, signal analysis, and various branches of engineering. A great many algorithms exist for quickly computing different variations of the transform – fast Fourier transforms, or FFTs. Because of their practical importance, even small improvements to the running time of an FFT implementation are important and much research has gone into improving the constant factors of algorithms, both in a theoretical and an applied context. In this thesis I have sought to discover whether modern general-purpose compilers have become so advanced that they can generate optimal code for an FFT implementation automatically or if the programmer must perform compiler transformations manually in order to create an efficient implementation. To answer this question, a number of FFT implementations were developed, primarily using simple transformations, which could have been performed by the compiler but were not. During the development process, I discovered a new variation of the CooleyTukey factorization of the DFT matrix, which corresponds to a different evaluation order of the same computation as the normal Cooley-Tukey FFT. This evaluation order is shown to be faster than the ordinary evaluation order in practice, even though the theoretical complexity is the same. I finally conclude that mid-low level optimizations can significantly improve an FFT implementation and that a modern general-purpose compiler does not generate optimal code without significant support from the programmer.