Implementation of Fast Fourier Transform on General Purpose Computers Tianxiang Yang

FFT Formulation Basically a matrix-vector product:

FFT - What do we already have? A history of theoretical ideas: –Gauss (1805). First but largely unnoticed. –Cooley-Tukey (1965). Reduces the order of the number of operations from N 2 to Nlog 2 (N). Also suitable for any length of FFT computation. –Yanve (1968). Requires the least known number of multiplications, as well as additions for length 2 n FFTs. –Almost uncountable others.

Motivation: Divide and Conquer Map the original problem into several sub- problems in such a way the the following inequality is satisfied : sum(cost(subproblems)) + cost(mapping) < cost(original problem)

Main Categories of FFT Algorithms Original Cooley-Tukey. Split-radix. Prime factor. Winograd FFT algorithms. Many techniques were invented such as: DFT computation as a convolution, computation of the cyclic convolution, etc.

Implementation Issues General Purpose Computers Digital Signal Processors Vector and Multi-Processors VLSI

Fewer operations always better?

FFT implementations on GPP Algorithms under survey include: –FFTPACK, Temperton, SUNPERF, Sorensen, Bailey, Oorua, Krukar, QFT, Green, Singleton, NRF, FFTW –Special interest: FFTW (Fast Fourier Transform in the West)

Overview of FFTW Planner + Executor –FFTW has collected a sea of small combinable small programs called “codelets” –Planner tries to minimize the actual execution time, not the number of floating point operations. –A dedicated FFTW compiler is used to combine codelets by the plan by wisely allocating register and memory usage and by taken advantages of the processor pipeline.

FFTW Generates unexpected code specific optimized for the current machine. An adaptive approach. Performance results: –Significant faster than most proposed implementations. –Faster or equivalent to some machine specific optimized library –Best FFT on GPP ever.

Reference A.V. Oppenheim and R.W. Schafer, Discrete-time Signal Processing. Englewood Cliffs, NJ Prentice-Hall, P. Duhamel and M. Vetterli, “Fast Fourier Transforms: A Tutorial Review and a State of the Art”, Signal Processing, vol. 19, Apr (official FFTW site).