Implementation of Fast Fourier Transform on General Purpose

Computers

Tianxiang Yang

FFT Formulation

• Basically a matrix-vector product:

1

2

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0

)1)(1()1(21

)1(26

42

132

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2

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)( /2 NjN eW

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 N2 to Nlog2(N). Also suitable for any length of FFT computation.

– Yanve (1968). Requires the least known number of multiplications, as well as additions for length 2n 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 07632. Prentice-Hall, 1989.

• P. Duhamel and M. Vetterli, “Fast Fourier Transforms: A Tutorial Review and a State of the Art”, Signal Processing, vol. 19, Apr. 1990

• http://www.fftw.org (official FFTW site).