1 / 11

Implementation of Fast Fourier Transform on General Purpose Computers

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.

ula
Download Presentation

Implementation of Fast Fourier Transform on General Purpose Computers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Implementation of Fast Fourier Transform on General Purpose Computers Tianxiang Yang

  2. FFT Formulation • Basically a matrix-vector product:

  3. 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.

  4. 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)

  5. 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.

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

  7. Fewer operations always better?

  8. 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)

  9. 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.

  10. 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.

  11. 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).

More Related