1. Fast (finite) Fourier Transforms (FFTs) Shirley Moore svmoore@utep.edu CPS5401 Fall 2013 svmoore.pbworks.com December 5, 2013 1

2. Learning Objectives After this lesson, you should be able to – Define and describe the finite Fourier transform – List applications of FFTs – Describe how the FFT is used for periodic time series analysis – Describe the idea behind the fast finite Fourier transform algorithm – Locate and use software and documentation for FFTs and parallel FFTs 2

3. Finite Fourier Transform The finite, or discrete, Fourier transform of a complex vector y with n elements is another complex vector Y with n elements where ω is a complex nth root of unity 3

4. Inverse Fourier Transform 4

5. Applications of Fourier Transforms Signal processing – When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Examples – X-ray crystallography to reconstruct a crystal structure from its diffraction pattern – Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field. – Many other forms of spectroscopy also rely upon Fourier Transforms to determine the three-dimensional structure and/or identity of the sample being analyzed, including Infrared and Nuclear Magnetic Resonance spectroscopies. – Generation of sound spectrograms used to analyze sounds. 5

6. Applications of Fourier Transforms 2D FFTs are used for synthetic aperture radar image processing. 3D FFTs are used in molecular dynamics codes for computing long-range Coulombic effects via the particle-mesh Ewald method, and in density- functional electronic structure codes for alternating between real-space and reciprocal- space operator representations. Solution of the discrete Poisson equation (see http://www.cs.berkeley.edu/~demmel/cs267/lect ure24/lecture24.html ) http://www.cs.berkeley.edu/~demmel/cs267/lect ure24/lecture24.html 6

7. Periodic Time Series Analysis The tones generated by a touch-tone telephone and the Wolfer sunspot index are two examples of periodic time series, that is, functions of time that exhibit periodic behavior, at least approximately. Fourier analysis allows us to estimate the period from a discrete set of values sampled at a fixed rate. The following table shows the relationship between the various quantities involved in this analysis. 7

8. Fast Finite Fourier Transform Direct application of the definition requires n multiplications and n additions for each of the n components of Y for a total of 2n 2 floating-point operations. This does not include the generation of the powers of ω. A computer capable of doing one multiplication and addition every microsecond would require a million seconds, or about 11.5 days, to do a million-point FFT. Several people discovered fast FFT algorithms independently and many people have since contributed to their development, but it was a 1965 paper by John Tukey of Princeton University and John Cooley of IBM Research that is generally credited as the starting point for the modern usage of the FFT. Modern fast FFT algorithms have computational complexity O(n log 2 n) instead of O(n 2 ). 8

9. Fast FFT Algorithms 9

10. FFT Resources Fastest Fourier Transform in the West (FFTW) – www.fftw.org www.fftw.org Steve Plimpton’s parallel FFT research – http://www.sandia.gov/~sjplimp/algorithms.html #ffts http://www.sandia.gov/~sjplimp/algorithms.html #ffts Evaluation of Parallel FFTs by Evangelos Brachos (Master’s thesis) – http://www.epcc.ed.ac.uk/wp- content/uploads/2011/11/EvangelosBrachos.pdf http://www.epcc.ed.ac.uk/wp- content/uploads/2011/11/EvangelosBrachos.pdf 10