Dft2

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Fast Fourier Transform
(FFT)

Outline
 Purpose
 Background
 Fourier Transform
 Continuous Fourier Transform
 Discrete Fourier Transform (DFT)
 Fast Fourier Transform (FFT)
 FFTW
 Examples

Purpose
 Analyze the frequencies contained in a
sampled signal
 Used in:
 Digital Signal Processing
 Optics
 Oceanography
 Acoustics
 Physics
 Number Theory

Fourier Transform
 Named after Jean Baptiste Joseph Fourier
 an integral transform that re-expresses a
function in terms of sinusoidal basis functions
 transforms linear differential equations with
constant coefficients into ordinary algebraic
ones
 turns the complicated convolution operation
into simple multiplication

Continuous Fourier Transform
 decomposes a function into a continuous
spectrum of the frequencies that comprise that
function
 Function:

 Inverse:

Discrete Fourier Transform
 Also called Finite Fourier Transform
 The n complex numbers x0, ..., xn-1 (ie. xi=xreal + i*ximag)
are transformed into the n complex numbers f0, ..., fn-1
according to the formula:

 Inverse (IDFT)

DFT cont.
Written in matrix form, the DFT is:

where

Fast Fourier Transform (FFT)
 An efficient algorithm to compute the DFT and its
inverse
 Divide and conquer algorithm
 Reduces time from O(n2) to O(n log n)
 Seismic Data contains hundreds of thousands of
samples and would take months to evaluate the DFT
 Cooley-Tukey is most popular implementation
 Developed in 1965
 Re-invented work done by Carl F. Gauss in 1805

FFT Definitions
 sinusoid
the curve of y=sin x
 radix
the size of an FFT decomposition.
 twiddle factors
the coefficients used to combine results from a previous
stage to form inputs to the next stage.
 “in place” FFT
an FFT that is calculated entirely inside its original sample
memory.

FFT Definitions cont.
 bit reversal
0, 1, 2, 3, 4, 5, 6, 7 is reordered to 0, 4, 2, 6, 1, 5, 3, 7
Decimal Binary Binary Decimal
0 000 000 0
1 001 100 4
2 010 010 2
3 011 110 6
4 100 001 1
5 101 101 5
6 110 011 3
7 111 111 7

FFT Implementation
 Radix-2
 N, the sample size, must be a power of 2

 Mixed-radix
 Can use different powers at different stages
 Ex. 1000 = 2*2*2*5*5*5 (six stages) or
1000 = 10 * 10 * 10 (three stages)

FFT Implementation (cont.)
1. Pad input sequence, of N samples, with ZERO's until the
number of samples is the nearest power of two.
e.g. 500 samples are padded to 512 (2^9)
2. Bit reverse the input sequence.
e.g. 3 = 011 goes to 110 = 6
3. Compute (N / 2) two sample DFT's from the shuffled inputs.
4. Compute (N / 4) four sample DFT's from the two sample
DFT's.
5. Compute (N / 2) eight sample DFT's from the four sample
DFT's.
...
6. Combine back into one N-sample DFT

Fast Fourier Transform in the West
(FFTW)
 Developed at MIT
 Free collection of fast C routines for
computing the DFT in 1 or more dimensions
 Includes complex, real, symmetric, and
parallel transforms
 MPI parallel transforms are only available in
v. 2.1.5

FFTW cont.
 Serial/Parallel, Share/Distributed Memory

 Faster than most “non-free” implementations

 Portable, automatically adapt to machine

Two Phases of FFTW
 Hardware dependent algorithm
 Planner
 ‘Learn’ the fast way on your machine
 Produce a data structure --‘plan’
 Reusable
 Executor
 Compute the transform
 Apply to all FFTw operation modes
 1D/nD, complex/real, serial/parallel

MPI FFTW Routines
 fftwnd_mpi_plan fftwnd_create_plan(mpi_comm comm,
int rank, const *int n, fftw_direction dir, int flags);

 void fftwnd_mpi_local_size(fftwnd_mpi_plan p,
int *local_first, int *local_first_start,
int *local_second_after_transpose,
int *local_second_start_after_transpose,
int *total_local_size);

 local_data = (fftw_complex*) malloc(sizeof(fftw_complex)
* total_local_size);

 work = (fftw_complex*) malloc(sizeof(fftw_complex)
* total_local_size);

 void fftwnd_mpi(fftwnd_mpi_plan p, int n_fields,
fftw_complex *local_data, fftw_complex *work,
fftw_mpi_output_order output_order);

 void fftw_mpi_destroy_plan(fftwnd_mpi_plan p

Examples
 View C source code
fftw_mpi.c
fftw_mpi_test.c
 To compile:
cc -o ffmpi fftw_mpi_test.c –lmpi -lfftw_mpi -lfftw –lm
 Matlab Example

References
 http://en.wikipedia.org/wiki/Fourier_transform
 http://en.wikipedia.org/wiki/Continuous_Fourier_transform
 http://en.wikipedia.org/wiki/Discrete_Fourier_transform
 http://astonomy.swin.edu.au/~pbourke/analysis/dft
 http://www.dspguru.com/info/faqs/fftfaq.htm
 http://www.arc.unm.edu/~gbma/Workshop/FFTW/slides/outlinee.htm
 http://astron.berkeley.edu/~jrg/ngst/fft/fourier.html
 http://www.spd.eee.strath.ac.uk/~interact/fourier/fft/fftalgrm.html
 http://www.ni.com/swf/presentation/us/fft/
 http://www.fftw.org

Dft2

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