EADS: Accelerator Project

Rohit Prakash (2003CS10186)Anand Silodia (2003CS50210)

Speed up scientific application

Application

Candidate Partition

Performance Prediction

Choose next partition

Time lines (tentative)

28th January : Figure out the best algorithm of FFTCompare the algos on the following parameters – - Execution Time - No. of multiplications - No. of additions

19th February : Study hardware implementation of FFT.....

Terminologies

radix : The "radix" is the size of an FFT decomposition twiddle factors: "Twiddle factors" are the coefficients used to combine results from a previous stage to form inputs to the next stage

First Implementation

Implemented Recursive radix-4 FFT analysed this using gprof Looked into other FFT implementations

iterative parallel split radix

Analysis of the implementation

Considered FFT of 1024 random points (double)

Results from gprof -> No. of Complex multiplications : 21760 No. of Complex additions : 7680

(Each complex multiplication consists of 4 real multiplications and 2 real additions)

(Each complex addition/subtraction consists of 2 real additions/subtractions)

Problems with this implementation

Inefficient use of memory (recursive procedure)

Wasted computations (some factors computed multiple times)

Maximum time utilized in computing Twiddle factors (complex number multiplications)

2nd Implementation

Radix-4 iterative in-place implementation -iterativeFFT(a)

BitReversal(a,A)

n length(a)

for(s 1 to log4(n)) // logarithm is of base 4

{

do m 4s

ω e2Лi/m

for(k0 to n-1 by m)

{

do τ 1

for(j0 to m/4)

{

tA[k+j]

u τ A[k+j+m/4]

v τ2A[k+j+2*m/4]

x τ3A[k+j+3*m/4]

A[k+j]t+u+v+x

A[k+j+m/4]t+(i)u-

v-(i)x

A[k+j+2*m/4]t-u+v-

x

A[k+j+3*m/4]t-

(i)u-v+(i)x

τ τ* ω

}

}

}

Analysis of this implementation

Considered FFT of 1024 random points (double)

Results from gprof -> No. of Complex multiplications : 14080 No. of Complex additions/subtractions : 7680

(Each complex multiplication consists of 4 real multiplications and 2 real additions)

(Each complex addition/subtraction consists of 2 real additions/subtractions)

Improvements

Precompute twiddle factors Trade additions for multiplications

(it’s possible to multiply with 3 real multiplies and 5 real adds rather than usual 4 real multiplies and 2 real adds)

use compiler flags (10%-15% execution time on some systems) -O3 -march=pentiumpro -ffast-math -fomit-frame-pointer

Some results

Precomputing twiddle factors: No. of multiplications : 8960 5120 less multiplications (complex)

Trading multiplications for additions Did not show any appreciable decline in

execution time Using compiler flags

Drastic improvement in execution time

Comparative Analysis

User time for 1024 points

0

5

10

15

20

25

30

recursive fft inplace fft inplace twiddleprecompute

inplace twiddlecompiler

final fftw

tim

e (m

illi

seco

nd

s)

User time for 4096 points

0

10

20

30

40

50

60

70

recursive fft inplace fft inplace twiddleprecompute

inplace twiddlecompiler

final fftw

tim

e (m

illi

seco

nd

s)

User time for 262144 points

0

500

1000

1500

2000

2500

recursive fft inplace fft inplacetwiddle

precompute

inplacetwiddle

compiler

final fftw

tim

e (

mil

lis

ec

on

ds

)

Further enhancements possible

Use higher radix – 8,16,32, etc. Use split-radix or Winograd algorithms If data is real, we can have great

improvements Use Fast Bit-Reversal method (IEEE D.M.W.

Evans)

Resources

Rivest, Cormen Numerical Recipes in C IEEE papers

Conversion of Digit-Reversed to Bit-Reversed order in FFT algorithms (Panos E. and C.S. Burrus)

The Design and Implementation of FFTW3 (Matteo Frigo and Steven G. Johnson)

cnx.org Other fft implementations on the net

Best: fftw

Thank You