Download presentation

Presentation is loading. Please wait.

Published byRaymundo Purington Modified over 2 years ago

1
**Parallel Processing (CS 730) Lecture 7: Shared Memory FFTs***

September 4, 1997 Parallel Processing (CS 730) Lecture 7: Shared Memory FFTs* Jeremy R. Johnson Wed. Feb. 14, 2001 *Parts of this lecture was derived from chapters IX in Lipson. Feb. 14, 2001 Parallel Processing

2
September 4, 1997 Introduction Objective: To derive and implement a shared-memory parallel program for computing the fast Fourier transform (FFT). Topics Derivation of the FFT Recursive version Iterative version A parallel divide & conquer algorithm using threads A parallel loop version using OpenMP Obtaining additional parallelism Feb. 14, 2001 Parallel Processing

3
**FFT as a Matrix Factorization**

Compute y = Fnx, where Fn is n-point Fourier matrix. Feb. 14, 2001 Parallel Processing

4
**Matrix Factorizations and Algorithms**

function y = fft(x) n = length(x) if n == 1 y = x else % [x0 x1] = L^n_2 x x0 = x(1:2:n-1); x1 = x(2:2:n); % [t0 t1] = (I_2 tensor F_m)[x0 x1] t0 = fft(x0); t1 = fft(x1); % w = W_m(omega_n) w = exp((2*pi*i/n)*(0:n/2-1)); % y = [y0 y1] = (F_2 tensor I_m) T^n_m [t0 t1] y0 = t0 + w.*t1; y1 = t0 - w.*t1; y = [y0 y1] end Feb. 14, 2001 Parallel Processing

5
Rewrite Rules Feb. 14, 2001 Parallel Processing

6
**FFT Variants Cooley-Tukey Recursive FFT Iterative FFT**

Vector FFT (Stockham) Vector FFT (Korn-Lambiotte) Parallel FFT (Pease) Feb. 14, 2001 Parallel Processing

7
Tensor Permutations A natural class of permutations compatible with the FFT. Let be a permutation of {1,…,t} Mixed-radix counting permutation of vector indices Well-known examples are stride permutations and bit-reversal. Feb. 14, 2001 Parallel Processing

8
**Example (Stride Permutation)**

Feb. 14, 2001 Parallel Processing

9
**Example (Bit Reversal)**

Feb. 14, 2001 Parallel Processing

10
**Iterative Cooley-Tukey Algorithm**

September 4, 1997 Iterative Cooley-Tukey Algorithm R Stage 0 Stage 1 Stage 2 Stage 3 Feb. 14, 2001 Parallel Processing

11
**Iterative Cooley-Tukey Algorithm**

September 4, 1997 Iterative Cooley-Tukey Algorithm R Stage 0 Stage 1 Stage 2 Stage 3 Feb. 14, 2001 Parallel Processing

12
**Modified Pease Algorithm**

September 4, 1997 Modified Pease Algorithm Stage 0 Stage 1 Stage 2 Stage 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 4 6 8 10 12 14 1 3 5 7 9 11 13 15 Feb. 14, 2001 Parallel Processing

13
**Iterative Implementation**

function y = ifft2(x) % Input: x a vector of length n. n = 2^t, t an integer, t >= 0. % Output: y = F_{2^t} x % Algorithm: Iterative. % F_{2^t} = { Prod_{c=1}^t I_{2^{t-c}}) % T^{2^{t-c+1}}_{2^{t-c}}) } R^{2^t} n = length(x); t = ceil(log2(n)); xt = bitreversal(x); yt = zeros(n,1); for c=t:-1:1 m = 2^(c-1); p = 2^(t-c); % W = W_p(omega_{2p}) W = exp((2*pi*i)/(2*p)*-(0:p-1)'); % yt = I_p)xt for j=0:m-1 % y^{2p}_{j*2p+1} = I_p)T^{2p}_p x^{2p}_{j*2p+1} % = I_p)(I_p $ W) x^{2p}_{j*2p+1} xt((j*2+1)*p+1:(j+1)*2*p) = W .* xt((j*2+1)*p+1:(j+1)*2*p); yt(j*2*p+1:(j*2+1)*p) = xt(j*2*p+1:(j*2+1)*p) + xt((j*2+1)*p+1:(j+1)*2*p); yt((j*2+1)*p+1:(j+1)*2*p) = xt(j*2*p+1:(j*2+1)*p) - xt((j*2+1)*p+1:(j+1)*2*p); end xt = yt; y = yt; Feb. 14, 2001 Parallel Processing

14
**Iterative Implementation**

function y = ipfft2(x) % In-place Pease FFT algorithm. % Input: x a vector of length n. n = 2^t, t an integer, t >= 0. % Output: y = F_{2^t} x % Algorithm: Conjugated Pease. % F_{2^t} = { Prod_{c=1}^t F_2)T_c L^n_{2^c} R^{2^t} % n = length(x); t = ceil(log2(n)); y = bitreversal(x); w = exp(-2*pi*i/n); for c=t-1:-1:0 for r=0:2^(t-1)-1 r0 = mod(r,2^c); r1 = floor(r/2^c); a0 = r0*2^(t-c) + r1; a1 = a0 + 2^(t-c-1); y0 = y(a0+1); y1 = w^(r1*2^c) * y(a1+1); y(a0+1) = y0 + y1; y(a1+1) = y0 - y1; end Feb. 14, 2001 Parallel Processing

Similar presentations

OK

Applied Symbolic Computation1 Applied Symbolic Computation (CS 567) The Fast Fourier Transform (FFT) and Convolution Jeremy R. Johnson TexPoint fonts used.

Applied Symbolic Computation1 Applied Symbolic Computation (CS 567) The Fast Fourier Transform (FFT) and Convolution Jeremy R. Johnson TexPoint fonts used.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on statistics in maths pie Ppt on magic maths Ppt on corporate etiquettes meaning Ppt on fdi in india 2013 Ppt on library management system using java Ppt on self awareness definition Ppt on railway reservation system in java Ppt on airport automation Ppt on schottky diode characteristics Ppt on natural numbers vs whole numbers