92006629

(1)FFTFFTFFTFFTFFT

FFTFFT

(2)FFT

FFT

FFTCooley-Tukey FFTStockham FFTFFTFFTFFTFFTFFT

.FFTFFTCooley-Tukey FFTStockham FFTFFTFFTFFTFFT

1.1 (1)DFTN a0a1 aN-1 c0c1 cN-1Discrete Fourier Transform

DFTO(N2)

DFTDFT

DFT

1.1 (2)DFT [0, 2]N f (xn) = an exp (ikx)k=0, 1, , N1 cnDFT cn N an DFT

FFT

1.2FFT (1)DFTFFTNDFT

N = N/2,ej = a2j,oj = a2j+1

kexp(2i(k+N/2)/N) = exp(2ik/N)

N DFT N/2 DFT exp(2ik) DFTFFT; Fast Fourier Transform(*)

1.2FFT (2)FFTFFTNDFTT(N) exp(2ik) 2N3N5N

T(1) = 0

FFT 5N log2 N NDFTT(N) = 2T(N/2) + 5NT(N) = 5N log2 N

1.3Cooley-Tukey FFT (1)DFT*N/2DFTc0c1cN1 (*)

1.3Cooley-Tukey FFT (2)Cooley-Tukey FFTN/2DFT* Cooley-Tukey FFT Cooley & Tukey, 1965

c0c1c7c2c3c4c5c6a0a4a7a2a6a1a5a3N=8Cooley-Tukey FFT0123

1.3Cooley-Tukey FFT (3)Cooley-Tukey FFTCooley-Tukey FFT L+1 L FFTin-place FFT c0c1c7c2c3c4c5c6a0a4a7a2a6a1a5a3

1.3Cooley-Tukey FFT (4)Cooley-Tukey FFT{aj}aj j jp-1j1j0 p = log2Najip-1i1i0

ip-1= j0ip-2 = j1 i0 = jp-1{aj}

FFTc0c1c7c2c3c4c5c6a0a4a7a2a6a1a5a3j0 = 0 ip-1 = 0j0 = 1 ip-1 = 1j1 = 0 ip-2 = 0j1 = 1 ip-2 = 1

jp-1 = 0 i0 = 0jp-1 = 1 i0 = 1

1.4Stockham FFT (1)self-sortingFFT

XL (j, k) L= 2L L= 2pL1 XL 2LLXL (j, *) 2LL ajaj+2L aj+2(L1 )L DFT

XL (j, k)L = 0 X0 (j, 0) = ajL = p Xp (0, k) = ck X0 (j, 0) Xp (0, k) X0 X1X2Xp self-sortingFFT

1.4Stockham FFT (2) XL (j, k)DFT*XLXL+1 XL (j, k)

Stockham FFT X0 X1 Xp-1 Xp Stockham FFT

Stockham FFTSelf-sorting In-place N

XL+1 (j, k) = XL (j, k) + XL (j+L, k)NkLXL+1 (j, k+L) = XL (j, k) XL (j+L, k)NkL j = 0, 1, , L1k = 0, 1, , L1

1.4Stockham FFT (3)Stockham FFT

DO 20DO 30

DO 10 L = 0, p1 = 2L = 2pL1 DO 20 k = 0, 1DO 30 j = 0, 1 XL+1 (j, k) = XL (j, k) + XL (j+, k)Nk XL+1 (j, k+) = XL (j, k) XL (j+, k)Nk30 CONTINUE20 CONTINUE10 CONTINUE

1.4Stockham FFT (4)L+1N =128L = 3

N = 8

XL (j, k)XL+1 (j, k)2LLLL2LXL (j+L, k)XL (j, k)XL+1 (j, k)XL+1 (j, k+L)

1.5FFTFFT (1)FFT (I)1.1(1)DFTDFT

FFTFFTN = exp(2i/N) N* N

FFTFFT(1) ck ck* (2) ck* FFT(3) (4) 1/N

DFTDFT

1.5FFTFFT (2)FFT (II)Stockham FFT XL XL+1 L FFT

FFTDO 10 L = p1, 0, 1 = 2L = 2pL1DO 20 k = 0, 1DO 30 j = 0, 1 XL (j, k) = (XL+1 (j, k) + XL+1 (j, k+)) / 2 XL (j+, k) = (XL+1 (j, k) XL+1 (j, k+))Nk / 230 CONTINUE20 CONTINUE10 CONTINUE

1.5FFTFFT (3)FFTDFTN N* N DFTFFT (II) N N* N FFTFFT*FFTFFT

FFTFFTFFT

1.6FFTDFTNxNy {ajx, jy} {ckx, ky} DFT

Ny Nx FFT Nx Ny FFT 5 NxNy log2 (NxNy)

FFTFFTFFT12xy

1.7FFT (1)DFTDFTN = lm jk

NDFT

l m DFT m l DFTj = rl + s s = 0l1 r = 0m1 k = pm + qq = 0m1 p = 0l1

1.7FFT (2)DFTN = lm DFT(1)l m FFT(2) s q exp(2iqs / N) (3)m l FFT

FFT

01234567891011121314150481215913261014371115l mFFTm lFFTajckarl+scpm+q

.

2.1

NEC SX-7 VPP5000 SR2201SR8000Intel Pentium 4

01215SR8000

2.2

Intel Pentium IIIIBM Power 4AMD AthlonDEC Alpha

8128KMB

2.3Stockham FFT N/2N/41

> DO 20 DO 30 O(N 1/2)DO 10 L = 0, p1 = 2L = 2pL1 DO 20 k = 0, 1DO 30 j = 0, 1 XL+1 (j, k) = XL (j, k) + XL (j+, k)Nk XL+1 (j, k+) = XL (j, k) XL (j+, k)Nk30 CONTINUE20 CONTINUE10 CONTINUE

2.4kStockham FFT XL (j, k) XL (j +L, k) L*L = N/2 XL NA NA*L LNAXL (j+L, k)XL (j, k)LXL+1 (j, k+L)XL+1 (j, k)LL

2.5 (1)FFTStockham FFT FFTFFTFFT

FFTXL (j, k)XL+1 (j, k)XL+2 (j, k)LL+1

2.5 (2)FFTL = 2LL = 2 pL2XL+1 (j, k) = XL (j, k) + XL (j+2L, k)2kLXL+1 (j, k+L) = XL (j, k) XL (j+2L, k)2kLXL+1 (j +L, k) = XL (j +L, k) + XL (j+3L, k)2kLXL+1 (j +L, k+L) = XL (j +L, k) XL (j+3L, k)2kLXL+2 (j, k) = XL+1 (j, k) + XL+1 (j+L, k)kLXL+2 (j, k+2L) = XL+1 (j, k) XL+1 (j+L, k)kLXL+2 (j, k +L) = XL+1 (j, k +L) + XL+1 (j+L, k +L) ( k+L ) LXL+2 (j, k+3L) = XL+1 (j, k +L) XL+1 (j+L, k +L) ( k +L ) LLL+1

2.5 (3)FFT L+1L

XL+1 (j, k) = XL (j, k) + XL (j+2L, k)2kLXL+1 (j, k+L) = XL (j, k) XL (j+2L, k)2kLYL+1 (j +L, k) = XL (j +L, k)kL+ XL (j+3L, k)3kLYL+1 (j +L, k+L) = XL (j +L, k) ( k +L ) L XL (j+3L, k) ( 3k +L ) L= iXL (j +L, k)kL + iXL (j+3L, k)3kLXL+2 (j, k) = XL+1 (j, k) + YL+1 (j+L, k)XL+2 (j, k+2L) = XL+1 (j, k) YL+1 (j+L, k) XL+2 (j, k +L) = XL+1 (j, k +L) + YL+1 (j+L, k +L)XL+2 (j, k+3L) = XL+1 (j, k +L) YL+1 (j+L, k +L)LL+1LL = exp (i/2) = i

2.5 (4)FFTFFTByte/Flop

6 / 44 / 46.4

22 / 128 / 83.76 24 / 16

66 / 3216 / 162.61 72 / 48Byte/Flop

2.6 (1)Stockham FFT XL (j, k) N M M < N O(N) FFT O(N log2 N)

FFTM = N 1/2 1.7 (2) NFFT(1) M = N 1/2 M FFT(2) s q exp(2iqs / N) (3) M M FFT

(1)(3) FFT(1)(2)(3) O(N) FFT O(N)

2.6 (2)N =16M = 4

N M r r1FFT MFFT

.FFTFFT

3.1 FFT (1)yxFFTall-to-all broadcast xyFFT

p 5 NxNy log2 (NxNy) / p NxNy / p p1

3.1 FFT (2)xFFTyy

PU0PU2PU1PU1PU1PU3PU0202PU0PU21313

3.2FFT (1)FFTN = NxNy NFFT(1) Ny Nx FFT y(2) jy kx exp(2i kx jy / N) (3) all-to-all broadcast (4) Nx Ny FFT x

FFT01234567891011121314150481215913261014371115kxkykxkyNy NxFFTNx NyFFT+ ajckajxNy+jyckxNx+kyFFT

3.2FFT (2) p 5 N log2 N / p N / p p1

SR80008GFLOPS0.0625Gword/sN = 230

FFTFFTCooley-Tukey FFTXy