main.dvi

Similar documents
(5 B m e i 2π T mt m m B m e i 2π T mt m m B m e i 2π T mt B m (m < 0 C m m (6 (7 (5 g(t C 0 + m C m e i 2π T mt (7 C m e i 2π T mt + m m C m e i 2π T

main.dvi

1 1.1 Excel Excel Excel log 1, log 2, log 3,, log 10 e = ln 10 log cm 1mm 1 10 =0.1mm = f(x) f(x) = n

2

SOWC04....

211 ‚æ2fiúŒÚ

untitled


EPSON PX-G920 基本操作ガイド

スタートアップガイド_応用編

橡68-honbun.PDF

数値計算：フーリエ変換

p03.dvi

PDF



³ÎΨÏÀ

‘¬”R.qx

数理.indd

TSP信号を用いた音響系評価の研究

power.tex

Korteweg-de Vries

No.20 2 / 32

Microsoft Word - 信号処理3.doc

untitled

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

CPU Levels in the memory hierarchy Level 1 Level 2... Increasing distance from the CPU in access time Level n Size of the memory at each level 1: 2.2

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =

. (.8.). t + t m ü(t + t) + c u(t + t) + k u(t + t) = f(t + t) () m ü f. () c u k u t + t u Taylor t 3 u(t + t) = u(t) + t! u(t) + ( t)! = u(t) + t u(

CDMA (high-compaciton multicarrier codedivision multiple access: HC/MC-CDMA),., HC/MC-CDMA,., 32.,, 64. HC/MC-CDMA, HC-MCM, i

main.dvi

(1) (2) (3) (4) (1) 1 1

-- Blackman-Tukey FFT MEM Blackman-Tukey MEM MEM MEM MEM Singular Spectrum Analysis Multi-Taper Method (Matlab pmtm) 3... y(t) (Fourier transform) t=

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

untitled

　

impulse_response.dvi

5


[1] 1.1 x(t) t x(t + n ) = x(t) (n = 1,, 3, ) { x(t) : : 1 [ /, /] 1 x(t) = a + a 1 cos πt + a cos 4πt + + a n cos nπt + + b 1 sin πt + b sin 4πt = a

Relaxation scheme of Besse t t n = n t, u n = u(t n ) (n = 0, 1,,...)., t u(t) = F (u(t)) (1). (1), u n+1 u n t = F (u n ) u n+1 = u n + tf (u n )., t

untitled

(interferometer) 1 N *3 2 ω λ k = ω/c = 2π/λ ( ) r E = A 1 e iφ1(r) e iωt + A 2 e iφ2(r) e iωt (1) φ 1 (r), φ 2 (r) r λ 2π 2 I = E 2 = A A 2 2 +

(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n

2. 2 I,II,III) 2 x expx) = lim + x 3) ) expx) e x 3) x. ) {a } a a 2 a 3...) a b b {a } α : lim a = α b) ) [] 2 ) f x) = + x ) 4) x > 0 {f x)} x > 0,


1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m

統計学のポイント整理

Part () () Γ Part ,

取扱説明書 基本ガイド

SFGÇÃÉXÉyÉNÉgÉãå`.pdf

AHPを用いた大相撲の新しい番付編成

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.

Fourier Niching Approach for Multi-modal Optimization 2 Yan Pei Hideyuki Takagi 2 Graduate School of Design, Kyushu University 2 2 Faculty of Design,

LMS NLMS LMS Least Mean Square LMS Normalized LMS NLMS AD 3 1 h(n) y(n) d(n) FIR w(n) n = 0, 1,, N 1 N N =

. ż ż 57 a v i ż ż v o b a ż v i ż v i ż v o ż v o a b 57. v i ż ż v o v o = Ġ v i (86) = ż ż + ż v i (87) v o v i Ġ = ż ż + ż (88) v i v o?? Ġ 6

(DFT) 009 DFT: Discrete Fourier Transform N x[n] DFT N 1 X[k] = x[n]wn kn, k = 0, 1,, N 1 (6 ) n=0 1) W N = e j π N W N twidd



II 2 II

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4

untitled

( ) g 900,000 2,000,000 5,000,000 2,200,000 1,000,000 1,500, ,000 2,500,000 1,000, , , , , , ,000 2,000,000

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta

2D-RCWA 1 two dimensional rigorous coupled wave analysis [1, 2] 1 ε(x, y) = 1 ε(x, y) = ϵ mn exp [+j(mk x x + nk y y)] (1) m,n= m,n= ξ mn exp [+j(mk x

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

xy n n n- n n n n n xn n n nn n O n n n n n n n n

34号　目　次


I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1

untitled

00_cover1_2.pm6

アプリケーションソフトウェアガイド PM2200C用

Report C: : ( )

untitled

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4


untitled


C言語による数値計算プログラミング演習


29

Sample function Re random process Flutter, Galloping, etc. ensemble (mean value) N 1 µ = lim xk( t1) N k = 1 N autocorrelation function N 1 R( t1, t1

5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

1

30

7

pp d 2 * Hz Hz 3 10 db Wind-induced noise, Noise reduction, Microphone array, Beamforming 1

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier

2 Poisson Image Editing DC DC 2 Poisson Image Editing Agarwala 3 4 Agarwala Poisson Image Editing Poisson Image Editing f(u) u 2 u = (x

m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P

OHP.dvi

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i

チュートリアル：ノンパラメトリックベイズ

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

Transcription:

4 DFT DFT Fast Fourier Transform: FFT 4.1 DFT IDFT X(k) = 1 n=0 x(n)e j2πkn (4.1) 1 x(n) = 1 X(k)e j2πkn (4.2) k=0 x(n) X(k) DFT 2 ( 1) 2 4 2 2(2 1) 2 O( 2 ) 4.2 FFT 4.2.1 radix2 FFT 1 (4.1)

86 4. X(0) w 0 w 0 w 0 X(1) w 0 w 1 w 1 =....... X( 1) w 0 w 1 w 1 x(0) x(1). x( 1) (4.3) w = e j2π w 0,w1,,w 1 kn = m + r, 0 r 1 w kn = wr (4.4) (4.3) X() =F ()x() (4.5) X() =[X(0),X(1),,X( 1)] T (4.6) x() = [x(0),x(1),,x( 1)] T (4.7) (4.8) [] T F () DFT 2 Decimation-in-frequency F () X() F () 2k k 2k +1 k ˆX() = ˆF ()x() (4.9) ˆX() = [ ˆX 0 (/2), ˆX 1 (/2)] T (4.10) ˆX 0 (/2) = [X(0),X(2),,X( 2)] T (4.11) ˆX 1 (/2) = [X(1),X(3),,X( 1)] T (4.12) ˆX 0 (k) = X(2k), 0 k 2 1 (4.13)

4.2 FFT 87 3 ˆX 1 (k) = X(2k +1), 0 k 2 1 (4.14) x() = [x 0 (/2), x 1 (/2)] T (4.15) x 0 (/2) = [x(0),x(1),,x(/2 1)] T (4.16) x 1 (/2) = [x(/2),x(/2+1),,x( 1)] T (4.17) ˆF () = F 1(/2) F 2 (/2) (4.18) F 3 (/2) F 4 (/2) 1. F 1 (/2) F 2 (/2) /2 DFT k n w kn /2, w /2 = e j2π /2 (4.19) F 1 (/2) k n w 2kn = e j2π(2kn)/ = e j2πkn/(/2) = w kn /2 (4.20) 0 k, n 2 1 F (/2) k n F 2 (/2) k n w 2k(n+/2) = e j2π(2k(n+/2))/ = e j2πkn/(/2) j2πk = e j2πkn/(/2) = w kn /2 (4.21) 0 k, n 2 1 F (/2) k n 2. F 3 (/2)F 4 (/2) F 3 (/2) = F 4 (/2) (4.22)

88 4. F 3 (/2) k n w (2k+1)n = e j2π(2k+1)n/, 0 k, n 2 1 (4.23) F 4 (/2) k n w (2k+1)(n+/2) = e j2π(2k+1)(n+/2)/ = e j2π(2k+1)n e j2π(/2)/ = e j2π(2k+1)n, 0 k, n 2 1 (4.24) F 3 (/2) k n 1 3. F 3 (/2) F 3 (/2) = F (/2)D(/2) (4.25) D(/2) = diag(w 0,w 1,,w /2 1 ) (4.26) F 3 (/2) k n w (2k+1)n = e j2π(2k+1)n/ = e j2πkn/(/2) e j2πn/ 0 k, n 2 1 (4.27) F (/2) k n e j2πn/ F (/2) diag (e j2π 0,e j2π 1,e j2π 2 ),,e j2π /2 1 ) (4.28) (4.26) 4 DFT F ()

4.2 FFT 89 ˆF () = F (/2) F (/2) F (/2)D(/2) F (/2)D(/2) (4.29) = F (/2) 0 I(/2) 0 0 F (/2) 0 D(/2) I(/2) I(/2) I(/2) I(/2) (4.30) I(/2) /2 4.1 D(/2) /2 F (/2) k n 4.1 (4.30) w kn /2, w /2 = e j2π/(/2), 0 k, n 2 1 (4.31) F () DFT F () 4.1 F (/2) FFT 5 (4.11) (4.12)

90 4. 0 1 2 3 4 5 6 7 000 001 010 011 100 101 110 111 000 010 100 110 001 011 101 111 000 100 010 110 001 101 011 111 0 4 2 6 1 5 3 7 (4.32) =8 DFT F (8) X(0) X(1) X(2) X(3) X(4) X(5) X(6) X(7) = w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 0 w 2 w 4 w 6 w 0 w 2 w 4 w 6 w 0 w 3 w 6 w 1 w 4 w 7 w 2 w 5 w 0 w 4 w 0 w 4 w 0 w 4 w 0 w 4 w 0 w 5 w 2 w 7 w 4 w 1 w 6 w 3 w 0 w 6 w 4 w 2 w 0 w 6 w 4 w 2 w 0 w 7 w 6 w 5 w 4 w 3 w 2 w 1 x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) (4.33) w = e j2π/8 (4.34)

X(0) X(2) X(4) X(6) = X(1) X(3) X(5) X(7) 4.2 FFT 91 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 x(0) w 0 w 2 w 4 w 6 w 0 w 2 w 4 w 6 x(1) w 0 w 4 w 0 w 4 w 0 w 4 w 0 w 4 x(2) w 0 w 6 w 4 w 2 w 0 w 6 w 4 w 2 x(3) (4.35) w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7 x(4) w 0 w 3 w 6 w 1 w 4 w 7 w 2 w 5 x(5) w 0 w 5 w 2 w 7 w 4 w 1 w 6 w 3 x(6) w 0 w 7 w 6 w 5 w 4 w 3 w 2 w 1 x(7) ˆX 0 (4) = F 1(4) F 2 (4) x 0(4) (4.36) ˆX 1 (4) F 3 (4) F 4 (4) x 1 (4) ˆX 0 (4) = [X(0),X(2),X(4),X(6)] T (4.37) ˆX 1 (4) = [X(1),X(3),X(5),X(7)] T (4.38) x 0 (4) = [x(0),x(1),x(2),x(3)] T (4.39) x 1 (4) = [x(4),x(5),x(6),x(7)] T (4.40) w 0 w 0 w 0 w 0 w F 1 (4) = F 2 (4) = 0 w 2 w 4 w 6 w 0 w 4 w 0 w 4 (4.41) w 0 w 6 w 4 w 2 F 4 (4) = w 4 F 3 (4) = F 3 (4) w 0 w 1 w 2 w 3 w = 0 w 3 w 6 w 1 w 0 w 5 w 2 w 7 w 0 w 7 w 6 w 5 (4.42)

92 4. w 0 w 1 w 2 w 3 w F 3 (4) = 0 w 3 w 6 w 1 w 0 w 5 w 2 w 7 w 0 w 7 w 6 w 5 w 0 w 0 w 0 w 0 w 0 0 0 0 w = 0 w 2 w 4 w 6 0 w 1 0 0 w 0 w 4 w 0 w 4 0 0 w 2 0 w 0 w 6 w 4 w 2 0 0 0 w 3 ˆX 0 (4) ˆX 1 (4) = F 1 (4)D(4) (4.43) F = 1 (4) F 1 (4) x 0(4) (4.44) F 1 (4)D(4) F 1 (4)D(4) x 1 (4) = F 1(4) 0 I(4) 0 0 F 1 (4) 0 D(4) I(4) I(4) x 0(4) (4.45) I(4) I(4) x 1 (4) 4.2 4.2 (4.45) F 1 (4) (4.34) w = e j2π/8 w = e j2π/4

X(0) X(2) = X(4) X(6) X(1) X(3) = X(5) X(7) 4.2 FFT 93 w 0 w 0 w 0 w 0 x 1 (0) w 0 w 1 w 2 w 3 x 1 (1) w 0 w 2 w 0 w 2 (4.46) x 1 (2) w 0 w 3 w 2 w 1 x 1 (3) w 0 w 0 w 0 w 0 x 2 (0) w 0 w 1 w 2 w 3 x 2 (1) w 0 w 2 w 0 w 2 (4.47) x 2 (2) w 0 w 3 w 2 w 1 x 2 (3) w = e j2π/4 (4.48) x 1 (0) x(0) x(4) x 1 (1) x(1) x(5) = + (4.49) x 1 (2) x(2) x(6) x 1 (3) x(3) x(7) x 2 (0) w 0 x(0) x(4) x 2 (1) w = 1 x(1) x(5) x 2 (2) w 2 (4.50) x(2) x(6) x 2 (3) w 3 x(3) x(7) F 1 (4) DFT F (4) 4.3 F (4) F (8) X(0) w 0 w 0 w 0 w 0 x 1 (0) X(4) w = 0 w 2 w 0 w 2 x 1 (1) X(2) w 0 w 1 w 2 w 3 (4.51) x 1 (2) X(6) w 0 w 3 w 2 w 1 x 1 (3) ˆX 00 (2) = F 1(2) F 2 (2) x 10(2) (4.52) ˆX 01 (2) F 3 (2) F 4 (2) x 11 (2)

94 4. 4.3 (4.46) (4.50) ˆX 00 (2) = [X(0),X(4)] T (4.53) ˆX 01 (2) = [X(2),X(6)] T (4.54) x 10 (2) = [x 1 (0),x 1 (1)] T (4.55) x 11 (2) = [x 1 (2),x 1 (3)] T (4.56) F 1 (2) = F 2 (2) = w0 w 0 = 1 1 (4.57) w 0 w 2 1 1 F 4 (2) = w 2 F 3 (2) = F 3 (2) = w0 w 1 (4.58) w 0 w 3 F 3 (2) = w0 w 0 w0 0 w 0 w 2 0 w 1 = F 1 (2)D(2) (4.59) (4.52) (4.45) ˆX 00 (2) = F 1(2) F 1 (2) x 10(2) ˆX 01 (2) F 3 (2) F 3 (2) x 11 (2)

4.2 FFT 95 = F 1(2) 0 I(2) 0 0 F 1 (2) 0 D(2) I(2) I(2) x 10(2) (4.60) I(2) I(2) x 11 (2) F 1 (2) F 1 (4) DFT 4.4 x(n) X(k) (4.32) X(0) X(1) X(3) X(7) 4.4 DFT F (8) FFT w = e j2π/8 4.2.2 FFT 1

96 4. 4.5 4.5 FFT 2 D(/2 m ),m =2,,L = log 2 /2 2 2 + L = log 2 4.1 log 2 FFT 4.1 radix2-fft 2 log 2 2 log 2 log 2 3 log 2 O() DFT O( 2 ) FFT O() O( 2 ) FFT O()

4.2 FFT 97 4.2.3 FFT FFT x() DFT DFT (4. F () F (/2) D(/2)F (/2) (4. F (/2) D(/2)F (/2) = I(/2) I(/2) I(/2) 0 F (/2) 0 I(/2) I(/2) 0 D(/2) 0 F (/2) x() DFT X() ˆx 0 (/2) = [x(0),x(2),,x( 2)] T (4.63) ˆx 1 (/2) = [x(1),x(3),,x( 1)] T (4.64) X 0 (/2) = [X(0),X(1),,X(/2)] T (4.65) X 1 (/2) = [X(/2+1),X(/2+1),,X( 1)] T (4.66) 4.6 FFT FFT 4.6 FFT

98 4. 4.2.4 radix4-fft FFT radix4-fft DFT ˆX() = ˆF ()x() (4.67) ˆX() = [ ˆX 0 (/4), ˆX 1 (/4), ˆX 2 (/4), ˆX 3 (/4)] T (4.68) ˆX 0 (/4) = [X(0),X(4),,X( 4)] T (4.69) ˆX 1 (/4) = [X(1),X(5),,X( 3)] T (4.70) ˆX 2 (/4) = [X(2),X(6),,X( 2)] T (4.71) ˆX 3 (/4) = [X(3),X(7),,X( 1)] T (4.72) ˆX 0 (k) = X(4k), 0 k 4 1 ˆX 1 (k) = X(4k +1), 0 k 4 1 ˆX 2 (k) = X(4k +2), 0 k 4 1 ˆX 3 (k) = X(4k +3), 0 k 4 1 x() = [x 0 (/4), x 1 (/4), x 2 (/4), x 3 (/4)] T (4.73) x() /4 DFT F ( /4) F ( /4) F ( /4) F ( /4) F ˆF () = 1 (/4) jf 1 (/4) F 1 (/4) jf 1 (/4) F 2 (/4) F 2 (/4) F 2 (/4) F 2 (/4) F 3 (/4) jf 3 (/4) F 3 (/4) jf 3 (/4) (4.74) ˆf(k, n) = f(4k, n) (4.75) ˆf(k + /4,n) = f(4k +1,n) 0 k 4 1 (4.76)

4.2 FFT 99 ˆf(k + /2,n) = f(4k +2,n) 0 n 1 (4.77) ˆf(k +3/4,n)=f(4k +3,n) (4.78) F i (/4) /4 DFT F (/4) D i (/4) F i (/4) = F (/4)D i (/4) (4.79) d i (k, k) = e j2πki/ D i (/4) (k, k) (4.80) ˆX 0 (/4) F (/4)D 0 (/4) ˆX 1 (/4) F (/4)D = 1 (/4) ˆX 2 (/4) F (/4)D 2 (/4) ˆX 3 (/4) F (/4)D 3 (/4) I(/4) I(/4) I(/4) I(/4) x 0 (/4) I(/4) ji(/4) I(/4) ji(/4) x 1 (/4) I(/4) I(/4) I(/4) I(/4) x 2 (/4) I(/4) ji(/4) I(/4) ji(/4) x 3 (/4) (4.81) 4.7 ˆX i (/4) F (/4) F () radix4-fft 1 radix4-fft 4.2 4.2 radix4-fft 3 4 log 4 3 log 4 2 log 4 (4 + 3 2 ) log 4

100 4. 4.7 radix4-fft radix2-fft radix4-fft 4.3 radix-4-fft =4 L 4.3 radix2-fft radix4-fft radix2-fft radix4-fft 64 768 1152 576 1056 128 1792 2688 - - 256 4096 6144 3072 5632 512 9216 13824 - - 1024 20480 30720 15360 28160 4.3 FFT (4.1) (4.2) DFT Inverse DFT 1/ IDFT DFT exp( jωt) exp(jωt) 1/ IDFT FFT 1/ 1/2

4.3 FFT 101 1. DFT FFT w 0 w 0 w 0 w 0 w F (4) = 0 w 1 w 2 w 3 w 0 w 2 w 0 w 2, w = e j2π/4 (4.82) w 0 w 3 w 2 w 1 (a) F (4) F 1 (2) F 2 (2) F 3 (2) F 4 (2) ˆF (4) = F 1(2) F 2 (2) (4.83) F 3 (2) F 4 (2) (b) (c) F 1 (2) F 2 (2) F 3 (2) F 4 (2) F 3 (2) F 1 (2) (d) ˆF (4) F 1 (2) (e) FFT F 1 (2) 2. DFT FFT w 0 w 0 w 0 w 0 w F (4) = 0 w 1 w 2 w 3 w 0 w 2 w 0 w 2, w = e j2π/4 (4.84) w 0 w 3 w 2 w 1 (a) F (4) F 1 (2) F 2 (2) F 3 (2) F 4 (2) ˆF (4) = F 1(2) F 2 (2) (4.85) F 3 (2) F 4 (2) (b) F 1 (2) F 3 (2) F 2 (2) F 4 (2)

102 4. (c) F 2 (2) F 1 (2) (d) ˆF (4) F 1 (2) (e) FFT F 1 (2) 3. DFT FFT x(n) x(n) DFT X(k) DFT =2 L DFT X(k) 1 DFT X(k),k =0 1 FFT ( ) = 128 = 1024 L = log 2 FFT DFT % % (1) (2) 1 (3) /2 (4) log 2 (5) L (6) 2 (7) ( 1) (8) (/2)L =(/2) log 2 (9) L = log 2 (10) 7 (11) 8 (12) 9 (13) 10 (14) 9.4 (15) 5.5 (16) 3.1 (17) 1.8 (18) 1 4.

2024 © docsplayer.net プライバシー | 利用規約 | フィードバック