E x(n) µ = E[x(n)] (4 1) x(n) x(n) x(0), x(1), x(2),, x(n 1) ˆµ = 1 N 1 x(n) N n=0 (4 2) (4 1) x(n) R xx (m) = E[x(n)x

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1 -- 5 4 2011 2 1 4-1 4-1-1 4-1-2 4-1-3 4-1-4 4-2 4-2-1 4-2-2 4-2-3 4-3 4-3-1 4-3-2 4-4 c 2011 1/(41)

1 -- 5 -- 4 4--1 4--1--1 2008 3 E x(n) µ = E[x(n)] (4 1) x(n) x(n) x(0), x(1), x(2),, x(n 1) ˆµ = 1 N 1 x(n) N n=0 (4 2) (4 1) x(n) R xx (m) = E[x(n)x(n + m)] (4 3) x(n) y(n) R xy (m) = E[x(n)y(n + m)] (4 4) c 2011 2/(41)

x(0), x(1), x(2),, x(n 1) ˆR xx (m) = N m 1 n=0 x(n)x(n + m) m 0 R xx ( n) m < 0 (4 5) (4 3) E[ ˆR xx (m)] = (N m )R xx (m) (4 6) (4 5) N m N m ˆR xy (m)/(n m ) (4 3) 1 R xx (m) m R xx (m) = R xx ( m) (4 7) 2 R xx (m) m = 0 R xx (0) R xx (m) (4 8) x(0), x(1), x(2),, x(n 1) ˆR xx (m) = N m 1 1 x(n)x(n + m) m 0 (4 9) N m n=0 x(n), n = 0, 1, 2,, N = 1 x(n + m), n = 0, 1,, N m 1 x(n) x(n + m) m x(n) x(n + m) m m x(n) x(n) x(n + m) m = 0 x(n) 2 1 x(n) x(n + m) m c 2011 3/(41)

x(n) 1 Hz 1 s x(n) R xx (m) x(n) 0 R xx (m) S xx (ω) = R xx (m)e jωm m= (4 10) S xx (ω) x(n) ω f R xx (m) = 1 2π π π S xx (ω)e jωm dω (4 11) (4 11) m = 0 E[x(n) 2 ] E[x(n) 2 ] = 1 2π π π S xx (ω)dω (4 12) (4 12) x(n) [ π, π] [ω, ω + dω] S xx (ω)/2π S xx (ω)/2π (4 10)(4 11) x(n) S xx (ω) 1 S xx (ω) 1 (4 10) R xx ( m) = R xx (m) c 2011 4/(41)

S xx (ω) = R xx (m)(cos(ωm) j sin(ωm)) m= = R xx (0) + = R xx (0) + 2 1 m= R xx (m)(cos(ωm) j sin(ωm)) + R xx (m) cos(ωm) m=1 R xx (m)(cos(ωm) + j sin(ωm)) m=1 (4 13) 2 S xx (ω) S xx (ω) = S xx ( ω) (4 14) 1 (4 14) S xx (ω) = R xx (m) cos(ωm) m= (4 15) (4 13)(4 15) R xx (m) S xx (ω) ω = 0 ω = 0 3 S xx (ω) S xx (ω) 0 (4 16) 4) 4--1--2 1 x(n), n = 0, 1, 2,, N 1 x(n) ˆr xx (τ) = 1 N 1 x(n)x(n + τ) N n=0 (4 17) c 2011 5/(41)

Ŝ xx (ω) = N 1 τ= (N 1) ˆr xx (τ)e jωτ (4 18) ˆr xx (τ) 0 (4 17) ˆr xx (τ) = 1 N N τ 1 n=0 x(n)x(n + τ), τ = 0, 1, 2,, N 1 (4 19) ˆr xx ( τ) = ˆr xx (τ), τ = 1, 2,, N 1 (4 20) τ = 0 N τ 1 τ = N 1 1 τ M Ŝ xx (ω) = w(τ)ˆr xx (τ)e jωτ τ= M (4 21) (4 21) Blackman-Tukey method (4 21) w(τ) 4 2 S xx (ω) = r xx (τ)e jωτ τ= (4 22) r xx (τ) τ =, 2, 1, 0, 1, 2, S xx (ω) (4 22) S xx (ω) = lim 2M M τ= 2M 1 = lim M 2M + 1 ( 1 ) τ r xx (τ)e jωτ 2M + 1 2M τ= 2M (2M + 1 τ )r xx (τ)e jωτ (4 23) c 2011 6/(41)

4 1 1 n M w(n) = 0 n > M 1 n M w(n) = n M 0 n > M 1 2 w(n) = + 1 πn 2 cos M n M 0 n > M 054 + 046 cos πn M w(n) = n M 0 n > M 2(1 n M )3 (1 2 n M )3 n M 2 w(n) = 2(1 n M )3 M 2 < n M 0 n > M 2M M M (2M + 1 τ )r xx (τ) = r xx (m n) τ= 2M m= M n= 2M (4 24) S xx (ω) 1 S xx (ω) = lim M 2M + 1 M M m= M n= 2M r xx (m n)e jωτ (4 25) r xx (τ) r xx (τ) = E[x(m)x(n)] S xx (ω) = lim E 1 M M x(m)x(n)e M 2M + 1 jω(m n) m= M n= M = lim E 1 M M x(m)e jωm x(n)e jωn M 2M + 1 (4 26) m= M n= M m n 2 S xx (ω) = lim E 1 M M 2M + 1 x(n)e jωn (4 27) n= M S xx (ω) x(n) (4 27) x(n), n = 0, 1, 2,, N 1 (4 27) c 2011 7/(41)

Ŝ xx (ω) = 1 N 1 2 N x(n)e jωn n=0 (4 28) (4 28) periodogram 2 0 (4 18) (4 18) (4 17) (4 22) Ŝ xx (ω) = = N 1 τ= (N 1) 1 N ˆr xx (τ)e jωτ N 1 2 x(n)e jωn n=0 (4 29) w(τ) = 1 M = N 1 (37) E[ˆr xx (τ)] = N k N r xx(τ), τ N 1 (4 30) N N k (4 29) c 2011 8/(41)

E[Ŝ xx (ω)] = = N 1 τ= (N 1) N 1 τ= (N 1) = F[w B (τ)r xx (τ)] = π π E[ˆr xx (τ)]e jωτ N k N r xx(τ)e jωτ W B (ω ξ)s xx (ξ)dξ (4 31) F w B (τ) W B (ω) W B (ω) W B (ω) = 1 N ( ) 2 sin(ωn/2) (4 32) sin(ω/2) (4 31) N W B (ω) δ dirac (ω) w B (τ) = 1 lim E[Ŝ xx (ω)] S xx (ω) N (4 33) Ŝ xx (ω) = F[w(τ)ˆr xx (τ)] = π W(ω ξ)ŝ xx (ξ)dξ π π E[Ŝ xx (ω)] = W(ω ξ)e[ŝ xx (ξ)]dξ π (4 34) (4 35) N (4 33) E[Ŝ xx (ω)] = π π W(ω ξ)s xx (ξ)dξ (4 36) c 2011 9/(41)

( ) 2 sin(ωn) var[ŝ PE (ω)] = S xx (ω) 2 1 + N sin ω (4 37) var[ŝ BT (ω)] = S xx(ω) 2 N M w(τ) τ= M (4 38) Ŝ PE (ω) Ŝ BT (4 38) (4 37) (4 37) ω, 0, ± π var[ŝ PE (ω)] S xx (ω) 2 (4 39) f N f 1 N (4 40) 2M + 1 f 1 M (4 41) c 2011 10/(41)

M (4 35) W(ω) (4 38) M M = N/5 (4 35) E[Ŝ xx (ω)] 0 4 2 x(n) = sin(2π(015)n) + sin(2π(020)n) + w(n) (4 42) w(n) 5dB N = 20 50 M = N/5 = 4 4 1 50 4 2 4 1 4 2 3dB 20dB 1),,,, 1987 2),,,, 1989 3),,,, 1991 4),, MATLAB,, 2004 5) SM Kay, Modern Spectral Estimation: Theory and Application, Prentice Hall, 1988 6) P Stoica and R Moses, Introduction to Spectral Analysis, Prentice Hall, 1997 c 2011 11/(41)

1 群 5 編 4 章 ver1/201134 図 4 1 ペリオドグラムによるスペクトル解析結果 図 4 2 ブラックマン チューキー法によるスペクトル解析結果 c 電子情報通信学会 2011 電子情報通信学会 知識ベース 12/(41)

4--1--3 2009 8 1 2 3 AR Autoregressive Model MA Moving Average Model ARMA Autoregressive Moving Average Model AR x(t) = MA x(k) = p i=1 q i=0 i=1 a i x(k i) + w(k), x(k) = 1 A(z) w(k) b i w(k i), x(k) = B(z)w(k) p q ARMA x(k) = a i x(k i) + b i w(k i), x(k) = B(z) A(z) w(k) i=0 (4 43) (4 44) (4 45) w(k) E{w(k)} = 0, E{w(k)w(l)} = σ 2 wδ kl A(z) = p a i z i, B(z) = i=0 q b i z i, a 0 = b 0 = 1 i=0 AR (4 43) x(k) x(k 1),, x(k p) w(k) E{w 2 (k)} AR {a i } r xx (l) + p σ 2 w l= 0 a i r xx (l i) = 0 l 0 i=1 (4 46) r xx (l) x(k) r xx (l) = E{x(k)x(k + l)} l 0 Yule-Walker r xx (0) r xx ( 1) r xx ( p + 1) r xx (1) r xx (0) r xx ( p + 2) r xx (p 1) r xx (p 2) r xx (0) a 1 a 2 a p = r xx (1) r xx (2) r xx (p) (4 47) {x(0), x(1),, x(n 1)} r xx (n) c 2011 13/(41)

ˆr xx = 1 N N n 1 t=1 x(t)x(t + n) (4 48) r xx (n) ˆr xx (n) (4 47) AR (4 46) l = 0 σ 2 w AR P AR ( f ) = σ 2 w 1 + p i=1 a i exp( j2π f i) 2 (4 49) (4 47) O(p 3 ) O(p 2 ) Levinson Lattice Levinson (m 1) AR {a (m 1) i } m AR {a (m) i } κ m a (m) m = 1 m 1 ρ m 1 r xx(m) + a (m 1) i r xx (m i) (4 50a) i=1 a (m) i = a (m 1) i + κ m a (m 1) m i, i = 1, 2,, m 1 (4 50b) ρ m = (1 κm)ρ 2 m 1 (4 50c) κ 1 = r xx (1)/r xx (0), ρ 1 = (1 κ1 2)r xx(0) AR (4 50a) {κ m } m AR e (m) f (k) = x(k) + m m 1 = x(k) + a (m) i=1 i=1 m 1 i x(k i) = x(k) + ( a (m 1) i i=1 a (m) i x(k 1) + κ m x(k m) ) + κ m a (m 1) m i x(k i) + κm x(k m) (4 50b) m 1 m 1 = x(k) + a (m 1) i x(k i) +κ m x(k m) + a (m 1) m i x(k i) i=1 i=1 } {{ }} {{ } e (m 1) f (k) e (m 1) b (k 1) AR (m 1) m f (k) = e (m 1) f (k) + κ m e (m 1) b (k 1) (4 51a) (k) = e(m 1) (k 1) + κ m e (m 1) f (k) (4 51b) e (m) e (m) b b e (0) f (k) = e (0) b (k) = x(k) (m 1) m c 2011 14/(41)

κ m κ m e (m 1) f (k) e (m 1) b (k 1) Burg Itakura Burg κ m = 2 N 1 k=m e(m 1) f (k)e (m 1) N 1 k=m [(e(m 1) f b (k 1) (k)) 2 + (e (m 1) b (k 1)) 2 ] (4 52) (4 52) AR AR {r 0, r 1,, r p } S ( f ) {r 0, r 1,, r p, r p+1, r p+2, } ARMA (4 43) IIR x(k) = h j z j w(k) = H(z 1 )w(k) (4 53) j=0 (4 45) x(k n) AR Yule-Walker r xx (q) r xx (q 1) r xx (q p + 1) 1 r xx (q + 1) r xx (q + 1) r xx (q) r xx ( p + 1) a 1 r xx (q + 2) = r xx (q + p 1) r xx (q + p 2) r xx (q) a p r xx (q + p) (4 54) AR Yule-Walker MA q k r xx (l) = σ 2 w b ( j)b( j + l) for l = 0, 1,, q (4 55) j=0 {r xx (l)} MA MA {x(k)} AR 1/C(z 1 ) AR c 0, c 1,, c L (p L < N 1) B(z 1 )/A(z 1 ) 1/C(z 1 ) c p c p 1 c p q+1 b 1 c p+1 c p+1 c p c p q+2 b 2 c p+2 = c L c L 1 c L q+1 b q c L (4 56) c 2011 15/(41)

MA MA ARMA (a) AR 1/C(z 1 ) w(k) ŵ(k) L ŵ(k) = c i x(k i) i=0 (4 57) (b) {x(k), ŵ(k)} {a i, b j } MA MA ŵ(k) x(k) z(k), y(k) (c) ARMA 3 ARMA P ARMA ( f ) = σ2 w 1 + q i=1 b i exp( j2π f i) 2 1 + p i=1 a i( j2π f i) 2 (4 58) p, q AIC AIC(i, j) = N ˆσ 2 w(i, j) + 2(i + j) (4 59) i, j AR MA ˆσ 2 w(i, j) w(k) N AIC(i, j) i, j AR MA 4--1--4 2009 8 x(k) = p A i exp( j2π f i k + jϕ i ) + w(k), k = 0,, N 1 (4 60) i=1 {A i, f i, ϕ i } i p w(k) σ 2 w (4 60) c 2011 16/(41)

x(0) x(1) x(n 1) = 1 1 exp( j2π f 1 ) exp( j2π f p ) exp( j2π f 1 (N 1)) exp( j2π f p (N 1)) a 1 a 2 a p + w(0) w(1) w(n 1) S = [ s 1, s 2,, s p ] s i = (1, exp( j2π f i ),, exp( j2π f i (N 1))) T a = (a i, a 2,, a p ) T a i = A i exp( jϕ i ) x = (x(0), x(1),, x(n 1)) T w = (w(0), w(1),, w(n 1)) T x=sa+ w (4 61) w(k) x Sa ( 1 p(x Sa) = exp 1 ) (x Sa) H (x Sa) (4 62) π N σ 2N w σ 2 w { f i } {a i } (4 62) L(a, f) = (x Sa) H (x Sa) (4 63) a â = [S H S] 1 S H x (4 64) (4 64) (4 63) L(â, f) = x H x x H S[S H S] 1 S H x (4 65) 2 (4 60) p r xx (l) = P i exp( j2π f i l) + σ 2 wδ(l) i=1 (4 66) P i = A 2 i i M M(M > p) R xx = E{x M x H M } = p P i s i si H + σ 2 wi = R ss + R ww (4 67) i=1 R ss p R ww M R ss {λ i } R xx c 2011 17/(41)

R xx = p (λ i + σ 2 w)u i ui H + i=1 M σ 2 wu i ui H i=p+1 U s = [u i,, u p } {s 1, s 2,, s p } s H i M α i u j = 0, for i = 1, 2,, p (4 68) j=p+1 Pisarenko M = p + 1 (4 69) si H u p+1 = 0 p (u p+1 ) n+1 exp( j2π f i n) = 0, for i = 1, 2,, p (4 69) n=0 (u p+1 ) n+1 p (u p+1 ) n+1 z n = 0 (4 70) n=0 2π f i f i s1 Hu 1 2 s2 Hu 1 2 s H p u 1 2 P 1 λ 1 λ p+1 s1 Hu 2 2 s2 Hu 2 2 s H p u 2 2 P 2 λ 2 λ p+1 = (4 71) s1 Hu p 2 s2 Hu p 2 s H p u p 2 P p λ p λ p+1 λ i ( p + 1) (p + 1) R xx λ p+1 = σ 2 w MUSIC P MUSIC ( f ) = 1 Mi=p+1 s H ( f )u i 2 (4 72) s( f ) = (1, exp( j2π f ),, exp( j2π f (M 1))) T f { f i } s H ( f j )ν i = 0 (4 72) AR AR R xx M R xx c 2011 18/(41)

a PC = R 1 xx r xx p i=1 1 λ i u i u H i r xx (4 73) p Capon AR a MV = 1 s H ( f )R 1 xx s( f ) (4 74) a T x(k) 2 s H ( f )a = 1 ESPRIT x(k) = (x(k),, x(k + m 1)) T y(k) = (x(k + 1),, x(k + m)) T (4 61) x(k) = Sa + w(k), y(k) = SΦa + w(k + 1) (4 75) Φ = diag[e j2π f1,, e j2π fp ] S ESPRIT { f i } Φ x(k) y(k) R xx = E{x(k)x H (k)} = SPS H + σ 2 wi (4 76a) R xy = E{x(k)y H (k)} = SPΦ H S H + σ 2 wz (4 76b) Z = E{w(k)w H (k + 1)} 1 2 C xx = R xx σ 2 wi = SPS H C xy = R xy σ 2 wz = SPΦ H S H Φ C xx ξc xy = SP(I ξφ H )S H ξ = exp( j2π f i ) C xx ξc xy ESPRIT R xx U s S U s = ST T (4 75) J 1 SΦ = J 2 S (4 77) J 1 1 (M 1) J 2 2 M S U s c 2011 19/(41)

E x = J 1 U s E y = J 2 U s S U s R xx E x E x = J 1 ST J 1 S = E x T 1 E y E y = J 2 U s = J 2 ST = J 1 SΦT E y = J 2 U s = J 2 ST = J 1 SΦT = E x T 1 ΦT S Φ T 1 ΦT Ψ E y = E x Ψ Ψ Ψ AIC(i) = 2N ln Σ i (a, f) + 6i i i AIC(i) = (M i) ln 1 M i Mj=i+1 λ j Mj=i+1 λ (M i) j + i(2m i) (4 78) c 2011 20/(41)

1 -- 5 -- 4 4--2 2010 5 4--2--1 4--2--2 4--2--3 f x (t) = 1 d 2π dt arg{x(t)} (4 79) x R (t) x H (t) x(t) = x R (t) + jx H (t) (4 80) 4--2--1 1 STFT X w (ω, τ) = + w(t τ)x(t)e jωt dt (4 81) x(t) w(t) (4 81) X w (ω, τ) 2 2 STFT STFT 2 ω 0 c 2011 21/(41)

rad/s ω 0 + ω rad/ s t 0 s t 0 + t s STFT (4 81) X w (ω, τ) = 1 jωτ e 2π + X(ω )W(ω ω )e jτω dω (4 82) + σ 2 t = (t t ) 2 w(t) 2 dt (4 83) + σ 2 ω = (ω ω ) 2 W(ω) 2 dω (4 84) t ω w(t) W(ω) (4 83) (4 84) σ t σ ω 1 2 (4 85) (4 85) STFT STFT STFT x(τ) = 1 2πw(0) + X w (ω, τ)e jωτ dω (4 86) w(0) 0 (4 86) 3 STFT (4 81) x(t) STFT x(n) STFT X w (Ω, m) = + n= w(n m)x(n)e jωn (4 87) Ω X w (Ω, m) 2 c 2011 22/(41)

STFT - x(m) = 1 2πw(0) +π π X w (Ω, m)e jωm dω (4 88) STFT m Ω k (4 87) X w (Ω k, m) = h(n) [ x(n)e jωkn] (4 89) h(n) = w( n) (4 89) STFT h(n) 4 3 x(n) h(n) X w ( Ω k, m) j e Ω n k 4 3 STFT (4 87) X w (Ω k, m) = e jωkm [ x(n) h(n)e jωkn] (4 90) 4 4 STFT h(n)e jωkn x(n) j n h(n) e Ω k X w ( Ω k, m) j e Ω m k 4 4 STFT Ω 0 2π K Ω k = 2πk/K k= 0, 1,, K 1 STFT DFT STFT STFT M K DFT c 2011 23/(41)

X w (k, m) = + n= h(m n)x(n)e j 2πnk K, k = 0, 1,, K 1 (4 91) 4 5 M DFT K STFT L (4 91) L = 1 4 5 DFT 4 STFT 1 L = M X w (k, Mm) = e j 2πMm K [ ] M x(n) h(n)e j 2πkn K (4 92) M M 4 3 4 4 L = M L < M L > M IIR STFT K h(n) FBS OLA FBS c 2011 24/(41)

K 1 H(Ω 2πk/K) = Kh(0) k=0 (4 93) M = 1 ˆx(m) = 1 K 1 X w (k, m)e j 2πkm K Kh(0) k=0 (4 94) OLA M ˆx(m) = M H(0) + p= 1 K 1 X w (k, Mp)e j 2πkm K K k=0 (4 95) IDFT OLA M = 1 + p= h(mp n) = H(0) M (4 96) g(n) K 1 ˆx(m) = + k=0 p= g(m Mp)X w (k, Mp)e j 2πkm K (4 97) (4 97) FBS OLA g(m) = δ(m) FBS g(m) = 1/H(0) OLA (4 97) K 1 ˆx(m) = k=0 e j 2πMm K [ Xw (k, Mn) g(n)e j 2πkn K M ] (4 98) M M FBS + m= g(n Mm)h (Mm n + sk) = δ(s) (4 99) M = L FFT c 2011 25/(41)

4--2--2 - (4 81) + + X w (ω, τ) 2 = w(t τ)x(t)e jωt 2 = r xw,x w (u, τ)e jωu du + r xw,x w (u, τ) = w(t τ)x(t)w (t τ u)x (t u)dt (4 100) (4 101) x(t) w(t τ) x w (t) r xw,x w (u, τ) (4 100) (4 101) (4 102) r xx (τ) = + x(t)x(t τ)dt (4 102) τ (4 100) W x (t, ω) = + ( x t + τ ) ( x t τ ) e jωτ dτ 2 2 (4 103) (4 103) (4 103) W x (t, ω) = 1 2π + ( X ω + ξ ) ( X ω ξ ) e jξt dξ 2 2 (4 104) 1) x(t) = 0 t (t 1, t 2 ) W x (t, ω) = 0 t (t 1, t 2 ) X(ω) = 0 ω (ω 1, ω 2 ) W x (t, ω) = 0 ω (ω 1, ω 2 ) 2) Wx(t, ω) = W x (t, ω) 3) W x (t, ω) = W x (t, ω) + 4) W x(t, ω)dω = x(t) 2 + W x(t, ω)dt = X(ω) 2 + + W x(t, ω)dωdt = + x(t) 2 dt = 1 + 2π X(ω) 2 dω 5) x(t) x(t t 0 )e jω0t W x (t, ω) W x (t t 0, ω ω 0 ) 6) x(t) = 1 2πx (0) + W x ( t 2, ω ) e jωt dω (4 105) c 2011 26/(41)

2) (4 80) h(τ) + ( WPS x (t, ω) = h(τ)x t + τ ) ( x t τ ) e jωτ dτ 2 2 (4 106) 4) Φ(t, ω) S W x (t, ω) = + + Φ(t t, ω ω )W x (t, ω )dt dω (4 107) X w (ω, τ) 2 = + + W x (t, ω )W w (t t, ω ω)dt dω (4 108) (4 103) A x (τ, ν) = + ( x t + τ ) ( x t τ ) e jνt dt 2 2 (4 109) (4 109) A x (τ, ν) = + ( X ω + ν ) ( x ω ν ) e jτω dω 2 2 (4 110) (4 109) ν = 0 (4 110) τ = 0 A x (τ, ν) = + + W x (t, ω)e j(νt τω) dtdω (4 111) W x (m, Ω) = 2 + n= x(m + n)x (m n)e j2ωn (4 112) c 2011 27/(41)

Cohen class S (t, ω) = + ( x u + τ ) ( x u τ ) φ(ν, τ)e j(νt+ωτ uν) dνdτdu 2 2 (4 113) φ(ν, τ) 4 2 4 2 φ (θ, τ) Wigner φ(θ, τ) = 1 Spectrogram Margenau-Hill Kirkwood Rihaczek Born-Jordan Cohen Page W w (t, ω) cos 1 2 θτ e jθτ/2 sin 1 2 θτ/ 1 2 θτ e jθ τ Choi-Williams e θ2 τ 2 /σ Zhao-Atlas-Marks sin aθτ g(τ) τ aθτ + + ( t ) t AS W x (t, ω) = Φ a, aω W x (t, ω )dt dω (4 114) - + + ( t W x,ψ (a, b) 2 = W x (t, ω ) b )W Ψ a, aω dt dω = 1 + ( ) t b 2 x(t)ψ dt (4 115) a a 4 6 (4 115) a b j k a = 2 j b= 2 j k 2 c 2011 28/(41)

ω 1 a a σ 1 t ω 1 σ t σ ω 1 ω1 a 1 σ a ω 1 ω 0 σt σ ω ω 1 0 a 0 a0σ t σω a 0 τ τ 1 0 t b 0 t b 1 a b 4 6 - W x,ψ ( j, k) 2 = + x(t)2 j/2 Ψ (2 j t k)dt 2 (4 116) 2 1) Jont B Allen and Lawerence R Rabiner, A Unified Approach to Short-Time Fourier Analysis and Synthesis, Proc of the IEEE, vol65, no11, pp1558-1564, 1977 2) Michael R Portnoff, Time-Frequency Representation of Digital Signals and Systems Based on Short- Time Fourier Analysis, IEEE Trans Acoustics, Speech and Signal Processing, volassp-28, no1, pp55-69, 1980 3) TACM Claasen and WFG Mecklenbrauker, The Wigner Distribution A Tool for Time-Frequency Signal Analysis Part I: Continuous-Time Signals, Philips J Res 35, pp217-250, 1980 4) TACM Claasen and WFG Mecklenbrauker, The Wigner Distribution A Tool for Time-Frequency Signal Analysis Part II: Discrete-Time Signals, Philips J Res 35, pp276-300, 1980 5) TACM Claasen and WFG Mecklenbrauker, The Wigner Distribution A Tool for Time-Frequency Signal Analysis Part III: Relations with Other Time-Frequency Signal Transformations, Philips J Res 35, pp372-389, 1980 6) O Rioul and P Flandrin, Time-Scale Energy Distributions: A General Class Extending Wavelet Transforms, IEEE Trans on Signal Processing, vol40, no7, pp1746-1757, 1992 7) Boualem Boashash, Estimating and Interpreting the Instantaneous Frequency of a Signal Part 1: Fundamentals, Proc of the IEEE, vol80, no4, pp520-538, 1992 8) Boualem Boashash, Estimating and Interpreting the Instantaneous Frequency of a Signal Part 2: Algorithm and Applications, Proc of the IEEE, vol80, no4, pp540-568, 1992 9) F Hlawatsch and GF Boudreaux-Bartels, Linear and Quadratic Time-Frequency Signal Representations, IEEE SP Magazine, pp21-67, 1992 c 2011 29/(41)

10) RE Crochiere and LR Rabiner, Multi-Rate Digital Signal Processing, Prentice-Hall, Inc, Englewood Cliffs, NJ, 1983 11) L Cohen, Time-Frequency Analysis, Prentice-Hall, Inc, Englewood Cliffs, NJ, 1995 1998 12), 2009 4--2--3 2009 3 1 1 2 (a) (b) Case 1 Case 2(a) (b) 2 1) u(k) y(k) 4 7 Time-Varying Model 4 7 u(k) c 2011 30/(41)

2, 3, 4, ARMA 5) n m y(k) = a i (k)y(k i) + u(k) + b j (k)u(k j) + w(k) i=1 j=1 (4 117) a i (k) b j (k) w(k) 6, a i (k) MA FIR 7) b j (k) 8, 9, AR IIR 10) AR AR Case 1 (4 117) 8, 9, 11) MEM MLM Case 2 Case 2(a) (4 117) 14) Case 2(b) (4 117) AR 11) ARMA 12, 13) 15, 16, 17) p(k + 1) = A(k)p(k) + q(k) y(k) = h(k) t p(k) + w(k) (4 118) p(k) k h(k) A(k) q(k) (4 117) A(k) q(k) p(k) (4 117) 6, M-1 LMS 7) 1, 2, 3, M-2 RLS 4) M-3 c 2011 31/(41)

5, 10) 3 M-1 M-2 M-3 (4 118) 3 18) y(k) = s c i (k) f (i, k) + w(k) i=1 (4 119) f (i, k) c i (k) c i (k) c i (k) f (i, k) y(k) u(k) 4 8 Time-Varying Model Network (4 119) Radial Base Function (RBF) 19, 20, 21) y(k) = s t c i (k) f (i, k) + g i,k (u(k)) + n(k) i=1 i=1 (4 120) f (i, k) (4 119) g i,k (u(k)) RBF 4 8 c 2011 32/(41)

RBF RBF 22) (4 120) 4 1) Yoshikazu Miyanaga, Eisuke Horita, Jun ya Shimizu, Koji Tochinai, Design of Time-Varying ARMA Models and Its Adaptive Identification, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vole77-a, no5, pp760-770, May 1994 2) Hiroyoshi Morikawa, Adaptive estimation of time-varying model order in the ARMA speech analysis, IEEE Transactions on Acoustics, Speech and Signal Processing, vol38, no7, pp1073-1083, July 1990 3) Yoshikazu Miyanaga, Nobuhiro Miki, Nobuo Nagai, Kozo Hatori, A Speech Analysis Algorithm Which Eliminates the Influence of Pitch Using the Model Reference Adaptive System, IEEE Transaction on Acoustic, Speech and Signal Processing, volassp-30, no1, pp88-96, Feb 1982 4) Yoshikazu Miyanaga, Nobuhiro Miki, Nobuo Nagai, Adaptive Identification of a Time-Varying ARMA Speech Model, IEEE Transaction on Acoustic, Speech and Signal Processing, volassp-34, no3, pp423-433, June 1986 5) EN Demiris, SD Likothanassis, BG Konstadopoulou, DG Karelis, Real time nonlinear ARMA model structure identification, IEEE Proceedings of 2002 14th International Conference on Digital Signal Processing, no2, pp869-872, July 2002 6) Saeed Gazor, Prediction in LMS-type adaptive algorithms for smoothly time varying environments,, IEEE Transactions on Signal Processing, vol47, no6, pp1735-1739, June 1999 7) T Naito, K Hidaka, Xin Jingmin, H Ohmori, and A Sano, Adaptive equalization based on internal model principle for time-varying fading channels, IEEE Proceedings of 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium, pp363-368, Oct 2000 8) JR Bellegarda and DC Farden, Time-varying modelling of arbitrary non-stationary signals, IEEE Proceedings of 1988 International Conference on Acoustics, Speech, and Signal Processing, volicassp-88, no4, pp2200-2203, April 1988 9) MK Emresoy and A El-Jaroudi, Evolutionary Burg spectral estimation, IEEE Signal Processing Letters, vol4, no6, pp173-175, June 1997 10) M Arnold, XHR Milner, H Witte, R Bauer, C Braun, Adaptive AR modeling of nonstationary time series by means of Kalman filtering, IEEE Transactions on Biomedical Engineering, vol45, no5, pp553-562, May 1998 c 2011 33/(41)

11) A Harma, M Juntunen, A method for parametrization of time-varying sounds, IEEE Signal Processing Letters, vol9, no5, pp151-153, May 2002 12) Yoshikazu Miyanaga, Nobuo Nagai, Nobuhiro Miki, ARMA Digital Lattice Filter Based on New Criterion, IEEE Transaction on Circuits and Systems, volcas-34, no6, pp617-628, June 1987 13) M Haseyama, N Nagai, and N Miki, N, An adaptive ARMA four-line lattice filter for spectral estimation with frequency weighting, IEEE Transactions on Signal Processing, vol41, no6, pp2193-2207, June 1993 14) KS Nathan, HF Silverman, Time-varying feature selection and classification of unvoiced stop consonants, IEEE Transactions on Speech and Audio Processing, vol2, no3, pp395-405, July 1994 15) P Falcone, M Tufo, F Borrelli, J Asgari, HE Tseng, HE, A linear time varying model predictive control approach to the integrated vehicle dynamics control problem in autonomous systems, IEEE Proceedings of 2007 46th IEEE Conference on Decision and Control, pp2980-2985, Dec 2007 16) J Chauvin, G Corde, P Moulin, M Castagne, N Petit, P Rouchon, Real-time combustion torque estimation on a diesel engine test bench using an adaptive Fourier basis decomposition, IEEE Proceedings of 2004 43rd IEEE Conference on Decision and Control, no2, pp1695-1702, Dec 2004 17) Junqing Wang and Tsu-Chin Tsao, Laser Beam Raster Scan Under Variablew Process Speed An Application of Time Varying Model Reference Repetitive Control System, IEEE Proceedings of 2005 International Conference on Advanced Intelligent Mechatronics, July 2005 18) Yi-Teh Lee and HF Silverman, On a general time-varying model for speech signals, IEEE Proceedings of 1988 International Conference on Acoustics, Speech, and Signal Processing, 1988 volicassp- 88, no1, pp95-98, April 1988 19) Hideaki Imai, Yoshikazu Miyanaga, Koji Tochinai, A Nonlinear Spectrum Estimation System Using RBF Network Modified for Signal Processing, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vole80-a, no8, pp1460-1466, Aug 1997 20) NM Haan, SJ Godsill, A time-varying model for DNA sequencing data, IEEE Proceedings of the 11th IEEE Signal Processing Workshop on Statistical Signal Processing, pp245-248, Aug 2001 21) Hui Peng, H Shioya, Xiaoyan Peng and K Sato, Nonlinear MPC based on the state-space form of RBF-ARX model, Proceedings of the 2004 IEEE International Conference on Control Applications, no2, pp1679-1684, Sep 2004 22) C Andrieu, M Davy, A Doucet, Efficient particle filtering for jump Markov systems Application to time-varying autoregressions, IEEE Transactions on Signal Processing, vol51, no7, pp1762-1770, July 2003 c 2011 34/(41)

1 -- 5 -- 4 4--3 2010 7 0 1) 0 4--3--1 1 n x 1, x 2,, x n p = m 1 + m 2 + + m n Mom ( x m1 1, xm2 2,, ) [ xmn n = E x m 1 1 xm2 2 ] xmn n (4 121) E[ ] j Φ(ω 1, ω 2, ω n ) = E [ exp ( j(ω 1 x 1 + ω 2 x 2 + + ω n x n ) )] Mom ( x m1 1, xm2 2,, ) xmn n = ( j) p p Φ(ω 1, ω 2,, ω n ) ω m1 1 ωm2 2 ωmn n ω1=ω 2= =ω n=0 (4 122) (4 123) Ψ(ω 1, ω 2,, ω n ) = ln ( Φ(ω 1, ω 2,, ω n ) ) p (4 124) c 2011 35/(41)

Cum ( x m1 1, xm2 2,, ) xmn n = ( j) p p Ψ(ω 1, ω 2,, ω n ) ω m1 1 ωm2 2 ωmn n ω1=ω 2= =ω n=0 (4 125) 2 k = 0, ±1, ±2, x(k) x(k) k Mom ( x(k), x(k + l 1 ),, x(k + l n 1 ) ) = Mom ( x(0), x(0 + l 1 ),, x(0 + l n 1 ) ) (4 126) n n R(l 1, l 2,, l n 1 ) = Mom ( x(k), x(k + l 1 ),, x(k + l n 1 ) ) (4 127) 1 R = µ = E[x(k)] n n R c (l 1, l 2,, l n 1 ) = Mom ( x(k) µ, x(k + l 1 ) µ,, x(k + l n 1 ) µ ) (4 128) 2 R c (l) x(k) n n C(l 1, l 2,, l n 1 ) = Cum ( x(k), x(k + l 1 ),, x(k + l n 1 ) ) (4 129) : : 2 C(l) = 2 R(l) µ 2 = 2 R c (l) (4 130) 3 C(l 1, l 2 ) = 3 R(l 1, l 2 ) µ( 2 R(l 1 ) + 2 R(l 2 ) + 2 R(l 2 l 1 )) + 2µ 3 = 3 R c (l 1, l 2 ) (4 131) µ = 0 2 R(l) = 2 R( l), (4 132) 3 R(l 1, l 2 ) = 3 R(l 2, l 1 ) = 3 R( l 2, l 1 l 2 ) = 3 R( l 1, l 2 l 1 ) (4 133) 0 l 2 l 1 3 n (n 1) 1, n 2) c 2011 36/(41)

n S (ω 1,, ω n 1 ) = n C(l 1,, l n 1 ) l 1= l n 1= exp ( ( jω 1 l 1 + + jω n 1 l n 1 ) ), π ω i π (4 134) P(ω) = 2 S (ω) = 2 C(l) exp( jωl) (4 135) l= 3 S (ω 1, ω 2 ) 4 f (x(k + l 1 ),, x(k + l n )) = ( 1 exp 1 ) (2π) n/2 Γ 1/2 2 xt Γ 1 x (4 136) k Γ ( ) γ i j = 2 R c li l j (4 137) n n, x x = ( x(l 1 ) µ, x(l 2 ) µ,, x(l n ) µ ) t (4 138) (4 136) n R c (l 1,, l n 1 ) 0 ; n : n R c (l 1,, l n 1 ) = (4 139) 2 R c (l i l m ) ; n : allpairing (i,m) 3) (1,, n) n/2 (i, m) 0 c 2011 37/(41)

0 0 4) 4--3--2 1 B(ω 1, ω 2 ) = 3 S (ω 1, ω 2 ) = l 1= l 2= 3 C(l 1, l 2 ) exp ( ( jω 1 l 1 + jω 2 l 2 ) ) (4 140) (4 131) 3 C(l 1, l 2 ) 3 R c (l 1, l 2 ) 0 a P(ω) = P( ω), 0 ω π (4 141) B(ω 1, ω 2 ) = B(ω 2, ω 1 ) = B ( ω 2, ω 1 ) = B( ω 1 ω 2, ω 2 ) = B(ω 1, ω 1 ω 2 ), 0 ω 1, ω 2 π (4 142) (ω 1, ω 2 ) [ π, π] [ π, π] (0 ω 1 ω 2, ω 1 05π, ω 2 π ω 1 ) 1) 3 S (ω 1, ω 2 ) b (ω 1, ω 2 ) 2 θ a, θ b [ π, π] c 2011 38/(41)

x(k) = cos(ω a k + θ a ) + cos(ω b k + θ b ) + cos(ω c k + θ c ), (4 143) ω c = ω a + ω b θ c = θ a + θ b x(k) B(ω 1, ω 2 ) = 1 ( 2 π2 δ(ω a ω 1 )δ(ω b ω 2 ), 0 ω 1 ω 2, ω 1 π ) 2, ω 2 π (4 144) 1) θ c θ a θ b 0 c B u (ω 1, ω 2 ) u(k) H(ω) x(k) B x (ω 1, ω 2 ) B x (ω 1, ω 2 ) = H(ω 1 )H(ω 2 )H (ω 1 + ω 2 )B u (ω 1, ω 2 ) (4 145) H H d B u (ω 1, ω 2 ) u(k) g(k) x(k) = u(k) + g(k) B x (ω 1, ω 2 ) B u (ω 1, ω 2 ) 0 2 (4 145) bic(ω 1, ω 2 ) = B(ω 1, ω 2 ) P(ω1 )P(ω 2 )P(ω 1 + ω 2 ) (4 146) 1, 5) (4 146) 6, 2 7) 3 L x(k), k = 0, 1,, L 1 N M x i (k), k = 0, 1,, N 1 i = 1, 2,, M W(k), k = 0, 1,, N N 1 X i (ω) = W(k) ( x i (k) µ ) exp ( jωk) k=0 (4 147) c 2011 39/(41)

µ = 1 L 1 x(k) L k=0 (4 148) B(ω 1, ω 2 ) = 1 M M X i (ω 1 )X i (ω 2 )Xi (ω 1 + ω 2 ) i=1 (4 149) 3 R c (l 1, l 2 ) = 1 L 0 k, k+l 1, k+l 2<L ( x(k) µ ) ( x(k + l1 ) µ ) ( x(k + l 2 ) µ ) (4 150) W(l 1, l 2 ), ( K l 1, l 2 K) B(ω 1, ω 2 ) = K l 1= K l 2= K K W(l 1, l ) 3 2 R c (l 1, l 2 ) exp ( j(ω 1 l 1 + ω 2 l 2 )) (4 151) (4 145) 8) 9) 10) P(ω) = 1 M M X i (ω)xi (ω) i=1 (4 149) B(ω 1, ω 2 ) (4 152) bic(ω 1, ω 2 ) = B(ω 1, ω 2 ) P(ω 1 ) P(ω 2 ) P(ω 1 + ω 2 ) (4 153) 1 11, 14) 7) 11) 12) 13) 14) 15) 16) 17) c 2011 40/(41)

18) 1) CL Nikias, AP Petropulu, Higher-order spectra analysis, Prentice Hall, 1993 2) DR Brillinger and M Rosenblatt, Asymptotic theory of estimation of k-th order spectra, Spectral Analysis of Time Series(Ed by B Harris), John Wiley & Sons, INC, pp153-188, 1967 3),,, 1978 4),,, 1970 5),,, 1982 6) MJ Hinich, Tests for Gaussianity and linearity of a stationary timeseries, Journal of Timeseries Analysis, vol3, no3, pp169-176, 1982 7) A, volj91-a, no9, pp861-868, 2008 8) MR Raghuveer and CL Nikias, Bispectrum estimation: a parametric approach, IEEE Trans ASSP, vol33, no5, pp1213-1230, 1985 9),, A, volj79-a, no6, pp1212-1220, 1996 10),, A, volj88-a, no8, pp974-982, 2005 11) YC Kim and EJ Powers, Digital Bispectral Analysis and Its Appliction to Nonlinear Wave Interactions, IEEE Trans Plasma Science, volps-7, no2, pp120-131, 1979 12) 3 31, Journal of Plasma and Fusion Research, vol81, no12, pp978-982, 2005 13) Tong H, Non linear time series, Oxford Science Publications, 1990 14) JWA Fackrell, Bispectral Analysis of Speech Signals, PhD Thesis, The University of Edinburgh, 1991 15) A Pillepich, C Porciani and S Matarrese, The bispectrum of redshifted 21 centimeter fluctuations from the dark ages, The Astrophysical J, vol662, pp1-14, 2007 16) D Kocur and R Stanko, Order Bispectrum: A New Tool for Reciprocated Machine Condition Monitoring, Mechanical Systems and Signal Processing, vol14, no6, pp871-890, 2000 17),, ME, vol11, no4, pp284-292, 1973 18) PS Glass, etal, Bispectral Analysis Measures Sedation and Memory Effects of Propofol, Midazolam, Isoflurame, and Alfentanil in Healthy Volunteers, Anesthesiology, vol86, no4, pp836-847, 1997 c 2011 41/(41)

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