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

Transcription

1 c /(41)

2 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 /(41)

3 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 /(41)

4 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 /(41)

5 S xx (ω) = R xx (m)(cos(ωm) j sin(ωm)) m= = R xx (0) + = R xx (0) 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) 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 /(41)

6 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 /(41)

7 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 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 /(41)

8 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 /(41)

9 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 /(41)

10 ( ) 2 sin(ωn) var[ŝ PE (ω)] = S xx (ω) 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 /(41)

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

12 1 群 5 編 4 章 ver1/ 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)

13 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 /(41)

14 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 /(41)

15 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 /(41)

16 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) 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 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 /(41)

17 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 /(41)

18 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 /(41)

19 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 /(41)

20 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 /(41)

21 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) 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 /(41)

22 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 /(41)

23 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 /(41)

24 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 = 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 L = M L < M L > M IIR STFT K h(n) FBS OLA FBS c /(41)

25 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 /(41)

26 (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) 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 /(41)

27 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 /(41)

28 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) 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 /(41)

29 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 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, pp , ) 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, ) TACM Claasen and WFG Mecklenbrauker, The Wigner Distribution A Tool for Time-Frequency Signal Analysis Part I: Continuous-Time Signals, Philips J Res 35, pp , ) TACM Claasen and WFG Mecklenbrauker, The Wigner Distribution A Tool for Time-Frequency Signal Analysis Part II: Discrete-Time Signals, Philips J Res 35, pp , ) 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, pp , ) O Rioul and P Flandrin, Time-Scale Energy Distributions: A General Class Extending Wavelet Transforms, IEEE Trans on Signal Processing, vol40, no7, pp , ) Boualem Boashash, Estimating and Interpreting the Instantaneous Frequency of a Signal Part 1: Fundamentals, Proc of the IEEE, vol80, no4, pp , ) Boualem Boashash, Estimating and Interpreting the Instantaneous Frequency of a Signal Part 2: Algorithm and Applications, Proc of the IEEE, vol80, no4, pp , ) F Hlawatsch and GF Boudreaux-Bartels, Linear and Quadratic Time-Frequency Signal Representations, IEEE SP Magazine, pp21-67, 1992 c /(41)

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

31 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 /(41)

32 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 /(41)

33 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, pp , May ) Hiroyoshi Morikawa, Adaptive estimation of time-varying model order in the ARMA speech analysis, IEEE Transactions on Acoustics, Speech and Signal Processing, vol38, no7, pp , July ) 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 ) 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, pp , June ) EN Demiris, SD Likothanassis, BG Konstadopoulou, DG Karelis, Real time nonlinear ARMA model structure identification, IEEE Proceedings of th International Conference on Digital Signal Processing, no2, pp , July ) Saeed Gazor, Prediction in LMS-type adaptive algorithms for smoothly time varying environments,, IEEE Transactions on Signal Processing, vol47, no6, pp , June ) 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, pp , Oct ) 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, pp , April ) MK Emresoy and A El-Jaroudi, Evolutionary Burg spectral estimation, IEEE Signal Processing Letters, vol4, no6, pp , June ) 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, pp , May 1998 c /(41)

34 11) A Harma, M Juntunen, A method for parametrization of time-varying sounds, IEEE Signal Processing Letters, vol9, no5, pp , May ) Yoshikazu Miyanaga, Nobuo Nagai, Nobuhiro Miki, ARMA Digital Lattice Filter Based on New Criterion, IEEE Transaction on Circuits and Systems, volcas-34, no6, pp , June ) 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, pp , June ) KS Nathan, HF Silverman, Time-varying feature selection and classification of unvoiced stop consonants, IEEE Transactions on Speech and Audio Processing, vol2, no3, pp , July ) 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 th IEEE Conference on Decision and Control, pp , Dec ) 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 rd IEEE Conference on Decision and Control, no2, pp , Dec ) 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 ) 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 ) 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, pp , Aug ) 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, pp , Aug ) 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, pp , Sep ) 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, pp , July 2003 c /(41)

35 ) n x 1, x 2,, x n p = m 1 + m 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 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 /(41)

36 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 /(41)

37 n S (ω 1,, ω n 1 ) = n C(l 1,, l n 1 ) l 1= l n 1= exp ( ( jω 1 l 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 /(41)

38 0 0 4) 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 /(41)

39 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 /(41)

40 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 /(41)

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