Stochastic Processes stoprocess@yahoo.com.cn 111111
➀ ➁ ➂ ➃ Lecture on Stochastic Processes (by Lijun Bo) 2
(Stationary Processes) X = {X t ; t I}, n 1 t 1,..., t n I, n F n (t 1,..., t n ; x 1,..., x n ) = F n (t 1 + τ,..., t n + τ; x 1,..., x n ), τ R t i + τ I (i = 1,..., n) : 3
( ) : X = {X n ; n N}, X 1,..., X n,... E[X 1 ] = 0 D(X 1 ) = σ 2 > 0, X = {X n ; n N} ( ) (Strong Sense White Noise) ( ) k 1,..., k n m (k i + m N, i = 1,..., n), P(X k1 x 1,..., X kn x n ) = n P(X k i=1 i x i ) = n P(X k i=1 i +m x i ) = P(X k1 +m x 1,..., X kn +m x n ). 4
严格意义下的(离散时间)白噪声的样本轨道: X1 N (0, 1) 5
, X = {X t ; t 0}, F 1 (t 1 ; x 1 ) = F 1 (x) t 1, τ, F 2 (t 1, t 2 ; x 1, x 2 ) = F 2 (t 1 + τ, t 2 + τ; x 1, x 2 ) = F 2 (0, t 2 t 1 ; x 1, x 2 ), t 2 > t 1. 6
Proof: τ = t 1, F 1 (t 1 ; x 1 ) = P(X t1 x 1 ) = P(X t1 +τ x 1 ) = P(X 0 x 1 ). F 2 (t 1, t 2 ; x 1, x 2 ) = F 2 (t 1 + τ, t 2 + τ; x 1, x 2 ) = F 2 (0, t 2 t 1 ; x 1, x 2 ). 7
X = {X t ; t 0},, t, m X (t) = xf 1 (t; dx) = xf 1 (dx) := m X, R X (s, t) = x 1 x 2 F (s, t; dx 1, dx 2 ) = R 2 = R X (0, t s) := R X (t s). R 2 x 1 x 2 F (0, t s; dx 1, dx 2 ) 8
(Wide-Sense Stationary Process) X = {X t ; t I}, (1) m X (t) = m X ( ), t I, (2) R X (s, t) = R X (t s), s, t, t s I, X 9
(1) (2) X = {X t ; t I}, X Proof: X,, X τ, ϕ X (t 1 + τ,..., t n + τ; u 1,..., u n ) n = exp i u k m X (t k + τ) 1 n 2 = exp i k=1 n u k m X (t k ) 1 2 k=1 = ϕ X (t 1,..., t n ; u 1,..., u n ). n k,l=1 k,l=1 u k u l C X (t k + τ, t l + τ) u k u l C X (t k, t l ) 10
(3) : X = {X t ; t 0}, X m X (t) = 0, t 0, σ 2, s = t, C X (s, t) = 0, s t, 11
Proof: R X (u) = σ 2, u = 0, 0, u 0, R X (s, t) = R X (t s). 12
X = {X t ; t 1}, (Moving Averages): Y t = X t + X t 1 2 Y = {Y t ; t 1}, 13
Proof: (1) m Y (t) = m X(t) + m X (t 1) 2 = 0, t 1. 14
(2) R Y (s, t) = [ Xt + X t 1 E 2 X t + X t 1 2 ] = 1 4 [R X(s, t) + R X (s, t 1) + R X (s 1, t) + R X (s 1, t 1)] σ 2 4, s = t 1 σ 2 2, s = t = σ 2 4, s = t + 1 0, s = t + 2 0, 15
R Y (s, t) = R Y (u) = σ 2 2, s = t σ 2 4, s t = 1 0, otherwise. σ 2 2, u = 0 σ 2 4, u = 1 0, otherwise. R X (s, t) = R X (t s). 16
X = {X t ; t I} ( ), : (1) (2) (3) R X (0) = E[ X t 2 ] m X 2 0. R X (τ) = R X ( τ). R X (τ) R X (0). (4) R X (t s), n 1, t 1,..., t n I α 1,..., α n n α k α l R X (t l t k ) 0. k,l=1 17
Proof: (1) R X (0) = E[X t X t ] = E[ X t 2 ] = D X (t) + m X 2 m X 2 0. (2) R X (τ) = E[X t X t+τ ] = E[X t+τ X t ] = R X (t (t + τ)) = R X ( τ). (3) R X (τ) = E[X t X t+τ ] E[ X t X t+τ ] E 1/2 [ X t 2 ]E 1/2 [ X t+τ 2 ] = R 1/2 X (0)R1/2 X (0) = R X(0). 18
(4) n n n α k α l R X (t l t k ) = α k α l E[X tk X tl ] = E α k X tk α l X tl k,l= k,l=1 [ n ] n = E α k X tk α k X tk k=1 k=1 n 2 = E α k X tk 0. k=1 k,l=1 19
X = {X t ; t I} ( ), X R X (τ) τ = 0, τ R X (τ) Proof:. R X (τ) τ = 0, E [ X t0 +τ X t0 2] [ ] = E (X t0 +τ X t0 )(X t0 +τ X t0 ) = R X (t 0 + τ, t 0 + τ) R X (t 0 + τ, t 0 ) R X (t 0, t 0 + τ) + R X (t 0, t 0 ) = R X (0) R X ( τ) R X (τ) + R X (0) (1) 0, τ 0. X t 0 20
. X, (1) E [ X t0 +τ X t0 2] = R X (0) R X ( τ) R X (τ) + R X (0) 0, τ 0. lim τ 0 R X (τ) = R X (0). R X (τ) τ = 0, τ R X (τ) R X (τ) τ = 0, X, τ 0 R, 21
R X (τ) R X (τ 0 ) = E[Xt X t+τ ] E[X t X t+τ0 ] E[ X t (X t+τ X t+τ0 ) ] ( E[ X t 2 ] ) 1/2 (E[ Xt+τ X t+τ0 ]) 1/2 = (R X (0)) 1/2 (E[ X t+τ X t+τ0 ]) 1/2 0, τ τ 0. 22
X = {X t ; t I} ( ), (1) X R X (τ) τ = 0, (2) X R X (τ) τ = 0, (3) X, X = { dx t dt ; t I}, m X (t) = 0 R X (τ) = R X (τ). 23
Proof: 3.3.2( Page71): X t 0 s R X(t 0, t 0 ), t R X(t 0, t 0 ), 2 s t R X(t 0, t 0 ) 2 t s R X(t 0, t 0 ) X, R X (s, t) = R X (t s), s R R X (t 0 + h, t 0 ) R X (t 0, t 0 ) X(t 0, t 0 ) = lim h 0 h R X ( h) R X (0) = lim = R h 0 h X(0). t R X(t 0, t 0 ) = R X (0). 24
, 2 t s R X(t 0, t 0 ) = lim h,k 0 = lim h,k 0 R X (t 0 + h, t 0 + k) R X (t 0 + h, t 0 ) R X (t 0, t 0 + k) + R X (t 0, t 0 ) hk R X (k h) R X ( h) R X (k) + R X (0) hk = lim h 0 R X ( h) R X (0) h 2 s t R X(t 0, t 0 ) = R X (0). = R X(0). 25
(2) (1) (3) m X (t) = 0 R X (t, t + τ) = R X (τ) = R X (τ). m X (t) = dm X(t) dt = dm X dt = 0. 2 R X (t, t + τ) = s t R X(t, t + τ) R X (t + k, t + τ + h) R X (t, t + τ + h) R X (t + k, t + τ) + R X (t, t + τ = lim h,k 0 hk = lim h,k 0 R X (τ + h k) R X (τ + h) R X (τ k) + R X (τ) hk = lim h 0 R X (τ + h) + R X (τ) h = R X(τ). 26
R X (τ k) R X (τ) k R X(τ), k 0. 27
X = {X t ; t I} ( ), t I, X t dx t dt, Cov(X t, dx t dt ) = 0. Proof: X = {X t ; t I} ( ), R X (τ) = R X ( τ). X, R X (τ), R X( τ) = R X(τ). R X (0) = R X (0) R X (0) = 0. E[X t dx t dt ] = t R X(s, t) s=t = t R X(t s) s=t = R X(0) = 0. dx t Cov(X t dt ) = E[X dx t t dt ] m Xm X (t) = 0. 28
X = {X t ; t I}, t I, X t dx t dt Proof:, X t dx t dt X t dx t dt, (X t, dx t dt ) h, ( X t, X ) t+h X t h = (X t, X t+h ) 1 1 h 0 1 h X, (X t, X t+h ) ( X t, X ) t+h X t h. 29
, l.i.m h 0 X t+h X t h = dx t dt,,(x t, dx t dt ) 30
X = {X t ; t [a, b]} ( < a < b < ),f(t), b a f(t)x t dt, g(t), [ b ] b E g(s)x s ds f(t)x t dt = a a b b a a g(s)f(t)r X (t s)dsdt. 31
Proof: f(s)f(t)r X (t s) [a, b] [a, b], b b a a f(s)f(t)r X (t s)dsdt 3.4.1 3.4.7( Page76 77), 32
X = {X t ; t I} Y = {Y t ; t I}, R XY (s, t) = R XY (t s), s, t I, (X, Y ) R Y X (s, t) = R Y X (t s), s, t I. 33
(X, Y ), Z t = X t + Y t Z = {Z t ; t I} Proof: (1) (2) τ R, m Z (t) = E[Z t ] = m X + m Y ( ). R Z (t, t + τ) = E [ Z t Z t+τ ] = E[(Xt + Y t )(X t+τ + Y t+τ )] = R X (t, t + τ) + R XY (t, t + τ) + R Y X (t, t + τ) + R Y (t, t + τ) = R X (τ) + R XY (τ) + R Y X (τ) + R Y (τ). R Z (τ) = R X (τ) + R XY (τ) + R Y X (τ) + R Y (τ), R Z (t, t + τ) = R Z (τ). 34
(X, Y ), R XY (τ) (1) R XY (τ) = R XY ( τ). (2) R XY (τ) 2 R X (0)R Y (0), R Y X (τ) 2 R X (0)R Y (0). 35
Proof: (1) R XY (τ) = E[X t Y t+τ ] = E[Y t+τ X t ] = R Y X (t + τ, t) = R Y X ( τ). (2) R XY (τ) 2 = E[Xt Y t+τ ] 2 E[ X t 2 ]E[ Y t+τ 2 ] = R X (0)R Y (0). 36
X = {X t ; t R}, l.i.m T 1 2T T T X t dt, ( X t ) X R 37
X = {X t ; t R}, l.i.m T 1 2T T T X t X t+τ dt, ( X t X t+τ ) X R 38
X = {X t ; t R}, P( X t = m X ) = 1, X X = {X t ; t R}, P( X t X t+τ = R X (τ)) = 1, τ R, X X, X, X 39
X = {X t ; t R}, X 1 2T ( lim 1 τ ) C X (τ)dτ = 0. T 2T 2T 2T 40
Proof: E[ X t ] = lim T [ 1 T 2T E T X t dt ] = m X, P( X t = m X ) = 1 E [ X t m X 2] = 0 D( X t ) = 0 ( lim D 1 T 2T T T X t dt ) = 0. 41
D ( 1 2T T T X t dt ) = = = = 1 4T 2 E [ 1 T 4T 2 E 1 4T 2 E 1 4T 2 = 1 2T T [ T T T T T T T T 2T 2T T 2 (X t m X )dt (X s m X )ds T T T (X t m X )dt (X s m X )(X t m X )dsdt C X (t s)dsdt ( 1 τ 2T ) C X (τ)dτ. ] ] 42
X = {X t ; t R}, X lim T 1 T 2T 0 ( 1 τ 2T ) C X (τ)dτ = 0. Proof: X,, C X (τ) = C X ( τ). 43
X = {X t ; t R}, lim τ C X (τ) = 0, X Proof: lim τ C X (τ) = 0, ε > 0, T 1 > 0, τ T 1, C X (τ) < ε. 2T > T 1, 1 2T ( 1 τ ) C X (τ)dτ 2T 2T 2T 1 2T C X (τ) dτ = 1 T1 C X (τ) dτ + 1 C X (τ) dτ 2T 2T 2T T 1 2T T 1 τ 2T 1 2T 2T 1C X (0) + 1 2T 2(2T T 1)ε T 1 T C X(0) + 2ε. 44
T = max{ T 1 2, T 1 ε C X(0)}, T > T, 1 2T ( 1 τ ) C X (τ)dτ 2T 2T T 1 T C X(0) + 2ε < 3ε. 2T 45
I = [0, ) X = {X t ; t 0}, X t = l.i.m T 1 T T 0 X t dt. I = R, X 1 T ( lim 1 τ ) C X (τ)dτ = 0. T + T T T X, lim T + 2 T T 0 ( 1 τ T ) C X (τ)dτ = 0. 46
X t = A cos(t) + B sin(t) (t R), A B E[A] = E[B] = 0 D(A) = D(B) = σ 2. X = {X t ; t R} Proof: (1) X 1. m X (t) = E[X t ] = cos(t)e[a] + sin(t)e[b] = 0 = m X. 2. R X (t, t + τ) = E[X t X t+τ ] = (E[ A 2 ]) cos(t) cos(t + τ) + (E[ B 2 ]) sin(t) sin(t + τ) +(E[AB]) cos(t) sin(t + τ) + (E[BA]) sin(t) sin(t + τ) = σ 2 cos(τ) := R X (τ). 47
C X (τ) = R X (τ) = σ 2 cos(τ), 1 2T = σ2 T 2T 2T 2T 0 = σ2 sin(2t ) T ( 1 τ 2T ( 1 τ 2T ) C X (τ)dτ ) cos(τ)dτ + σ2 (cos(2t ) 1) 2T 2 0, T. 48
X = {X t ; t R} τ R, Y t = X t X t+τ, t R. Y = {Y t ; t R}, X Y 49
X Y ( τ R), X 1 2T ( lim 1 u ) (RY (u) R X (τ) 2) du = 0. T + 2T 2T 2T X Y ( τ R), lim T + 1 2T 2T 0 ( 1 u 2T ) (RY (u) R 2 X(τ) ) du = 0. 50
Proof: C Y (u) = R Y (u) R X (τ) 2. C Y (u) = R Y (u) m Y 2 = R Y (u) E[Y t ] 2 = R Y (u) E[X t X t+τ ] 2 = R Y (u) R X (t, t + τ) 2 = R Y (u) R X (τ) 2. 51
I = [0, ) X = {X t ; t 0} Y = {X t X t+τ ; t 0, t + τ 0}, X X t X t+τ = Y t = l.i.m T + 1 T T 0 X t X t+τ dt. X 1 T ( lim 1 u ) (RY (u) R X (τ) 2) du = 0. T + T T T X, Y, lim T + 2 T T 0 ( 1 u T ) (RY (u) R 2 X(τ) ) du = 0. 52
( PAGE112 4.3.4) X = {X t ; t R}, lim R X(τ) = 0, τ X Proof: Y t = X t X t+τ = X t X t+τ τ R, m Y (t) = E[X t X t+τ ] = R X (τ) = m Y. R Y (t, t + u) = E[Y t Y t+u ] = E[X t X t+τ X t+u X t+u+τ ] = E[X t X t+τ ]E[X t+u X t+u+τ ] + E[X t X t+u ]E[X t+τ X t+u+τ ] +E[X t X t+u+τ ]E[X t+τ X t+u ] = RX(τ) 2 + RX(u) 2 + R X (u + τ)r X (u τ) := R Y (u). ( PAGE112 4.3.4) 53
lim C Y (u) = 0. u C Y (u) = R Y (u) m Y 2 = RX(τ) 2 + RX(u) 2 + R X (u + τ)r X (u τ) m Y 2 RX(τ) 2 m Y 2 = 0, u. 4.3.2( Page111) 54
( ) X = (X 1, X 2,..., X n ) N(b, C) n, b = (b 1,..., b n ) = (E[X 1 ],..., E[X n ]), C = (C ij ) = (E[X i X j ]) 1 i,j n. n X k : r 1 + r 2 + + r N = k. M 1,...,N (X) := E [X r 1 1 Xr 2 2 Xr N N ], (a) k, k M 1,...,N (X b) = 0. (b) k = 2m, k M 1,...,N (X b) = (C ij C κl C xz ). ( ) 55
b = 0. E[X 1 X 2 X 3 X 4 ] = C 12 C 34 + C 13 C 24 + C 14 C 23 = E[X 1 X 2 ]E[X 3 X 4 ] + E[X 1 X 3 ]E[X 2 X 4 ] (2m 1)! +E[X 1 X 4 ]E[X 2 X 3 ], 2 m 1 (m 1)!. m E[Xi 4 ] = E[X i X i X i X i ] = 3Cii. 2 E[Xi 3 X j ] = E[X i X i X i X j ] = 3CiiC 2 ij. E[Xi 2 Xj 2 ] = C ii C jj + 2Cij. 2 E[Xi 2 X j X k ] = C ii C jk + 2C ij C ik. 56
ϕ(τ) : R C,, n 1, t 1,..., t n λ 1,, λ n ϕ n k,l=1 α k α l ϕ(t k t l ) 0. Bochner-Khintchine( ) ϕ(τ) ϕ(τ), ϕ(0) = 1. 57
(Wiener-Khintchine ) X = {X t ; t R}, R X (τ) R X (τ) = 1 2π e ixτ df X (x), i = 1, τ R, F X (x) R,,, F X () = 0, F X (+ ) = 2πR X (0). X F X (x), S X (x) F X (x) = x S X (x) X S X (y)dy, 58
Proof: (1) R X (0) = 0, F X (0) = 0. (2) R X (0) > 0, f(τ) = R X(τ) R X (0). X 4.2.1( Page104) f(τ), f(0) = 1, f(τ). Bochner-Khintchine( ) G(x) f(τ) = R X(τ) R X (0) = e ixτ dg(x). F X (x) = 2πR X (0)G(x). 59
X = {X t ; t R} R X (τ) R, R X (τ) dτ <, S X (x) = e ixτ R X (τ)dτ, x R. Wiener-Khintchine R X (τ) = 1 2π e ixτ df X (x) = 1 2π e ixτ S X (x)dx. 60
Proof: R X (τ) R, e ixτ R X (τ)dτ R X (τ) dτ <. R X (τ) Fourier S X (x) = Fourier, e ixτ R X (τ)dτ. R X (τ) = 1 2π e ixτ S X (x)dx. Wiener-Khintchine, df X (x) = S X (x)dx. 61
Wiener-Khintchine {X n ; n = 0, ±1, ±2,... }, R X (n, n + m) = E[X n X n+m ] = R X (m), m Z, Wiener-Khintchine R X (m) = 1 2π π π e ixm df X (x), F X (x) [ π, π],,, F X ( π) = 0, F X (π) = 2πR X (0). 62
X F X (x), S X (x) F X (x) = x π S X (y)dy, x [ π, π], S X (x) X 63
{X n ; n = 0, ±1, ±2,... } R X (m) <, m Z S X (x) = e ixm R X (m), m Z Wiener-Khintchine R X (m) = 1 2π π π e ixm df X (x) = 1 2π π π e ixm S X (x)dx. 64
( 4.4.2PAGE116) {X n ; n = 0, ±1, ±2,... } E[X n ] = 0, E[X m X n ] = σ 2 σ 2, m = n δ m,n = 0, otherwise, m, n = 0, ±1, ±2,.... {C n ; n = 0, ±1, ±2,... } C n <, C n 2 <. n Z n Z Y n = C k X n k. k Z Y = {Y n ; n = 0, ±1, ±2,... } ( 4.4.2PAGE116) 65
Proof: (1) Y m Y (n) = E[Y n ] = C k E[X n k ] = 0. k Z [ ] R Y (n, n + m) = E[Y n Y n+m ] = E C k X n k C l X n+m l = k,l Z k Z C k C l E [ X n k X n+m l ] = σ 2 C k C l δ n k,n+m l k,l Z = σ 2 C k C l δ n k,n+m l l k=m,k,l Z l Z = σ 2 C k C k+m δ n k,n k = σ 2 C n C n+m = R Y (m). k Z n Z 66
(2) R Y (m) R Y (m) = σ 2 C n C n+m σ 2 m Z m Z n Z C n C n+m m,n Z = σ ( 2 2 C n C m = σ 2 C n ) <. m,n Z n Z (3) 4.4.4(Page116) Y S Y (x) = m Z e ixm R Y (m) = σ 2 m Z e ixm n Z C n C n+m = σ 2 2 e ix(l k) C k C l = σ 2 C k e ixk k,l Z k Z. 67
x = {x t ; t R} F x (y) = Parseval, e iyt x t dt, y R. x 2 t dt = 1 2π F x (y) 2 dy. F x (y) 2 68
x2 t dt =, x t R ( ) lim T 1 2T T T x 2 t dt <. Parseval, x t, t T x T (t) = 0, t > T R, Fourier F x (y; T ) = e iyt x T (t)dt = T T e iyt x t dt. 69
Parseval 2T, T, lim T 1 2T T T x 2 t dt = 1 2π lim T 1 2T F x(y; T ) 2 dy. S x (y) = lim T 1 2T F x(y; T ) 2 = lim T 1 2T T T e iyt x t dt 2. X = {X t ; t 0}, lim T 1 2T E [ F X (y; T ) 2] = lim T 1 2T E T T 2 e iyt X t dt 70
X [ lim E 1 T 2T X T T X t 2 dt ] 71
X = {X t ; t 0} R X (τ), X, S X (y) = = lim T e iyτ R X (τ)dτ 1 2T E T T 2 e iyt X t dt = lim T 1 2T E [ F X (y; T ) 2]. 72
Proof: 1 T 2 [ 2T E e iyt X t dt = 1 T 2T E T T e iys X s ds T T e iyt X t dt ] R T X(τ) = = 1 2T = = 2T 2T T T T T ( 1 u 2T e iy(t s) R X (t s)dsdt e iyτ R T X(τ)dτ, ( 1 τ ) R X (τ)1 { τ 2T }. 2T ) e iyu R X (u)du 73
(T ), e iyτ R X (τ), 1 T 2 lim T 2T E e iyt X t dt = lim e iyτ R X(τ)dτ T T T = = e iyτ = S X (y). lim T RT X(τ)dτ e iyτ R X (τ)dτ 74
X = {X t ; t 0}, R X (τ), X, Proof: R X (τ), X, S X (y) = e iyτ R X (τ)dτ = lim T 1 2T E [ F X (y; T ) 2]. 1 lim T 2T E [ F X (y; T ) 2], S X (y) X,R X (τ) S X (y) = e iyτ R X (τ)dτ = e iyτ R X ( τ)dτ = S X ( y). 75
1 2π S X (y)dy = R X (0). R X (τ), S X (0) = R X (τ)dτ. Proof: : S X (x) = R X (τ) = 1 2π e ixτ R X (τ)dτ e ixτ S X (x)dx. 76
R X (τ) = a cos(ατ) X = {X t ; t R} S X (y) 77
Dirac Delta Dirac Delta δ(x) +, x = 0 δ(x) = 0, x 0, δ(x)dx = 1. δ, ( ) 1 p σ (x) = exp x2 2πσ 2 2σ 2 δ(x), as σ 0. 78
Dirac Delta f(τ)δ(τ t)dτ = f(t), f(τ)δ(τ)dτ = f(0). δ(x) Fourier e iyτ δ(τ)dτ = 1, Fourier 1 2π e iyτ dy = δ(τ). 79
: R X (τ) = a cos(ατ) X = {X t ; t R} S X (y) Answer: S X (y) = = a 2 = aπ e iyτ R X (τ)dτ = a ( 1 2π e iyτ ( e iατ + e iατ ) dτ e i(y α)τ dτ + 1 2π = aπ[δ(y a) + δ(y + a)]. e iyτ cos(ατ)dτ ) e i(y+α)τ dτ 80
X = {X t ; t R} Y = {Y t ; t R} ( X, Y R XY (t, t + τ) = R XY (τ)) R XY (τ), S XY (y) = e iyτ R XY (τ)dτ, y R X Y 81
X = {X t ; t R} Y = {Y t ; t R} R XY (τ), S XY (y) = lim T 1 [ ] 2T E F X (y; T )F Y (y; T ), F X (y; T ) = T T e iyt X t dt, F Y (y; T ) = T T e iyt Y t dt. 82
(1) S XY (y) = S Y X (y). (2) R XY (τ) S XY (τ) Fourier (3) X Y, y Re(S XY (y)),y Im(S XY (y)) (4) S XY (y) 2 S X (y)s Y (y), S Y X (y) 2 S X (y)s Y (y). 83
Proof: (1) S XY (y) = lim T = lim T 1 [ ] 2T E F X (y; T )F Y (y; T ) 1 [ ] 2T E F Y (y; T )F X (y; T ) (2), S XY (y) Fourier = S Y X (y). R XY (τ) = 1 2π e iyτ S XY (y)dy. 84
(3), S XY (y) = = e iyτ R XY (τ)dτ R XY (τ) cos(yτ)dτ i R XY (τ) sin(yτ)dτ Re(S XY (y)) = R XY (τ) cos(yτ)dτ = R XY (τ) cos( yτ)dτ = Re(S XY ( y)) Im(S XY (y)) = R XY (τ) sin(yτ)dτ = R XY (τ) sin(yτ)dτ = Im(S XY ( y) 85
(4) Cauchy Schwarz S XY (y) 2 = lim T = lim T lim T = S X (y)s Y (y). 1 [ ] 2 2T E F X (y; T )F Y (y; T ) 1 [ ] 2 4T 2 E F X (y; T )F Y (y; T ) 1 2T E [ F X (y; T ) 2] lim T 1 2T E [ F Y (y; T ) 2] 86