. , :
X(t) = X×j(t), (17.2.1)
- , j(t) - . :
mx(t) = M{Xj(t)}= j(t)×M{X}= j(t)×mx, (17.2.2)
mx - . mx = 0 mx(t) t (17.2.1) . :
Kx(t1,t2) = M{X(t1)X(t2)}= j(t1)j(t2)×M{X2}= j(t1)j(t2)×Dx. (17.2.3)
Dx - .
0X(t) :
0X(t) = Xi×ji(t), (17.2.4)
Xi. 0X(t):
M{0X(t)}= M{ Xi×ji(t)}= 0.
Kx(t1,t2) = M{0X(t1) 0X(t2)}= M{ Xi×ji(t1)Xj×jj(t2)}= ji(t1)jj(t2)M{XiXj}.
XiXj M{XiXj}= 0 i ¹ j, , i = j, M{XiXj}= M{Xi2}= Di. :
Kx(t1,t2) = ji(t1)ji(t2)Di. (17.2.5)
X(t) = mx(t) + 0X(t) = mx(t) + Xi×ji(t), (17.2.6)
mx(t) (17.2.5) , 0X(t) - X(t). (17.2.6) X(t). Xi , ji - . t1 = t2 (17.2.5) X(t):
Dx(t) = [ji(t)]2×Di. (17.2.7)
, (17.2.6) X(t), (17.2.5) , . . , t ji(t), X(t) ji(t).
, , - , exp(jwt). .
. :
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Z(t) = X(t) + jY(t), (17.2.8)
X(t) Y(t) - . , :
mz(t) = mx(t)+j×my(t). (17.2.9)
, , , , . , , . . , :
Rz(t1,t2) = M{Z(t1)×Z*(t2}= M{[X(t1)+jY(t1)][(X(t2)-jY(t2)]}=
= M{X(t1)X(t2)+Y(t1)Y(t2)+j×[Y(t1)X(t2)-X(t1)Y(t2)]} =
= Rx(t1,t2) + Ry(t1,t2) + j×[Ryx(t1,t2) - Rxy(t1,t2)]. (17.2.10)
, Ryx = Rxy = 0 (17.2.10) .
. t1 = t2 = t :
Dz(t) = M{|Z(t)-mz(t)|2} = Dx(t) + Dy(t), (17.2.11)
t.
. , 0- xk(t) 0X(t) :
xk(t) = Vx,k(wi) exp(jwit), (17.2.12)
Vx,k(wi) = (1/T) xk(t) exp(-jwit) dt, (17.2.13)
, :
xk(t) = Ax,k(0) + 2 (Ax,k(wi) cos(wit) + Bx,k(wi) sin(wit)), (17.2.12')
Ax,k(wi) = (1/T) xk(t) cos(wit) dt, (17.2.13')
Bx,k(wi) = (1/T) xk(t) sin(wit) dt. (17.2.13'')
wi = i×Dw - , Dw = 2p/T - . (17.2.13) . (17.2.4) (17.2.12) , (17.2.12) , Vx,k(w), Ax,k(w) Bx,k(w), - Vx(w), Ax(w) Bx(w). , 0X(t) .
0X(t) 0X(n×Dt) n 0 N, , , ( wN = p/Dt), (17.2.13) n (17.2.12). .
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, , . 0X(t), , (17.2.12-13) , (17.2.12-13), k. , 0X(t), :
M{X(t)} = M{Vx(wi)} exp(jwit) = 0, (17.2.14)
M{Vx(wi)} = 0, .. . , . , , , .
X(t), X(t) = mx(t) + 0X(t), :
mx(t) + 0X(t) ó mx(w) + Vx(w) = mx(w),
.., , ( ) , , , . - , .
, ( ) , , .
. X(t), 0-, :
PT = [x2(t)/T] dt = [|XT(f)|2/T] df,
X(f) x(t). , :
P = [ |XT(f)|2] df,
:
W(f) = |XT(f)|2.
. , . , , . :
Dx = W(f) df.
. 0-, Dw = p/T, wi = i×Dw, :
Kx(t) = Dx(0)/2 + Dx(wi) cos(wit), (17.2.15')
Dx(wi) = (2/T) Kx(t) cos(wit) dt, (17.2.16')
Dx(wi) (17.2.5) - Vx(wi), Ax(wi) Bx(wi), (17.2.12). , :
Kx(t) = Dx(wi) exp(jwit), (17.2.15)
Dx(wi) = (2/T) Kx(t) exp(-jwit) dt, (17.2.16)
. 17.2.1. . |
(D(w) ¹ ¥) (D(w) ³ 0), (D(-w) = D(w)). - . 17.2.1.
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X(t) (17.2.15) t = 0:
Dx = Dx(wi), (17.2.17)
.. .
, :
Bk = (Dw/Dmax) Dx(wi) = Dw×Dx/Dmax, (17.2.18)
Dmax - Dx(wi). , , , . , Dmax , , Bk , . (17.2.18) , .
T Þ ¥ (17.2.15), D(wi) S(w), - G(w), - . - , D(wi), . .
:
Bk = Gx(f) df /Gx(f)max = Sx(f) df /Sx(f)max = Kx(0) /Sx(f)max. (17.2.18')
Bk Tk. BkTk (17.1.7) (17.2.18'):
BkTk = 2 |Kx(t)|dt /Sx(f)max. (17.2.19)
:
BkTk ³ 1/2. (17.2.20)
, , .
. X(t) Y(t) Kxy(t) Kyx(t). :
Sxy(wi) = (1/T) Kxy(t) exp(-jwit) dt, (17.2.21)
:
Sxy(-w) = Sxy*(w) = Syx(w).
(17.1.11) , :
gxy2(w) = |Sxy(w)|2/(Sx(w)Sy(w)), (17.2.22)
w
0 £ gxy2(w) £ 1. (17.2.23)
X(t) Y(t) ( ).
-. q(t), . q(t) Q(w). t , Q(w)exp(jwt). Q(w) = Q*(w)
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x(t) y*(t) dt = X(f) Y*(f) df.
:
q(t)q(t+t) dt = (1/2p) Q(w)Q*(w) exp(jwt) dw.
Þ ¥, , - :
q(t)q(t+t) dt = |Q(w)|2 exp(jwt) dw,
R(t) = (1/2p) W(w) exp(jwt) dw. (17.2.24)
, , :
W(w) = R(t) exp(-jwt) dt. (17.2.25)
-. W(w) R(t) , :
R(t) = 2 W(f)cos(2pft) df, W(f) = 2 R(t)cos(2pft) dt.
, , , , .
K(t=0) = s2 = (1/2p) W(w) dw,
.., .
, , . - . , , - , , , ( ) .