. , () X(t) - , , Z(t) - . X(t) Þ Z(t) - , , . :
Z(t) = T[X(t)].
:
z(t) = h(t) * x(t-t),
h(t) - . , , , . :
z(t) = h(t)×x(t-t) dt.
: h(t) ó H(w).
() . ( ) , ( , , .).
. , , () (). .
, , , :
s(t) = c ´ a(t), s(t) = a(t-Dt), s(t) = a(t)+b(t).
, . , , :
y(t) = [s(t)]2, y(t) = log[s(t)].
, ( ) ( ).
, :
T[a(t)+b(t)] = T[a(t)]+T[b(t)].
:
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T[c ´ a(t)]= c ´ T[a(t)].
, , .
:
1. : Z(t) = f(t)×Y(t).
2. : Z(t) = dX(t)/dt.
3. : Z(t) = X(v) dv.
, - () , .. Z(t) = T[X(t)] = To[X(t)] + f(t).
, , X(t) Y(t), Z(t). , , ..:
Z(t) = T[c×X(t), c×Y(y)] = c×T[X(t),Y(t)],
Z(t) = T[X1(t)+X2(t), Y1(t)+Y2(t)] = T[X1(t),Y1(t)]+T[X2(t),Y2(t)].
, , .
Z(t) = T[X(t)] Z(t) X(t).
:
mz(t) = M{Z(t)} = M{T[X(t)]}.
: . :
mz(t) = T[M{X(t)}] = T[mx(t)], (17.3.1)
.. Z(t) X(t):
mz(t) = h(t) * mx(t-t). (17.3.2)
:
Rz(t1,t2) = M{Z(t1)Z(t2)}= M{T1[X(t1)]}T2[X(t2)]},
1 2 - t1 t2, , :
Rz (t1,t2) = T1T2[M{X(t1)X(t2)}] =T1T2[Rx (t1,t2)], (17.3.3)
.. .
Rz(t) . z(t) z(t+t) :
z(t)×z(t+t) = h(a)h(b) x(t-a) x(t+t-b) da db.
, ,
M{x(t-a) x(t+t-b)} = -Rx(t-a-t-t+b) = Rx(t+a-b),
:
Rz(t) = h(a)h(b) Rx(t+a-b) da db º Rx(t) * h(t+a) * h(t-b). (9.3.4)
, , , , , . .
, , t-b = t, :
h(t+a) * h(t-b) = h(t+a+b) * h(t) = h(t) * h(t+g) = Rh(t),
Rh(t) - . :
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Rz(t) = Rx(t) * Rh(t). (17.3.5)
.. . , , , .
:
Rzx (t1,t2) = T1[Rx(t1,t2)], Rxz(t1,t2) = T2[Rx(t1,t2)]. (17.3.6)
Rxz :
x(t)×z(t+t) dt = h(a) x(t) x(t+t-a) da dt.
Rxz(t) = h(a) Rx(t-a) da º Rx(t) * h(t-a). (9.3.7)
.. .
Ryx :
Rzx(t) = Rxz(-t) º Rx(t) * h(t+a). (17.3.8)
, h(t) = 0 t<0 Rxz(t) 0 t<0, Rzx 0 t>0.
, , ( ) .
(17.3.5), :
Sz(f) = Sx(f) |H(f)|2. (17.3.9)
, . .
, (17.3.7-8) :
Sxz(f) = Sx(f) H(f). (17.3.10)
Szx(f) = Sx(f) H(-f). (17.3.10')
, .
, (17.3.10) :
H(f) = Sxz/Sx Û h(t).
(17.3.4,9) :
sz 2 = Kz(0) = Sx(f) |H(f)|2 df º Kx(0) h2(t) dt = sx2 h2(t) dt, (17.3.11)
, :
= = Rz(0) º h2(t) dt º Sx(f) |H(f)|2 df. (17.3.12)
, ( - ). . :
sz 2 = - 2 º ( - 2) h2(t) dt. (17.3.13)
. :
gxz2(f) = |Sxz(f)|2/[Sx(f)×Sz(f)]. (17.3.14)
Sx(f) Sz(f) -, f :
0 £ gxz2(f) £ 1.
- . 1, , (17.3.14) Sxz Sz, Sx (17.3.9-10). . 0 1 :
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1. x(t) Þ z(t), . , , , ( ).
2. . , , , ..
3. z(t) x(t) - .
1-gxz2(f) z(t) f, x(t).
x(t) y(t). , - - . gxy x(t) y(t), x(t) y(t).
/4/.
.
. , , b,
Z(t) = a×X(t) + b×Y(t)
Z(t):
mz(t)= M{Z(t)}= M{aX(t)+bY(t)}= a×M{X(t)}+b×M{Y(t)}= a×mx(t)+b×my(t). (17.3.15)
:
Rz(t1,t2) = M{Z(t1)×Z(t2)}= M{[aX(t1)+bY(t1)][(aX(t2)+bY(t2)]}=
= M{a2X(t1)X(t2)+b2Y(t1)Y(t2)+×ab[X(t1)Y(t2)+Y(t1)X(t2)]} =
= a2Rx(t1,t2)+b2Ry(t1,t2)+ab×[Rxy(t1,t2)+Ryx(t1,t2)]. (17.2.16)
X(t) Y(t) Rxy Ryx . ( ). . , Z(t) = aiXi(t) t2-t1 = t :
Rz(t) = ai2Rxi(t) + aiajRxixj(t). (17.3.16')
X(t) y(t) Z(t)=X(t)+y(t) :
mz(t) = mx(t) + y(t), Rz(t1,t2) = Rx(t1,t2). (17.3.17)
X(t) Y Z(t)=X(t)+Y:
mz(t) = mx(t) + my, Rz(t1,t2) = Rx(t1,t2) + Dy. (17.3.18)
X(t) f(t). :
mz(t) = M{Z(t)}= M{f(t)×X(t)}= f(t)×M{X(t)}= f(t)×mx(t). (17.3.19)
Rz(t1,t2)=M{f(t1)X(t1) f(t2)X(t2)}= f(t1)f(t2)M{X(t1)X(t2)}=
= f(t1)f(t2)×Rx(t1,t2). (17.3.20)
f(t) = const = C Z(t) = C×X(t), :
mz(t) = ×mx(t), Rz(t1,t2) = 2×Rx(t1,t2). (17.3.21)
Z(t) = dX(t)/dt. X(t) , :
mz(t) = M{Z(t)} = M{dX(t)/dt} = d(M{X(t)})/dt = dmx(t)/dt, (17.3.22)
.. . :
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Rz(t1,t2) = M{(dX(t1)/dt1)(dX(t2)/dt2)}= M{X(t1)X(t2)}= Rx(t1,t2), (17.3.23)
.. .
Z(t) = X(v)dv.
mz(t) = M{Z(t)} = M{ X(v)dv} = M{X(v)}dv = mx(v)dv, (17.3.24)
.. . :
Rz(t1,t2) = M{ X(t1)dt1 X(t2)dt2} = M{ X(t1)X(t2)dt1dt2} =
= M{X(t1)X(t2)}dt1dt2] = Rx(t1,t2)dt1dt2, (17.3.25)
.. .
( , ).
Z(t) X(t) (17.3.2):
mz = h(t) * mx = mx h(t) dt, (17.3.26)
, , ( ) , .. . ( ), .
X(t) Y(t) Z(t), :
mz = mx + my, Dz = Dx + Dy + 2Kxy(0). (17.3.27)
Kz(t1,t2) = Kz(t) = Kx(t) + Ky(t) + Kxy(t) + Kyx(t). (17.3.28)
X(t) y(t) :
mz(t) = mx + y(t), Kz(t) = Kx(t). (17.3.29)
X(t) y(t) - , :
mz(t) = y(t)×mx, Dz(t) = y2(t)×Dx. (17.3.30)
Kz(t,t) = y(t)y(t+t)Kx(t). (17.3.31)
- mz = 0 :
Kz(t1,t2) = Kx(t1-t2) = - Kx(t) = Kz(t). (17.3.32)
Kzx(t) = d(Kx(t))/dt, Kxz(t) = -d(Kx(t))/dt. (9.3.33)
(17.3.32) , X(t) , t.
- mz(t) = mx(t)dt :
Kz(t1,t2) = Kx(u1-u2) du1du2. (17.3.34)