osnovy teorii sluchaynykh protsessov

7/26/2019 Osnovy Teorii Sluchaynykh Protsessov

1/88

____________________

()

.. , ..

,

2005


2/88

519

27

:

.. , ..

.. :

/ .., ... .: , 2005.

88 .

ISBN 5-7046-1197-4

,

: , , -

, . , -

-

, ..

ISBN 5-7046-1197-4 , 2005


3/88

3

() , -

-

. , -

, . , -

, , -

, . -

, -

, . -

, ,

.

,

.-

,

,

,

.

, -

,

.

-

.

-

,

, -

-

.

,

, ,

.


4/88

4

1.

1.1.

,

, , -

.

. -

, -

().

,

x, P(x).

. -

:

; -

, .

p(x).

x

xxxxPxp

x

+


5/88

5

. x

:

= xxxpx d)( . (1.4)

--

.

= xxpxfxf d)()()( -

f(x).

.

0x

: xxx =0 .

-

:

( ) ( ) .d)( 222 xxpxxxx

== (1.5)

:

( ) ( ) ( )222222 2 xxxxxxxx =+== . (1.6) -

:

=

x

xxpxF d)()( . (1.7)

F(x) , -

x.

-

.


6/88

6

1.2.

(xmin, xmax) . -

. 1.1 , - . 1.1 .

:

>>, -

,

Fx()

, . . Fy() Fx() (. . 3.9). -(3.23) :

)(ee

)( 22 xxxy RR =

+

,

. . -

.

. 3.9. RC-

-, . 3.10.

0

Fx(), Fy()

Fx(0)

Fx(0)/2

|K()|2

1

0

0,5

x(t)

t

y(t)

t

)

)


45/88

45

,

RC-.

. 3.10. RC-

3.5.

(, -

, ).

Fx() . 3.11 a.

.

: 2)()()( KFF xy = .

. 3.11.

0t

x(t)y(t)

Fx()

0 0

|K()|2

Fx(0)

)

) Fy()

0 0


46/88

46

-

30100 0. -

-

Fx(0).

20 )()()( KFF xy = .

:

=

0

d)cos()()( yy FR .

=0

:

[ ]

=++=

000 d)(cos)()( yy FR

+=

000 d)cos()cos()(yF

=+

000 d)sin()sin()(yF

)sin()()cos()( 00 ba = , (3.25)

+=

00 d)cos()()( yFa ,

+=

00 d)sin()()( yFb .

, 0

, :

+= d)cos()()( 0yFa , (3.26)

+= d)sin()(( 0yFb . (3.27)

-

0 , Fy(0 + ) , b() = 0 -

:

)cos()()cos()()( 02

0 yy aR == , (3.28)


47/88

47

)(2y , )(

.

-

. 3.12.

. 3.12.

-

. , 202

000 )()()()()( KFKFF xxy =+=+ ,

() -

, :

)()(2d)cos()()()( 02

0 xx FKFa ==

,

() ;

= d)cos()(2

1)( 2 K . (3.29)

-

:)cos()()(2)( 00xy FR = . (3.30)

3.3.

:

1)(0

j

K

K += .

Ry()

2 ()2


48/88

48

:

1

)( 0j

KK

+

= , Q2

0= .

:

20 e

2

)( = K .

(3.30),

)cos(e)( 0

020

= FKRy . (3.31)

(3.31) . 3.13.

. 3.13.

= 0:

020

2 FKy= . (3.32)

(3.14), -

:

= . (3.33)

3.4.

-

.

(. 3.14).

Ry()

2


49/88

49

,

Q.

F0, ,

(3.14), -

C

kT

QkTrQkTrQy ===

2

2

2 p222 . (3.34)

:

)cos(e)( p2 = yyR . (3.35)

100 -

200293 . (3.34),

1110

232 1004,4

10

2931038,1

=

=y 2 ,

6,36 .

1. -

?

2.

?

3.?

4.

?

5.

?

6. -

?

7. RC--

?

. 3.14.

C

ry(t)

L


50/88

50

4.

4.1.

, -

(. 4.1).

. 4.1.

- ,

. -. 4.2.

:

)](cos[)()( 0 tttEtx += , (4.1)

E(t) , 0 , (t) -

; E(t) (t)

cos(0t).

. 4.2.

x(t) E(t)

t

Fx()

0

Fx(0)


51/88

51

(3.25):

)sin()()cos()()( 00 baRx = , (4.2)

a() b() cos(0) ,

Fx() ,

b() = 0.

x(t) , -

, :

2

2

2e2

1)(

x

xp

= . (4.3)

-

. -

, E(t),

. ,

E(t)

-

x(t).

, .

x(t) : E(t) (t). -

, y(t),

x(t) -

900. (4.1)

:

)](sin[)()( 0 tttEty += . (4.4)

, x2(t) +y2(t) =E2(t).

,

-

, .

(4.1) (4.4), -

:

( ) ( ) )sin()(sin)()cos()(cos)()( 00 tttEtttEtx = , (4.5)


52/88

52

( ) ( ) )cos()(sin)()sin()(cos)()( 00 tttEtttEty += . (4.6)

:

( ) )()(cos)( tAttE = , (4.7)

( ) )()(sin)( tBttE = . (4.8)(4.5), (4.6) :

)sin()()cos()()( 00 ttBttAtx = , (4.9)

)cos()()sin()()( 00 ttBttAty += . (4.10)

x(t) (4.9) -

,

A(t) B(t) . -

, . . cos0t -

, A(t) -

.

, . . cos0t

900. 900,

, . B(t)sin0t

, B(t) -.

A(t) B(t),

(t) (t):

)()()( 22 tBtAtE += ,)(

)(arctg)(

tA

tBt = . (4.11)

A(t) B(t) (t) , -

, A(t) = (t) B(t) = 0,.

A(t) B(t) -

x(t) c -

,

. 4.3. -. 4.3. (t) (t)

2

)(tA

cos(0 t)

x(t)

2

)(tB

sin(0 t)


53/88

53

x(t), -

cos0t sin0t. cos0t, -

)2sin(2

)(

)2cos(2

)(

2

)(

)sin()cos()()(cos)( 000002

t

tB

t

tAtA

tttBttA += .

. ,

A(t)/2. , -

, sin0t ,

(t)/2.

x(t) (t),

A(t) B(t). (4.9) cos0t , (4.10)

sin0t .

)sin()()cos()()( 00 ttyttxtA += . (4.12)

, B(t):

)sin()()cos()()( 00 ttxttytB = . (4.13)

,

x(t) (t). x(t)

y(t) /2.

(4.12), (4.13) -

A(t) B(t). (4.11) (4.7) (4.8) -

(t) (t). -

-

y(t), A(t) B(t)

(t)

(t).

y(t). -

y(t) x(t) , -

, , -

y(t) , x(t). -

:

)sin()()cos()()()( 00 baRR xy == . (4.14)


54/88

54

,

x(t) y(t).

, , .

. , x(t) -

, y(t) .

, , -

x(t) y(t) .

x(t) y(t), [1]:

)cos()()sin()()( 00 baRxy += . (4.15)

A(t)

B(t).

.

(4.12), (4.13) , A(t) B(t) -

. -

)()()( += tAtARA , )()()( += tBtBRB . (4.16)

(4.16) A(t) (4.12),

( ) ( )])(sin)()(cos)()][sin()()cos()([)( 0000 ++++++= ttyttxttyttxRA .

, -

:

( ) ( )++++++= )(sin)cos()()()(cos)cos()()()(0000

tttytxtttxtxRA

( ) ( ))(cos)sin()()()(sin)sin()()( 0000 ++++++ tttxtytttyty ,

)()()()()( xRtytytxtx =+=+ , )()()()()( xyRtxtytytx =+=+ ,

, :

=+= )sin()()cos()()( 00 xyxA RRR

[ ] [ ] =++= )sin()cos()()sin()()cos()sin()()cos()( 000000 baba)(a= . (4.17)


55/88

55

, RB() = a(); RAB() = b().

, A(t) B(t) ,

x(t): 2=

2= 2. ,

(4.12), (4.13) , x(t) y(t)

, A(t) B(t) ,

:

2

2

2e2

1)(

A

Ap

= ,2

2

2e2

1)(

B

Bp

= . (4.18)

, RAB(0) = b(0) = 0,

A(t) B(t) -

, , , , -

:

2

22

22

e2

1)()(),(

BA

BpApBAp

+

== . (4.19)

-

(, ) -

(, ) [1]:

IBApEp ),(),( = , (4.20)

=BAE

B

E

A

I .

(4.7) (4.8), -

: EI= . (4.19) (4.20):

2

2

2

22

22

22

e2

e2

),(

EBA

EEEp

+

== . (4.21)


56/88

56

(),

(4.21)

:

2

2

2

2

e

d),()(

E

EEpEp

== . (4.22)

(4.22) . -

. 4.4.

1 1.253

. 4.4. ,

:

:

2

d)(

0

== EEEpE , (4.23)

:

==

0

222 2d)( EEpEE , (4.24)

:

22222 43,02

2 =

== EEE . (4.25)

-

, -

, , :

, .

E

p(E)

0 E


57/88

57

, -

(4.21) . -

:

2

1

)( =p , )(


58/88

58

== IBBAApEEp ),,,(),,,(

+

=

)1(2

)cos(2exp

)1(4 22

22

242

EEEEEE. (4.29)

(, ) -(4.29) :

=

d),,,(),,( EEpEEp . (4.30)

=

0)cos(

)(de

2

10 xI

x,

I0(x) .

(4.30)

( )

+

=

)1(

)1(2exp

12),,(

22022

22

24

EEI

EEEEEEp . (4.31)

, , (4.31) :

==

d),,(),( EEpEEp

( )

+

=

)1(

)1(2exp

1 22022

22

24

EEI

EEEE. (4.32)

:

2)( EEERE = . (4.33)

-(4.23). -

EE :

=

0 0

dd),( EEEEpEEEE (4.34)

(4.34) (4.32)

, , -:


59/88

59

++++= ...

256

1

64

1

4

11

2

6422EE . (4.35)

(4.35) (4.23) (4.33), -

:

+++= ...

64

1

16

1

8)( 6422

ER . (4.36)

, :

...014,0057,0915,0

)()( 642

2 +++==

E

EE

Rr (4.37)

-

:

)()( 2Er . (4.38)

4.1.-

)cos(e)( 02 =xR .

e)( = .

(4.38), -

:2

e)( Er .

, , -

:

22e43,0)( ER . (4.39)

(4.39) . 4.5.

. 4.5.

2

() =

2

exp(|

|)

RE()

2

0

0,432

1

2

1


60/88

60

4.2.

, . . -

,

. -

, . -

.

u(t) = Umcos (0t) +x(t), (4.40)

x(t) (4.9) . (4.9) (4.40),

:

u(t) = (Um+ ) cos (0t) sin (0t). (4.41)

(4.41)

(. 4.6).

A=Ecos Um, B=Esin . (4.42)

, -

:

p(E,) =p(A,B) |I|, (4.43)

p(A,B)

(4.19), |I| =E. ,

A2+B

2=E

2cos

2 2EUmcos +

2mU +E

2sin

2=

= E2+ 2mU 2EUmcos ,

. 4.6.

B

AUm

E


61/88

61

p(E,):

+=

2

22

2 2

cos2exp

2),( mm

EUUEEEp . (4.44)

, :

+==

202

22

2

2exp

d),()( mm

EUI

UEEEpEp . (4.45)

(4.45) .

Um. -

Um

.4.7. Um

= 0

(4.22). -

, Um>>

,

( )

2

2

2exp

2

1)( m

UEEp . (4.46)

0 1 2 3 4 5 6 7

0,2

0,4

0,6

. 4.7. ,

Um Um

2, . 4.8.

EUmUm>> EUm(. 4.9).

, (4.44)

:

E/

p(E)

Um= 0

Um= Um= 3

Um= 5


62/88

62

,cos2

exp)cos(cos212

exp2

1d),()( 2

22

0

+

==

a

aaa

EEpp

(4.47)

a= Um/ , (x) .

0 2 4

2

0,432

Um

2

E

0 2 4 Um/

1,253

p()

. 4.10. Um= 0 , -

, . . Um0

, ; Um

>> -

:

2

2

2exp

2

1)(

p . (4.48)

-

. 4.6. ,

Um >>

mm U

B

U

B= arctg , (4.49)

= /Um. (4.50)

(4.46), (4.48) , -

, . .

. 4.8. -

. 4.9. --

0

Um= 0Um

Um>>

p()

. 4.10. -


63/88

63

Um> 3. (4.51)

(4.51). , -

. 4.6, :

EUm+A(t). (4.52)

Um , KE()

R():

RE() R() = 2(). (4.53)

, -

)cos(|)|exp()( 02

=xR , Um> 3

|)|exp()( 2 =ER . (4.54)

, , -

-

. . 4.11 -

, . 4.11 .

. 4.11 .

, ,

, ,

. -

, -

.

. 4.12.

, (t) (t) -

, ,

. (t) -

, . 4.12 , -

. , -

,

R. Um0 (. 4.12 ) (t)


64/88

64

, Um RUm+R, . . -

(t) , Um= 0.

) )

. 4.12. : ,

)

Umcos(0 t)

)

)

x(t)

t

x(t) + Umcos(0 t)

t

t

. 4.11. : , , -

E(t)E(t)

A(t)

B(t)

Um

2R


65/88

65

,

. , Um= 0 (. 4.12 )

, Um

>> (. 4.12 ) -

, , Um/, -

.

1..

2.?

3.

? ?

4.

?

5.

?

6.

?

7. -?


66/88

66

5.

5.1.

-

:

,

.

-

: -

.

-

. -

, , -

. -

, .1

y(t) -

(1.18).

.

1.6,

y = |x|.

, (1.18) -

,

-

(. 5.1). -

:

=

2

2

2exp

2

1)(

xxp . (5.1)

x(t) . 5.2 . -

x(t) y(t)R

. 5.1.


67/88

67

, -

. x(t)

, . .y(t) =x(t), -

x(t) . , -

(. 5.3)

>=

.0,0

0,

x

xxy

y(t) . 5.2 .

, x y -

p(y) p(x). x,

, y= 0.

p(y) - .

, p(y)

> Sx() (. 6.10 ).

.


83/88

83

. 6.10. ,

1.?

2.

?

3.? -

? .

4.?

?

5. t0 ?

?

6.?

?

7. , -

?

0.5

1

Sx()

Sn()

K()

)

)


84/88

84

=

0

)cos()()( dFR

=

0

d)cos()(

2)( RF

F0() F0

)exp(2 22

2

2

+

2

( )1)exp(2 + ( )222

23

4

+

4

++ 222

311)exp(

( )32225

316

+

163

2

exp

222

2

22

2

exp

2

2

)sin(2

>

,0

,/2

2

cos(0) 2

( 0) 0

)cos(e 02

2

02

2

)(

1

+

+

)sin(

)cos(e 0

00

2

20

22p

22p

222

2p

2

;)(4

4

+=

+

( ) )cos(1e 02 +

( )220223

)(

2

+

2

)cos(2

exp 0

222

2

20

2

2

)(exp

2

2

)cos(

)sin( 0

2

+

,0

)()(),2/( 002

2


85/88

85

1. ... .: . .,2000.

2. .., ...

.: , 1994.3. .. .

.: , 1989.4. .. . .

.: , 2000.5. .., ..-

. .: , 1991.6. .. .

. .: . ., 2002.


86/88

86

.. 3

1. 4

1.1. .. 41.2. ... 6

1.3. . 6

1.4. 7

1.5. . 8

1.6. ... 9

.......... 12

2. . 13

2.1. ... 13

2.2. .. 16

2.3.

... 17

2.4. .. 19

2.5. ... 242.6. 28

2.7. ... 30

.......... 33

3. . 34

3.1. . 34

3.2. ... 35

3.3. ... 373.4. RC-.. 38

3.5. -... 45

.......... 49

4. 50

4.1. 50


87/88

87

4.2. -. 60

.......... 65

5.

66

5.1. -.. 66

5.2. -.. 70

........ 73

6. .. 74

6.1. , -

... 746.2. . 79

6.3. , -.. 81

.. 83

.. 84

85


88/88

,

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