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Днепропетровский национальный университет им. Олеся Гончара

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x(t)

0 t 0 t

#. 1.7. # '*&( ! &( "(, #/ "( ! ( ) / #/ "( ! ( )

0 ! " #(

* + +

" . *, # ,

+ .

& ξ (t) #/ "( ! , $ $+ '$ !$'( (, #

N - "

τ :

F (t1, ,tN , x1, , xN ) = F (t1 + τ, ,tN + τ, x1, , xN ).

ξ (t) #/ "( ! , $ 2 ! '$ !$'( (, #

, " +

t1, τ + :

m(t) = const ,
K (t1,t1 + τ) = k (τ).	(1.6)

2 k (τ) / !( "(, %.

& τ , .

0, # ,

+ . 0 , ,

. % + () 3)

() &) ( . 1.8).

) &

) 3

#. 1.8. ( ( 2 '(3 '

! " # ' $ $+ '$ / 2 ! '$ !$'( 0

/ -

m(t) = m = const , D(t) = D = const ,

" " + – τ :

K (t1,t1 + τ) = k (τ).

1.1. * , #

. *

ξ (t) η(t) = ξ(t) − mξ (t) ,

. %, # +

, . - " :

1)k (τ) = k (−τ), " τ ;

2)k (0) = D , D = K (t,t) = k (t − t) = k (0) ;

3)k (0) ≥ 0, D ≥ 0;

4)k (τ) ≤ k (0);

5)’ 3.

' "

r (t	,t	+ τ) = r (τ) =	k (τ)	=	k (τ)	(1.7)

1	1		D		k (0)

, # , +

1)r (τ) = r (−τ);

2)r (0) = 1;

3)r (τ) ≤ 1;

4)’ .

2 r (τ) / ! & "(, %.

" + , #

" ( . 1.9, , ). - "

, -+, #

+ , - ,

. ( # ,

. !( ' ,

) .

4 , # .

! & #/ (#/+ , #

« »

;

3 . k (t − t′)ϕ (t)ϕ(t′)dtdt′ ≥ 0 ϕ(t) ,	ϕ2 (t)dt < ∞ ; B – -
(B) (B)	(B)
t .

k (τ)	r(τ)
D	1
τ	τ

#. 1.9. / !( "(, ( ) / / ! & "(, ( ) -$"(1 ! " #$

!, ξ1 (t),

x1(t) ( . 1.10, ), :

: m,

, . ( # -

, ,

T , , # T

# . *,

( ),

;

. . / , # .

ξ2 (t), x2 (t)

( . 1.10, ) . ( #

, # . % .

x1 (t)

x2 (t)

mm2

0 t 0 t

#. 1.10. # '*&( ! &( "(, ! ( ) / ! ( )

0 ! " #(

( ) ". ( #

" +		τ
,	. ( # "
τ → ∞ , .
( #		,
	( ,	, ,

, .

. , #

[0;T ] , ":

	1	T	1	T
m = lim		x(t)dt ≈		x(t)dt .	(1.8)
			T
T →∞ T		0		0

D = lim 1 (x(t)− m)2 dt

T →∞ T

	1	T
≈		(x(t)− m)2 dt .	(1.9)
	T
		0

- "

ξ (t) − m ξ (t + τ) − m:

			1	T
k (τ) = lim			1	(x(t)− m)(x(t + τ)− m)dt ≈
k (τ) = lim				(x(t)− m)(x(t + τ)− m)dt ≈
	T →∞ T			0
				0
	1	T −τ
≈	1			(x(t)− m)(x(t + τ)− m)dt .				(1.10)
≈	T − τ			(x(t)− m)(x(t + τ)− m)dt .				(1.10)
	T − τ		0
			0
& "
				r (τ) =	k (τ)		.	(1.11)
				r (τ) =			.	(1.11)
						D
+ ,
.
1.3. / / #/ "( ! " #$
/ .
! ' /! ' θ						! " #$
",						.
" .
	ˆ	",
"( ! ' /! θ		",
, θ .
'							ˆ
'				θ θ ,

, – . * :

1) #$(#/+, #

E{θˆ} = θ ;

2)# ! ' 3 (#/+

xk (ti )

- ε > 0 :

lim Ρ	{	ˆ		− θ		≤ ε		= 1,
lim Ρ		θ	N	− θ		≤ ε		= 1,
N →∞			N				}
N →∞
N – , ;
3) -/ (#/+, #								k +
:
ˆ					ˆ(k )		}.
θ : min D{					θ		}.
		k

& , "

, # . /

. -

+ . ) ,

, + #

. % , #

. ! n

t1,t2, ,tN

Ωn,N = {xk (t1), xk (t2 ), , xk (tN );k = 1,n},

– k - ti , k = 1,n , i = 1, N 0 (1.1)

Ωn,N

mˆ (ti ) = 1 xk (ti ), i = 1, N . n k=1

! (1.2) ":

ˆ	1	N	ˆ	2
	1			2


D(ti ) =	N −1	(xk (ti )− m(ti )) , i = 1, N ,
	N −1	k=1
		k=1

(1.3)

σ (ti ) =	ˆ
	ˆ	, i = 1, N .
	D ti	, i = 1, N .
ˆ

0 " (1.4) +

K (ti,t j )=

(xk (ti )− m(ti ))

(xk (t j )− m(t j )), i, j = 1, N .

N k =1

" (1.5) –

K (ti ,t j )

R(ti ,t j

, i, j = 1, N ,

N −1

σ (ti )

σ (t j )

	N	’ .

	N −1

' t1,t2, ,tN + x(t),

" Ω1,N = {x(ti );i = 1, N}. , #

h = ti+1 − ti . % ti = ih

Ω1,N = {xi = x(ih),i = 1, N}.

Ω1,N

+ .

% (1.8) "

mˆ = 1 xi . N i=1

0 (1.9) +:

Dˆ = N1−1 i=1(xi − mˆ )2 ,

–

σˆ = Dˆ .

0 " (1.10) Ω1,N

. & τ , # 0 L ≤ N . % Ω1,N ,

+ τ

kˆ(0) = covˆ {(x1, x2, , xN ),(x1, x2, , xN )} , kˆ(1) = covˆ {(x1, x2, , xN −1),(x2, x3, , xN )},

…

, xN − L ),(xL+1, xL+

(L) = cov{(x1, x2,

2, , xN )} ,

N −τ

(xi

τ = 0,L ,

k (τ) =

N − τ

− m1)(xi

+ τ − m2 ),

i=1

m1 =

N −τ

m2 =

N −τ

xi ;

xi+τ .

N − τ

i=1

& " (1.11)

(τ) =

N − τ kˆ(τ)

, τ = 0,L ,

N − τ −1

ˆ ˆ

D1D2

(1.12)

(1.13)

–

{xi;i =

} {xi;i =

}

1, N − τ

τ + 1, N

N −τ

N − τ −1

(xi − m1)

N − τ −1

(xi+ τ − m2 )

i=1

! .

1.4. /! &+ 4(&+ (#/+ ! " #$

. .

, ’ ,

2’ . $

#. / 2’

, # . * # /! &+ , &(. /

". *

, ,

!' ( ' # / '.

1.4.1. ( ! / ! 5$!’)

x(t) = x(t ± kT ),

k = 1,2,

(1.14)

(+ )

( 2’ ):

∞

x(t) = a0 + (ak cos(2πfkt)+ bk sin(2πfkt)),

(1.15)

k =1

= kf

– , k = 1,2,3, ;

a0 =

x(t)dt ;

x(t)cos(2πfkt)dt ;

bk = T x(t)sin(2πfkt)dt

, # " x(t) - [0;T ]

+ . 2’

x(t0 )

(x(t + 0)+ x(t

− 0))

* a0

x(t). .

(1.15) ω = 2πf .

5 (1.15) ,

{xi = x(ih); i =

}, h –

1, N

, # fc =

, # N – , (1.15)

N 2

N 2−1

ak cos 2π

t +

bk sin

2π

, t [0;Ts ] ,

x(t) = a0 +

k=1

Ts = Nh – .

* , # t = ih , 1, N

	N 2			N 2−1
xi = x(ih) = a0	+ ak cos	2πki	+	bk sin	2πki	.	(1.16)
xi = x(ih) = a0	+ ak cos		+	bk sin		.	(1.16)
	k=1	N		k=1	N
	k=1			k=1

' " ":

2πki

, k =

xi =

xi cos

;

1, N 2 −1;

i=1

xi cos(iπ);

xi sin

2πki

, k =

aN 2

1, N

2 −1.

i=1

( # N – ,

N −1

xi = x(ih) = a0 + (

)

ak cos

2πki

+ (

)

bk sin

2πki

k =1

k=1

xi =

xi cos

2πki

;

ak =

k = 1,

(N −1)

2 ;

i=1

xi sin

2πki

bk =

k = 1,

(N −1)

2 .

i=1

x(t) = sin(0,1t) + sin(0,2t) + sin(t),t =

1,200

2’

k = 7 k = 40 ( . 1.11). /

, #

( . 1.11, ). &

( . 11.1, ).

#. 1.11. 0( , / ( & , ! " # ( (&+ (#/% !' ( k = 7 ( ) / k = 40 ( )

* " ak , bk 2’ (1.15)

, ak2 + bk2 .

% " " 2’ . !

2’ (1.15), + ":

∞

x(t) = a0 + ak2 + bk2

(ak

ak2 + bk2 cos(2πfkt)+ bk

ak2 + bk2 sin(2πfkt)).

k=1

( # Pk Φk

P =

+ b2

2 ,

cosΦ

= a

2P ,

− sinΦ

= b

2P ,

∞

x(t) = a0 + 2 Pk cos(2πfkt + Φk ),

k=1

Pk " Φk k - .

! ' &(/$' # /! ',

Pk , – (kN N ).

5 , # /! " . 2 k -

Φk = arctg (bk ak ).

( # , +

Φk = arctg (bk	ak ) = arccos 1			2	= arccos(ak		),
		1+ bk	ak			ak + bk

bk ak .

! / , # /!

2’ , "

Pk2 = (ak2 + bk2 )4.

2’ , # + (1.16),

				O(N2 )
.		+		2’
(3 2),	N log2 N	.	&

N , #

. ( # ,

* " 3 2 ( , '– %, –6, 7–%, 7, & .),

. ! '–%.

						X = (x ,..., x	N −1	), N = 2n ,
						0	N −1
				2π
	W = exp − j				, j – ,
	W = exp − j				, j – ,
				N
" 2’						Cx (k) = Fd ( jωk ),
		1	N −1
Cx (k) =		1	X (m)W km,			k = 0,1,..., N −1
Cx (k) =			X (m)W km,			k = 0,1,..., N −1
		T
		a m=0

. *, #

sj t j 0 1, # « »

2n .

0. ! A[0] (tk−1,...,t0 ) = xt , t = (tk−1...t0 )2 .

A[1] (sn−1,tn−2,...,t0 ) ← A[0] (0,tn−2,...,t0 )+ W 2n−1sn−1 A[0] (1,tn−2,...,t0 ).

A[2](sn−1,sn−2,tn−3,...,t0 ) ← A[1](sn−1,0,tn−3,...,t0 )+ W 2n−2 (sn−2sn−1,)2 A[1] (sn−1,1,tn−3,...,t0 ).

A[n](sn−1,...,s1,s0 ) ← A[n−1](sn−1,...,s1,0)+ W 2n−2 (s0s1...sn−1,)2 A[n−1](sn−1,...,s1,1).

Ta A[n] (sn−1,...,s1,s0 ) = Cx (s), s = s0,...,sn−2,sn−1.

		2π

W W = exp	j		Ta .

		N

3 2

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