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Владимирский государственный университет им. Столетовых

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		pi, j,k =			1			×
		pi, j,k =	i+1, j,k + i 1, j,k		i, j+1,k + i, j 1,k			×
			i+1, j,k + i 1, j,k	+	i, j+1,k + i, j 1,k	+	i, j,k +1 + i, j,k 1
			2 x2	+	2 y2	+	2 z 2
			2 x2		2 y2		2 z 2
× (		i+1, j,k ( pi+1, j,k sc g(Di+1, j,k Di, j,k )) + i 1, j,k ( pi 1, j,k + sc g(Di, j,k Di 1, j,k ))							+
					2 x2
+	i, j+1,k ( pi, j+1,k sc g(Di, j+1,k Di, j,k )) + i, j 1,k ( pi, j 1,k + sc g(Di, j,k Di, j 1,k ))								+
					2 y2

+ i, j,k +1 ( pi, j,k +1 sc g(Di, j,k +1 Di, j,k )) + i, j,k 1 ( pi, j,k 1 + sc g(Di, j,k Di, j,k 1 ))) . (5.2)

2 z 2

- ! % " (5.2) . % ,

pi(,nj),k = F( pi(+n1,1j),k , pi(n1,1j),k , pi(,nj+11),k , pi(,nj 11),k , pi(,nj,k1)+1 , pi(,nj,k1) 1 ) + (1 ) pi(,nj,k1) , (5.3)

F – , . (5.2),

– , 0 < < 2 , n – .

F « » " , " "

" pi, j,k % .

+ " " , .

O ! "

max	pi(,nj),k pi(,nj,k1)	< ,	(5.4)

i, j,k

– ".

5.5..( (=(! ) 9 8 0 . '(. ' 0(&*%4 + 9 * & #%.%61:

. * %%23(*'

8 ! " . %

16V4-50, . 16 (:;9 4q106

267

& . 3 %', !, . %

" (:;9 % .

8 " % 16V4-50

. . ( " : % " %', .

. $ & " %' %

% % % " " %. ( &

%' % . %. $

%', .

" % . $ & " " %

% % " " " %. > ". 8 ! "

" (:;9, . %

% . 8 % % . (:;9 %'. - & (5.2)

% ":

A =

i+1, j,k + i 1, j,k

i, j+1,k + i, j 1,k

i, j,k +1 + i, j,k 1

2 x2

2 y 2

2 z 2

B = (

i+1, j,k ( pi+1, j,k sc g(Di+1, j,k Di, j,k )) + i 1, j,k ( pi 1, j,k + sc g(Di, j,k Di 1, j,k ))

2 x2

i, j+1,k ( pi, j+1,k sc g(Di, j+1,k Di, j,k )) + i, j 1,k ( pi, j 1,k + sc g(Di, j,k Di, j 1,k ))

2 y2

E =

i, j,k +1 ( pi, j,k +1 sc g(Di, j,k +1 Di, j,k )) + i, j,k 1 ( pi, j,k 1 + sc g(Di, j,k

Di, j,k 1 ))

. (5.5)

2 z2

. 5.9 ,%, ,

% %

" (:;9 " " 2, $, 9, c

(5.5).

268

	"
. 5.9. $ ($ F, H,	, I ! (5.5)

$ (:;9 %',

. 5.10.

. 5.10. & '

269

8 , "

% % % " " "

%. $" " ",

! " "

, . 5.11.

P C

Rxy

D Rx

D+

P+

P-

D+ P- P+ D-

P- P+ D+ D-

D+

P-

P B

P+

D-

Rxy

. 5.11. ( ( $ (

0 %' . % % ! "

" ! " " "

. ( .

%', " " ,

270

. &,

% % " &.

5.6. * %%23(*' 8 ) (=(! ) 9 8 0 " + % %4 %2 2%'* & ,! %

> % %

% b,

. " . ,

! % ! " " [19,98,100,107]. r "

. [100]

M 1
y(n) = h(k)x(n k) , n = 0,1,..., N 1.	(5.6)

k =0

2 % % b [98]

	N 1		W = e j	2
S(k) = s(n)W n k ,			W = e j	N	,	k = 0,1,..., N 1,		(5.7)
	n=0
	1	N 1			j	2
s(n) =		S(k)W nk , W = e					, n = 0,1,..., N 1	(5.8)
						N
						N
	N k =0

" " ,

% , "

. . 9 . % "

% b (-(b) (5.7).

- " % -(b &

– & % % b – R(b " -(b >R(b " % -(b. 2 R(b, , ,

N = 2 p , p = 1,2,3,... . R

! %% R(b N ,

% N . $ & " R(b,

N = 29 = 512 , % % " R(b512.

271

8 . 9

(5.6) . 5.12.

. 5.12. ( "

>", " % " , .

8 R(b. 0

(5.7) (5.8) . , " "

" % %. +

, % %. O . , " . &

{h} ,i 0,1,...M 1 . & {W n k } , n, k 0,1,...N 1

% " ,

, ! ". O .

. % "

. ( "

% ",

. - R(b512 .

, , R(b8.

( " R(b. R R(b 2 . "

				,
At+1	= At		+ Bt	,
						(5.9)
Bt+1	= (At Bt				)Wt ,	(5.9)

Wt = W		× W .

272

> " . :


Re[At+1	]= Re[At ]+ Re[Bt ]; Im[At+1			]= Im[At ]+ Im[Bt ],

Re[Bt +1 ]= (Re[At ] Re[Bt ]) Re[Wt ] (Im[At ] Im[Bt ]),

Im[Bt +1 ]= (Re[At ] Re[Bt ]) Im[Wt ]+ (Im[At ] Im[Bt ]),

Re[Wt ]= Re[W ] Re[W ] Im[W ] Im[W ],

								(5.10)
Im[Wt ]= Re[W ] Im[W ]+ Im[W ] Re[W ].
( & . . %
	%		R(b	[110]. -	&		%
-. &								.
					Wt .

" & Wt .
8		"		&
						Wt . & Wt .
	.		%			! (NO		%' N 2

. ( ",

% R(b, %

% . " " & .

( & . % " %

(NO.

- " " & . >

% ", [100], "

" & &

W = e j

! & N .

R(b %

[111,113]. &

% & W

W Wt = W ×

W .

(NOW.

" &

R(b

N ,

% –

273

% % & . & ,

% % & ,

% " :

W = exp( j 2 pN2 N ), W = exp( j 2 qN ),

p= 0,1,...N1 1, q = 0,1,...N2 1, N1× N2 = NM 2 ,

NM - . R(b, "

%'.

F % &

%' (NO W NM (2(N1 + N2 )) .

- NM = 216 , N1 = 27 N2 = 28 %' (NO 85 [110,111].

8 & Wt % %.

( ., " %

% & = ( ), n = 0, 1, … (N/2)-1, %

Wt exp j 2 n N

R(b N. >% & &

% ,	%
		Wt , F = n ,
, " &
		Wt [A]= exp( j 2 A N ). 2
" F	"	&

% % R(b. F "

R(b, %

, , " F

, [113].

$"	, % " %
, &		! ",
	Wt
. 5.13.

274

Re[At ]

Re[At+1 ]

Im[At ]

Im[At+1 ]

t+1

Re[Bt ]

Re[Ct

]

Im[B ]

Re[B

]

Im[Ct

]

t+1

Im[Bt+1 ]

Bt+1

Re[Wt ]

Im[W ]

. 5.13. H$ $ (

$" ,

% ,

& Wt

W ,

. 5.14.

Re[At ]

Re[At+1 ]

Im[A ]

Im[At+1 ]

At+1

Re[Bt ]

Re[Ct ]

Im[Bt ]

Re[B

]

Re[WHt ]

Im[Ct ]

t+1

Im[Bt+1 ]

t+1

WHt

Im[WH ]

Re[WVt ]

WVt

Im[WVt ]

. 5.14. H$ ! (

$ % R(b >R(b . &

. ( R(b

& Wt = 1, %

R(b . % .

& . 9 &

. % R(b. O % . &

Hi Hi+1 .:


At+1	= (At + Bt ) Hi	(5.11)
		(5.11)
Bt+1	= (At Bt ) Hi+1	,

275

, ":

Re[At+1

]= (Re[At ]+ Re[Bt

]) Re[Hi ] (Im[At

]+ Im[Bt ]) Im[Hi ],

(5.12)

Im[At +1

]= (Re[At ]+ Re[Bt

]) Im[Ht ]+ (Im[At

]+ Im[Bt ]) Re[Ht ],

Re[Bt +1

]= (Re[At ] Re[Bt

]) Re[Ht+1

] (Im[At ] Im[Bt ]) Im[Ht +1

Im[Bt +1

]= (Re[At ] Re[Bt

]) Im[Ht+1

]+ (Im[At ] Im[Bt ]) Re[Ht +1

O % " % " , "

& Wt

. ( & " Wt

= W

× W

= 1

. O %

& Hi

Hi+1

. & Hi

Hi+1

% % %

			R(b				.
& .
$" , %
. &
. &					Hi		Hi+1 ,
. 5.15.
	Re[H1t ]
H1t	Im[H1t ]
	Im[H1t ]
	Re[At ]	Re[Dt ]	Re[At+1 ]
	Re[At ]

					A
			Im[At+1 ]			t+1
At	Im[A ]		Im[At+1 ]
	t	Im[Dt ]
	Re[Bt ]	Im[Dt ]
	Re[Bt ]	Re[Ct ]
	Im[B ]	Re[Ct ]	Re[B	]
Bt	Im[B ]		Re[B	]
	t	Im[Ct ]	t+1
		Im[Ct ]
			Im[Bt+1 ]		Bt+1
			Im[Bt+1 ]
	Re[H 2t ]
H 2t

Im[H 2t ]

. 5.15. H$ ' (

8 "

" % ( $R), . .

" (5.10) (5.12). R :

276

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