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Copyright ??? л??? л??????╗ & ??? лA???????? K????-C?????╗
ISSN 0013-5771. л????????????╗, ? 11, 2009
35
??? 004.021.
?????? ?????????? ????????????? OFDM-???????? ?а????????? ???????
?.??.????????, ????????? ???????, ???? ???? ?? ?.??.?.; avb_o@mail.ru
?.??.????????, ??????? ???? ???? ??
?.??.???????????, ??????? ???? ???? ??
?.??.??????????, ??????? ???? ???? ??
?.??.?????????, ????????? ?????? ???? ???? ??, ?.??.?.
???????? ?????: ????????? ???????, ?????????????, ?????????????? ??????? ????????, Orthogonal Frequency Division Multiplexing (OFDM), DigitalаVideo Broadcastingа? Terrestrial (DVB-T).
??????? ?????? ??????? ?????????? ????????????? ??? ?????? OFDM-????????, ?а?????????, ?а????????? ????????? ????????? ??????????? ????????? ?а????? ?????? ?????????????
?????? ?а??????????????. ?????? ????????? ??? ??????? ?????? ????? ???????????? ????? ???????? ?а????????? ????????
??? ???????????? ??????????????? ??????? ?а?????????? ??????????. ????? ??????????????? ??????? ???????? ?а????????????? ????????? ??? ??????? ?а????????? ?????? ?????????????
?а?????? ?????????? ?????? ?а????????? ???????.
? ?????? ?????????? ?????? ??????? ?а????????????? ?????????? ?????? ?а??????????? ????????? ??????? ???????,
????????? ??? ???????????? ???????????????? ?а??????, ???
?а??????????? ????????, ???????????? ??а??????????????
?????? ?????????????.
????????? ????????? OFDM-???????. ???????? ???????
??а????????? ???????? ?????? (???.а1):
zz ????????????? ????????????, ??????????? ???????????? ?????????? ??????? ?а????????????? ????????? ???
?а????????????? ??а??????? ???????;
zz ????? ????????????? ??а????????? ???????, ??????????????? ?????? ?????? ??????? ?а????????????? ??????
???????? ????? ??????? [1, 2, 3];
zz ????? ???????? ?????????????? ????? (???), ????????????? ????????? ??????????;
zz ????? ????????????? ?а????????? ??????? (???.а2), ??????????????? ??????????????? ?????? ????????? ?а????????? ?????????????, ??????????? ????????? ?а?????? ?а??????? ?????????????;
zz ????? ??????????? ?а????????????? ????????? ???????.
?????? ??
?????????????
???????
????????????
???????????
????
?????????????
?? ?????????
???????
????
???
????
?????????????
? ?????????
???????
???????????? ?????
MPEG2
????
??????????? ?
?????????????
???. 1
?????? ???????????? ??????? ?а????????? ???????. ???
??????????? ?????????? ????????????? ?????????? ??????
???????????? ???????.
? ????????? DVB-T ????????????? ????????? ????????? OFDM-???????. ???????????? N?= 1705 (????? 2k) ???
N?= 6817 (????? 8k) ??????????; ??????????? ??? ?????
2048а?а8192а??????????????, ?а????????? ????????, ??а???????? ?а??????, ??????????? ??????? ????. ??? ????????
??????? ????????????? ????????????? ???????????? ???????? ???????? ?а????????????? 1/4, 1/8, 1/16а??? 1/32а???????????? ???????? ????? ???????.
OFDM-?????? ???????? ??? ???? ??????????: ?????????? ?а??????? (????????? QPSK, QAM16а??? QAM64), TPS?????????? (Transmission Parameter Signaling), ??????????
?????????? ?а????????? ??????? ???????? ?а???????????,
?а?????-??????????, ?????????????? ??????????????? ???????????????????, ????????? ??а???????? ??????? [12].
?????????? ??????????? ?????? ?а????????? ???????
?а????:
k
X r (l , k ) = H (k ) X t (l , k )exp{ j[?2pdt (l )
+
N FFT
l ?1
+2p? dw (i )(1 + D g ) + f0 ]};
(1)
i =1
k ?[?
N ?1 N ?1
;
],
2
2
??? l а? ????? ????????????? ???????; k а? ????? ?????????? ?а???????; X t (l , k ) а? ??????????? ????????? k-? ?????????? ?аl-?? ???????????? ???????; H (k ) а? ??????? ??????????? ????????? ?????????????? ??????; dt (l ) а? ??????
??????????? ?????? l-?? ??????? ??а????????? ???????, ?????????? ?а??????? ?????????????; NFFTа? ??????????? ???;
dw (i ) а? ??????????????? ????????? ?а??????????? ??а???????
?а???????аi-?? ???????, ?????????? ?а?????????? ????? ???????????; D g а? ????????????? ???????????? ????????? ?????????, f0 а? ????????? ????.
??? ???????? ?????? ???????????? ??????? (1) ????
??????? ????????? ?????????, ?????????????? ?????????????????? ??????????????:
zz ????????????? ????????? ??????????????? dw (i ) ?????
?????? ????????? ????????????? ???? ?а??????????????
????? ??????????? ????? ??????????;
zz ?????????? ????????????? ???????? ?????????? ???
???????????? ???????-????????? ??????????????? (???)
????????? ?????????? ??????????? ??????? ??????? ??-??
??????????????? ??а??????? ?????????????;
zz ???????? ????????? ?????????? ?????? ???? ??а????????? ?а?????????? ?????????, ?а??? ??????????? ?????????
?????????????? ??а??????? ??а?????? ??????? l.
???????? ?????? ????????? ?????????????. ???????? ?????? ??????? ?????? ?????????????? ????? ??????? ??а????????? ??????? ??а????? ????????? ?????? ?????????????
???????. ???????? ????????? (1), ?????????? ????????? ??????????????? ???????? ?а????????? ???????? ??????? ???????. ?????? ???? ????????? ???????????? ??????????? ????? ????????? ???????? ??? ????? ????????? ???????????
?а???????? OFDM-??????? [4, 5]. ???????? DVB-T ???????????? ??????? ???????????? (continual) ?а??????????????
(scattered) ?????-???????? [12]. ????????? ?????????????
?а??????? ?а???????? 12а?????????? ?а?????????? ???????? ???? ????????? ??а??????? ?а???????. ?? ?????????????
Copyright ??? л??? л??????╗ & ??? лA???????? K????-C?????╗
36
ISSN 0013-5771. л????????????╗, ??11, 2009
???????? ????????????? ??? ?????? ?????????? ???????????????.
? ???????? ?????? ?????????? ??????????????? ???????????? ???????????? ?????????:
N sp ?1
N
d t (l ) = FFT arg{ ? X r *[l , n(l ) +12(i ?1)]X r ╫ 24p
i =1
(2)
╫[l , n(l ) +12i ]P [12(i ?1)]P (12i )},
??? Nspа? ?????????? ?????????????? ??????? ?а???????; n
(l)а? ??????? ??????? ??????????????? ??????; P (k)а? ???????? ???????????? ??????????????? ??????????????????.
?????????? (1) ?а(2), ???????:
N sp
N
d t (l ) = dt (l ) + FFT arg{? H r *[n(l ) +12(i ?1)]H r [n(l ) +12i ]}.
24p
i =1
? ???? ????????? ???????? ??????
N
Ddt =
sp
N FFT
arg{? H r *[n(l ) +12(i ?1)]H r [n(l ) +12i ]}
24p
i =1
?? ??????? ??а??а???????? ?????????? ??????????????? dt (l ) ,
??а??а?????? ???????, ??? ??? ????????? ?????????????? ??????а? ???????? ?????????? ??????? ???????.
???????????? ??????? ????????? ???????? ??? ????? ???????????????? ?а?????????? ?а??????. ?????? ???????????
?????????? ?????? ????? ????? ?а????? ?а????????? (2)
?а????? ?????? ??а???????? ?????? dt (l ) .
????? ??????????? ?????????, ????????? ?????????
????????????????, ?????? ????????? ???:
X 'r (l , k ) = H (k ) X t (l , k )exp{ j[?2pDdt
l ?1
+2p? dw (i )(1 + D g ) + f0 ]}, k ? [?
i =0
k
+
N FFT
N ?1 N ?1
;
].
2
2
????????? ???????? ?????? ?????????? ???????????????
???????????? ?????? ????? ?? H (k ) , ?, ??? ???????? ????????????????? ????????????, ??????, ????????? ????????
???? ?а??????, ??? ?????????? ?????????????? ??????????
????? ???????? ?????????? ????????, ??а??? ????? ??????????????? ?????? ??????????? ??????? ??????. ?а?????
?а???? ?????? ?????? ??? ???????? ?????????? ?????? ????? ???????? ?а H (k ) . ????? ???????,
l ?1
X 'r (l , k ) = H (k ) X t (l , k )exp{ j[2p? dw (i )(1 + D g ) + f0 ]},
i =0
N ?1 N ?1
k ? [?
;
].
2
2
?? ???.а2а????????? ????????????????? ?????? ????????? ?????? ?????????? ??????????????? ??? ?????????
???????? ??????????????? ?а??????. ???????? ?????? ???????? ?????? ??? ?????? ?а?????????? ????? ???????????
????? (????), ?????????а? ??? ??????, ??????????? ??????? ?????, ?????-??????????а? ??????? ?????.
D(??t) 100
10-1
10-2
??????????. ????? ?а????? ????????????? ?????????????
?а????????????? ?????? ??? ????????? ?????????? ???????: ????? ????????а? 8k, ????? ????????? ?????????а? 1/4,
??? ????????? ??????????а? QPSK, ???????????? ?????????? ????????? ???????а? 1000аOFDM-????????. ???????????
????????? ???????а? ??а16а??? ??а?????????????? ?а??????
?????. ??? ?????????? ????????? ???????????? ????? ????????????? 16??? ???????? ???????? ????????? CORDIC [13].
?????????, ????????? ?а[4, 5], ???????????? ?????????
?????????? ??????????????? ??а????????? ???????. ??? ????? ?а[4] ???????????? ????????????? ??????? ????????? ???,
?а?а[5]а? ??????????? ????? ???? ??? ??а????? ????? ????????. ?????? ?????? ?????????? ??????, ?а?????? ???????????
???????????? ??????????? ??а???????? ??????? ?????????????. ????? ??????? ????????, ??? ???????? ?????? ??а????
??????? ?????? ??? ????????????? ?????? ??????? ?????
???????? ?а???????? ????????????? ?????????????.
???????????? ???????? ??а????? ???? ???????????: ????????? ????????? ???????? ??????? ??????? ??????????????
?????????? ??????, ???????? ??????? ????????????? ?????????? ?????? ??????????? ?????????????? ??????? ???????
?а????????? ????????, ?а????? ?????????????? ????????????? ????? ???????????.
???????? ?????? ????????? ?????????????. ???????? ??????? ??????? ?????? ????????????? ??а????????? ???????
??а????? ????????? ???? ??????????? ??????????, ????????? ?а?????? ????????????? ????????????. ???????? ?????? ????????? ????????????? ????????? ??????? ?????????? ????????? ??????????????? ?а?????????????? ????? ????
??а??????? ?а???????.
?????? ?????????? ??????????????? ???????????? ??????????? ????? ????????? ???????? ??? ????? ???????????????? ????????????? ?????-????????? ???? ???????????????? ????????:
d w (l ) =
1
arg[? X 'r * (l ?1, k ) X 'r (l , k )], 2p(1 + D g )
k ?I
???а Iа ? ? ?????? ?? ? ? ?????? ???? ?????? ? ? ? ? ?? ???.
??а???????? ?а(2) ?????? ??????????? ?????????? ?????? ????? ????? ?а????? ?а????????? (3), ??? ?????????
?????????? ????????? ??? ?????? ?а???????? ??????????
?????????.
?????? ???????? ????? ?????????????? ????????? ???????. ????? ????????? ??????? ????????????? ????????????
w(l ) ????????. ???? ?????????? ??????? Dw(l ) = w(l +1) ? w(l ),
????? ?????????????? ????????? ??????????????? ??а????
?????? ?????? d wpr (l +1) = d w (l ) + Dw(l ) ?а??????????? ?????????????? ?????? ?а??????????? ??а?????? ????????. ?????
?????? ????????? ????????? ??????????????? ??а???????, ????????? ?????????????? ???????, ?а???????? ???????? ???????? ??а????????? ?а?????????? ??????????? [4].
??? ????????? ????????? ????? ?????? ????????? ????????????? ??????? ????????? ???????? 2pd wpr (l ), ?.??. ???????
??????????? ????????? ??а(l-1)-?? ?аl-?? ???????.
??????????? ????? ????????????? ????????? ???????????? ??а???.а3, ??? K (z)а? ?????? ?????? ??????, ????????????
??(l)
10-3
.
X?r(l, k)
10-4
10-5
(3)
3
6
9
12
15
???. 2
18
21
24
27
30 q, ??
2???(l)
-
?
K(z)
???. 3
2??pr?(l+1)
?
?cor(l+1)
Copyright ??? л??? л??????╗ & ??? лA???????? K????-C?????╗
ISSN 0013-5771. л????????????╗, ? 11, 2009
37
?????????? ?????? ???????? ?????????? ???????????????;
?а? ????????????? ????????; jcor (l +1) а? ???????? ????
??? ????????? (l+1)-?? ???????.
????????? ??????? ??а????????? exp[- j jcor (l )] ???????? ??? ?а????:
????????????????? ?????? ????????? ?????? ????????? ???? ???????? ?????????? (4) (????????? ?????) ?а(5)
(????????) ??? ??????, ???????????? ??????? ?????, ????????? ??а???.а5.
?
D(?)
10-3
X ''r (l , k ) = H (k ) X t (l , k )exp[ j Df(l )]
N ?1 N ?1
;
].
2
2
????? Dj(l ) ???????????? ??? ????????? ????? j0 , ??? ?а??????? ????, ????????? ???????? ????????? ?????????????.
????????????? ????????????? ? ???????? ?????? ??
??? ????????????????? ??а???? ????????? ????????? ??????/??? (???), ??????????????? ?????????? DVB-T.?????????????????? ??????????? ????????? ?????? ?????????????? ?????????? ??????????????? ???????? ????????? (3) ???
????????? ??????? ??????????????? ??????? ????????????
??а???.а4. ???????? ?????? ???????? ?????? ??? ??????
?а????, ?????????а? ??? ??????, ??????????? ??????? ?????, ?????-??????????а? ??????? ?????.
???????? ??????? ?????????????. ????????????????? ??-
10-4
k ? [?
D(??0)
10-4
10-6
12
15
20
25
30
q, ??
?? ??????? ?????, ??? ???????????? ??????????? ????????? ??????? ????????????? ???????? ???????? ??? ??????
?а???????? ?????????? ????????? ?а2,5а????.
? ?????? ?????? ????????? ?????????? ??????? ????????????? [10, 11] ????? ?????????? ??а???????. ??????? ????
??а????? ?????????? ?а???? ?????? ???????? ?????????
Dj (l , k ) = arg[ X '''r (l , k ) X '''r (l , k )* ],
Dj (l ) = arg[? H (k ) X '''r (l , k ) X '''r (l , k )* ].
k
3
6
9
12
15
18
21
24
27
30
q, ??
???. 4
?????????? ???????????????? ????????? ???? ?????????? ????????, ??? ??? ?? ????????????? ???? Dj(l ) ???????? ???????? ??а????????? ?а?????????? ?????????. ???
????????? ?а??????? ????????? ??????? ?а??????????????
??????? ????????? ?????????????? (??) ?????? [8, 9]. ?????
???????, ?????? ??а????? ????? ??????? ????????????? ?????????????? ?а????:
N ?1 N ?1
X '''r (l , k ) = X t (l , k )exp[ j Dj(l )], k ? [?
;
].
2
2
??? ?????? ???? Dj(l ) ???????????? ???????? ???????
???????? ???? ???? ???????? ?????????? ???????:
Dj (l ) = arg[? X '''r (l , k )P (k )]. (4)
k ?I
????? ???????????? ???????????? ??а???????? ???? ???????? ?????????? ???????а? ???????????? ?а??????????????.
?????????? ?????????????? ?????????а? ?????? ????????
?а???????? ?????????? ????????? ?а????? ???.??????????? ???????? ????????? ????? ????????, ???? ???????????
??????? ?????? ??????? ??а??? ?????: ??????????? ??????????????? ??????? arg H (k ) ?а??????????? H (k ) ?????????.
??? ???? ?????? ????????? ???? ??????????? ????? ????????? ???????? ???????:
5
???. 5
10-7
10-7
??? X '''r (l , k ) а? ?????? ??????????? ????????? k-? ?????????? l-?? ??????? ??? ???????????.
? ????? ?????? ??????? ??????? ???????????? ????????
???????? ?а?????????
10
10-6
-5
10-8
10-5
Dj (l ) = arg{? X ''r (l , k )exp[? j arg H ( k )]P (k )}. (5)
k ?I
???????????? ?а????? ?????? ??? ?????? ????????? ???????? ???????? ??а???? ?????????? ??????, ????????? ?????? ???????????? ???????????.
??????? ???????? ??? ??? ??????? ????????? ???????
????????????? ??? ???????? ?а????????? DVB-T ????????????? ?????????? ??а??????? ??? ?????????? ?а??????? ?????????? ??????????? ??????? ???????? ?????????
(???16, ???64) ?а???????? ????????? ?а??????, ??? ???
??? ???? ???????? ?????????? ???????? ???????? ????????????? ?????????? ?????? ??? ???????????.
???????????? ?а?????? ????????? ????????????? OFDM???????? ?а????????? ??????? ???????? ????? ???????? ???????? ???????????????? ??а????????? ?а????? ?????????? ?а????????? ???????? ???????? ?????? ?а???????? ??????, ??? ????
???????????? ?????????????????? ??????????????.
??????????
1. Sandell?M., J.-J. van de Beek and BorjessenP.?O.?Timing and frequency synchronizationа in OFDM systems using the cyclic prefix/Proceedings.?? 1995.?? P. 16?19.
2. Schmidl?T.?M., Cox?D.?C.?Robust frequency and timing synchronization for OFDM//IEEE Trans.Commun.?? Dec. 1997.?? Vol. 45.??
?. 12.?? P. 1613?1621.
3. Seung Duk Choi, Jung Min Choi, Jae Hong Lee Anа Initial Timing
Offset Estimation Method for OFDM Systemsа in Rayleigh Fading
Channel/Vehicular Technology Conference.?? September 2006.??
P.а1?5.
4. McNair?B., Cimini?L.?J., Sollenberger Jr. and N. A Robust Timing
and Frequency Offset Estimation Scheme for orthogonal Frequency
Division Multiplexing (OFDM) Systems/Proceedings of theа Vehicular Technology Conference (VTC?99?Spring).?? Houston, Texas,
USA.?? 1999.?? P. 690?694.
5. Muller-Weinfurtner?S.?Frequency-Domain Frame Synchronization
for Optimum Frequency-Differential Demodulation of OFDM/IEEE
Copyright ??? л??? л??????╗ & ??? лA???????? K????-C?????╗
38
6.
7.
8.
9.
10.
ISSN 0013-5771. л????????????╗, ??11, 2009
Global Telecommunications Conference (GLOBECOM ?99).?? Rio
de Janeiro, Brazil. December 1999.?? P. 857?862.
Moose?P.?H. A technique for orthogonal frequency division multiplexing frequency offset correction//IEEE Trans. Commun.?? Oct.
1994.?? Vol. 42.?? P. 2908?2914.
Kim?J.-J., Ryu?Y.-J., Oh H.-S., Han?D.-S.?Frame selection algorithm
with adaptive FFTаinput for OFDM systems Communications, 2002.
ICC 2002/IEEEа International Conference onа Volume 1, Issue.??
2002.?? P. 187?191.
Vigelis Rui?F., Cavalcante?C.?Casimiro Robust Pilot-Aided Channel
Estimator for Time-Varying OFDM Channels/Proceedings of The
Ninthа IEEEа International Workshop on Signal Processing Advancesаin Wireless Communications (SPAWC2008), July 6аth-9аth, RecifePE, Brazil.?? 2008.
Auer?G., Karipidis?E.?Pilot-symbol aided channel estimation for
OFDM: a separated approach for smoothing andа interpolation/Proceedings of the 40а thа IEEEа International Conference on Communications (ICC). Seoul, Korea.?? May 16?20.?? 2005.?? P. 2173?2178
Anh Tai Ho, Boucheret?M.-L., Thomas?N. et al. OFDM synchronization scheme to be used on a non frequency selective satellite chan-
nel/Dans: European Signal andа Image Processing Conference (EUSIPCO 2008). Lausanne, Suisse, 25?AUG-08?29?AUG-08, EURASIP, (support electronique).?? 2008.
11. Abhayawardhanaа V.?S., Wassellа I.?J.?Common phase error correction
for OFDMа in wireless communication/Proceedings of the 3а rdа International Symposium on Communication Systems, Networks and
Digital Signal Processing (CSNDSP ?02). Staffordshire, UK.?? July
2002.
12. Digitalа Video Broadcasting (DVB).Framing structure, channel
coding and modulation for digital terrestrial television/ETSI EN
300?744а v.1.5.1 (2004?11). European standart (Telecommunication
series).
13. ?????????.??., ?????????.??.?????????? CORDIC.???????????? ????????? ?а???????????/??. ????????. ????. л??????????? ???????: ?????? ?а??????????╗, ??????????-?????????.??
?.: ?????????, 2004.?? ?.а1.?? ?.а353?372.
???????? 7.10.09
??? 621.396
?????????????? ??????????? ?а?????????? ????? ?????????? ?????????????? ?а ???????? ????????? ????????????????
?.??.????????, ??????? ??????? ????????? ???? ???? "?????", ?.??.?.; tarasov_ga_@mail.ru
?.??.?????????, ??????? ??????? ????????? ???? ???? "?????", ?.??.?.
?.??.????????, ??????? ??????? ????????? ???? ???? "?????", ?.??.?.
???????? ?????: ?????????????? ???????????, ?????? ?????
?????????? ??????????????, ????????? ???????????????? ?????????? ?????, ??????????? ????????????? ???????????? ?????? ?????????????.
???? ??а ?????????? ????? ?????? ???????????????????????? ????????? ????????а? ??????????? ?а?????????? ????? ?????????? ?????????????? (???), ???????????
???????? ???????? (??). ??????? ???? ???????? ????????
??? ?????????? [3?6, 10?16]. ???????? ?????????? ???????????? ?а??? ???????? ??????? ?а???, ??? ???????? ?????
?а????????? ??????? ?????? ?????????????? ???????????
?а???????? ??????????. ??а???????? ???????? ?????????,
??? ???????, ??а???????????. ????????, ???????? ?????
?а??????? ?????? ??а?????? ??????????, ??а?а????????. ????? ????, ??а?????? ??????????? ??? ?????????? ??а???????.
?а????? ?а???? ????????? ?????? ??????????? ?а??????????
????? ??? ??а???????? ???????? ?????? ?а???????? ????????? ???????????????? ????????? ????? ?а??????? ??????.
??????????? ?а?????????? ????? ??? ??? ???????? ??????????? ?а?????????? ????????? ????? ?а???????. ????????????? ???????? ??????????? ?а?????????? ????? ???,
?????????????? ??а??, ????????????? ??? ?????? ?????????? ???????????а???????? (??) ?????????? ??????????????
??????? (???) ??а?????????? ?а???????. ???????? ??????????????? ?????? ?а??????????? ?а?????????? ?????
??? ??????? ?а??????? ????????????? ?а??????? ????????
??????????? ???????? (??) ???.?
??????? ???? ?????? ????? ??? ??а????????????? ??
??? ???? ?????????? ?а[12]. ??? ????? ??????? ???????????
?а?????????? ????? ??? ???????? ?а?????????? ???????????
?????? ????????????? ?? ??? ?а??????? ????????? ????????
??? ??????? ??.????? ?????????? ??????????? ?а????????
?????????????? ?? ?а?? ??? ??а????????????? ?а??????? ???????? ?? ??? ?????????? ??а????????????? ?а?????????????? ???????????? ??????? ??????? ?? ??? ? ??????????? ?? ??????? ??? ?????????? ???.???? ???? ?????
??? ??а?????? ????????? (Nа? 1), ??? Nа? ????? ????????
????????? (??). ????? ??????, ??????????????, ??? ????? ??????????? ?????????????? ?????????????? ??????? ???????
?? ?а?????????? ???, ?а????? ???????????? ?????????? ?????????? ??? ??????? ??????? ??? ??????? ????????????
?? ??? ??? ??????? ???.
? ???? ????????????, ???????????? ??????? ????????????? ????????????? ?а??????????? ???????????? ????????????????? ???????????? ??????? ?????????? ?????????, ???? ???????????, ??? ?а?????? ????????? ???????????
?а?????????? ????? ??? ????? ???? ???????? ?????????????? ???????????? ???????? ???????? ??????? ??????? ??
??? [7]. ??????????? ???????????? ????????, ??? ??????
??????? ?? ??? ?а??????? ???????? ???????? ????? ????
??????????????? ???????????????? ??????
???
l n = kl exp {?(n ?1)a l } ,
al =
12 NL
n = 1 , ... , N , (1)
(2)
???????????? ??????????; k?а? ????????; Lа? ?????? ???????; Nа? ????? ??; ?? nа? ??????? ???????? ?? ???.???? ?????
??а(1), ???????????? ???????????????? ????????????? ???????? ???????????????????, ?????? ???????? ????????????
??????????? ??????? ??????? ?? ??? ??????????????? ??????????? ?????????? ?? , ?а??????????? k?а? ???????????-
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