Weiss-Weinstein bound for MIMO radar with colocated linear arrays for SNR threshold prediction

Texte intégral

(1)Weiss-Weinstein bound for MIMO radar with colocated linear arrays for SNR threshold prediction Duy Tran Nguyen, Alexandre Renaux, Remy Boyer, Sylvie Marcos, Pascal Larzabal. To cite this version: Duy Tran Nguyen, Alexandre Renaux, Remy Boyer, Sylvie Marcos, Pascal Larzabal. Weiss-Weinstein bound for MIMO radar with colocated linear arrays for SNR threshold prediction. Signal Processing, Elsevier, 2012, 92 (5), pp.1353-1358. �10.1016/j.sigpro.2011.10.018�. �hal-00771395�. HAL Id: hal-00771395 https://hal-supelec.archives-ouvertes.fr/hal-00771395 Submitted on 15 Mar 2020. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2)

(3)

(4)

(5) "!#$!#&%'( *),+-./0+!1 . 23!#4!#5!7628 9

(6) : ; *)

(7) )2.,<

(8)

(9) += */ >0? @*A BDCFE2@*AHGJI0K2>0LMNK0O BP*QRCNSTUB$IJVW>2KJX0Y2L5MNIJZD[\A^]0_a`

(10) VWIJL Mcb5A5O d5e BgfhK2I0iJ_$b5L M*jkQ lUmDQRO7nK2I

(11) o=K2]aK2n7p q r{su t wvxw¡ky(z{t y(|~a}W7 wWy(Rt z{U~R y( $t wxywJc Dw

(12) Ny/|¢y(s }

(13) c 1£#R¤t ¥ sR z#DsR JyR|}y(z{}x|3w¦xazc£3RvxUwsu w{w/-

(14) 0-y( ywWz{t w t su||~(§

(15) *t y(z(t w/s 7-¨ D©\-Nt {/z{¢y(Us^Nvxww{|~ |w(w/ªx*NyysRsR{{w w «¬1®D¯D°u±*²R¯ b5BDd BDTxQRO³´ TUµl2¶NQ d B·l@*? ? Bl/¸UBSF¸U¶NQu¸gQHf¹@*O ¸Ue º{»(C*¼*@¸½f¹@*O ¸Ue º_2@¸U¼*@¸¾f¹»/f_$¿0TxQ S*QRT

(16) lAl/¸UBD[´RÀ=BDTxl2e [¼*TUúd BD[BDÇ e CÁ¸UBDTU[Ál#´RÂ7¼cBDTÂÃ´ TU[ÁQRCNmBJe C^m´ [¼NQRTUe~l´ C³Je ¸U¶HmO~Q lUle~mDQRO*¼*¶NQ lBS5º(QRTUTxQ AgTxQ S*QRTÄ3Å0ú³BDd BDTM @*CNSBDT#¸U¶*B2³Je~SBDO AÁl¼*TUBQ S Q lUl@*[¼¸Ue ´ C´RÂaQF@*C*e ÂÃ´ TU[ÇÆ·È=ÉÊËuÉÊ$Se~l/¸UTUe Ì*@¸Ue ´ CÂÃ´ Tg´ C*B¼NQRTxQR[B¸UBDT$´RÂae ÇUBDTUBl/¸M7¸U¶*BDTUBe~lgC*´FTUBl@*O ¸½m´ CNmBDTUC*e C*? O ú³BDT½Ìc´ @*CNS*l\´ C&¸U¶*B^[BQRCº(lUÍ@NQRTUBÁBDTUTU´ T½e C&¸U¶*BHmDQ lB´RÂ2QÎ$QR@NlUle~QRC&´ ÌNlBDTUduQu¸Ue ´ CÏ[´SBDO³Je ¸U¶4¼NQRTxQR[B¸UBDTUe ÐDBS [BQRC7Ä4Ga¶*e~lÁÑNQ l/¸îW´ [[\@*C*e~mDQu¸Ue ´ C8ÒNO O~l·¸U¶*e~lO~Q mxµÏÌ5A8@Nle C*?Ó¸U¶*BFÔBDe~lUl/º{ÔBDe CNl/¸UBDe C4Ìc´ @*CNSÕ¾ÃÔÔÖ]J¿·³J¶*e~mx¶'mDQRC ÌcB^mDQRO~m@*O~Qu¸UBSÏ@*CNSBDT\¸U¶*e~l\Se ×m@*O ¸ÁlUmBDCNQRTUe ŃÄ¹KØmO ´lBS5ºÂÃ´ TU[ÙBP¼*TUBlUle ´ CÏ´RÂJ¸U¶*B^ÔÔÖ]ÚÂÃ´ T·f¹»/f_ØTxQ S*QRT½³Je ¸U¶ m´ O ´mDQu¸UBS·O e C*BQRTWQRTUTxQ Al1e~lWSBDTUe d BSQRCNSÁQRCNQRO A5ÐDBS\ÂÃ´ TWQ½SBle ? CÁ¼*@*TU¼c´lBJe CÁ¸UBDTU[Álk´RÂ7QRTUTxQ A½? BD´ [B¸UTUA Äkb5e [\@*O~Qu¸Ue ´ CNl m´ CÒNTU[Û¸U¶*B$? ´5´S^QRÌ*e O e ¸/AÁ´RÂ¸U¶*B$¼*TU´ ¼c´lBSÁÌc´ @*CNS^¸U´·¼*TUBSe~m¸a¸U¶*B$[BQRCHlUÍ@NQRTUB

(17) BDTUTU´ T

(18) ¾fÓb5V¿´RÂ¸U¶*B$[ÁQuPe [\@*[Æ ÈNË ÜÝ(ÞÉÊËuÉÊJ¾fhK0j¿#e C^QRO ONTxQRC*? BJ´RÂb>2IgÄjkQRT¸Ue~m@*O~QRTUO A Mu¸U¶*B0¸Ue ? ¶¸UC*BlUl#´RÂ=¸U¶*B2Ìc´ @*CNS¸U´½¼*TUBSe~m¸#¸U¶*B$b>2I'¸U¶*TUBl¶*´ O~S BÀ=Bm¸0e~lJl¶*ú³JC7Ä ßÞàuáWËuÉUâ ÜDãf¹»/f_äTxQ S*QRTM5ÔBDe~lUl/º{ÔBDe CNl/¸UBDe CFÌc´ @*CNSÏ¾ÃÔÔÖ]J¿M*¼NQRTxQR[B¸UBDTaBl/¸Ue [ÁQu¸Ue ´ C7Ä åcæÁçè ¯D°ué7ê3ë3²R¯ìé è b5e CNmBe ¸g³aQ l0ÒNTxl/¸$e ÇUTU´S@NmBS4í îï{M-¸U¶*BÁf¹»/f_ðTxQ S*QRT

(19) ¶NQ l

(20) ?QRe C*BS¹e ÇUBDTUBl/¸$e Ce C5d Bl/¸Ue ?Qu¸Ue ´ C¹¸U´He [¼*TUúd B\[\@*O º ¸Ue~lBDCNl´ TlAl/¸UBD[Ál\¼cBDTÂÃ´ TU[ÁQRCNmB Ä8Kñf¹»/f_ñTxQ S*QRT\@¸Ue O e ÐDBl[\@*O ¸Ue ¼*O BFlBDCNl´ Txl\e C4Ìc´R¸U¶4¸U¶*B^¸UTxQRCNl[e ¸¸UBDTÁQRCNS&¸U¶*B TUBmBDe d BDT½¸U´lBDCNS8´ @¸ÁQRCNS4m´ O O Bm¸QÓlB¸´RÂ2¼*TU´ Ì*e C*?Ó³aQ d BÂÃ´ TU[Álg³J¶*e~mx¶"mDQRC8ÌcBHÂÃ@*O O A&@*CNm´ TUTUBDO~Qu¸UBSÏ´ T·¼NQRT¸Ue~QRO O A m´ TUTUBDO~Qu¸UBS-Ä»{Â3QRO OçU¶*B2¸UTxQRCNl[e ¸¸UBSÁ³aQ d BÂÃ´ TU[Ál#QRTUB

(21) m´ TUTUBDO~Qu¸UBS-M e ¸JTUBÂÃBDTxlk¸U´\¸U¶*B$m´ C5d BDÇUe ´ CNQRO*¼*¶NQ lBS5º(QRTUTxQ A\TxQ S*QRTÄ Ga¶NQRC*µl$¸U´F¸U¶*BlB^Q S*Se ¸Ue ´ CNQROSBD? TUBDBl$´RÂaÂÃTUBDBS´ [FM1Q¹f¹»/f_ØTxQ S*QRTg¶NQ lg[ÁQRC5AÏQ SduQRÇxQR? Bl

(22) e C4m´ [¼NQRTUe~l´ C³Je ¸U¶ QF¼*¶NQ lBS5º(QRTUTxQ AFTxQ S*QRTÄHE2BD¼cBDCNSe C*?h´ ÇU¶*BHSe~l/¸xQRCNmBÁÌcB¸/³BDBDCÏBDO BD[BDÇxl½´RÂ0QRÇUBDC*CNQ lgQRCNS¸U¶*B¼c´le ¸Ue ´ CNlg´RÂa¸U¶*B ¸UTxQRCNl[e ¸¸Ue C*?FQRCNShTUBmBDe d5e C*?FQRTUTxQ Al\¾m´ O ´mDQu¸UBSh´ T$³Je~SBDO AhlBD¼NQRTxQu¸UBSN¿M-BQ mx¶m´ CÒN? @*TxQu¸Ue ´ Ch´RÂf¹»/f_ÛTxQ S*QRT

(23) ¶NQ l e ¸xl½ú³JC&e ÇUBDTUBl/¸xl\l@Nmx¶8Q l½e [¼*TUúd5e C*?¹¼NQRTxQR[B¸UBDT½e~SBDÇUe ÒcQRÌ*e O e ¸/A"í ò ï

(24) í óuï{ô#´RÀ=BDTUe C*?¹¶*e ? ¶*BDT½õNBPe Ì*e O e ¸/AÏe CÏ¸UTxQRCNl[e ¸ ÌcBQR[Û¼NQu¸¸UBDTUCHSBle ? CNlDö3e [¼*TUúd5e C*?g¸U¶*B$QRC*? @*O~QRTWTUBl´ O @¸Ue ´ C7MO ú³BDTUe C*?\le~SBDO ´ ÌcBlDM5´ TSBmTUBQ le C*?g¸U¶*B$l¼NQu¸Ue~QROc¼cú³BDT SBDCNle ¸/A4´RÂ

(25) ¸UTxQRCNl[e ¸¸UBS'le ? CNQRO~lFí òï{ô2e CNmTUBQ le C*?&SB¸UBm¸Ue ´ C'¼cBDTÂÃ´ TU[ÁQRCNmB MaQRCNS'QRO O ú³Je C*?&Se TUBm¸^QR¼*¼*O e~mDQRÌ*e O e ¸/A8´RÂ ÷7ø ù ú7û üýxþÿ xú ÿ Rý ø

(26) ü ù ý ÿ ÿ uü /ÿ 3ù ù ÿ/ý 7ÿ/úÿ xü {ø xü -ywRy(t su|-z{u/t |~|w{ J|JRt sR -y{wz{z{t s £-Ry(t xc ✩.

(27)

(28) .

(29)

(30) . !. " $#% &('. *). ,+.

(31) Q S*QR¼¸Ue d B2¸UBmx¶*C*e~Í@*BlÂÃ´ T0¼NQRTxQR[B¸UBDTaBl/¸Ue [ÁQu¸Ue ´ Cí uï{Ä »{Â½³BFÂÃ´m@Nl´ C'¸xQRTU? B¸O ´mDQRO e ÐQu¸Ue ´ C7M¸U¶*BÓl@*¼cBDTUe ´ TUe ¸/A4´RÂ½f¹»/f_,TxQ S*QRT l¼cBDTÂÃ´ TU[ÁQRCNmBF³gÄ TÄ ¸Ää¸U¶*Bh¼*¶NQ lBS5º QRTUTxQ A^TxQ S*QRT0¶NQ l2ÌcBDBDCl¶*ú³JChe Ch¸UBDTU[Ál2´RÂWO ú³BDT2Ìc´ @*CNS¹´ Ch¸U¶*B·[BQRCÓlUÍ@NQRTUB½BDTUTU´ T·¾fÓb5V¿Ä-G´^¸U¶*B\ÌcBl/¸$´RÂW´ @*T µ5C*ú³JO BS? B M QRO O*¸U¶*B0BPe~l/¸Ue C*?g³´ TUµl#Q duQRe O~QRÌ*O Bae ÇU¶*B2O e ¸UBDTxQu¸U@*TUB0m´ CNmBDTUC´ C*O A½¸U¶*B

(32) mDQ lBJ´RÂ7SB¸UBDTU[e C*e~l/¸Ue~mJ¼NQRTxQR[B¸UBDT ÂÃ´ T0¸U¶*B·O ´mDQu¸Ue ´ C¹´RÂ#¸U¶*B½¸xQRTU? B¸Ä2»(C8í îï{M=¼*TUBDO e [e CNQRTUAFmDQRO~m@*O~Qu¸Ue ´ CNl0´RÂk¸U¶*BiWTxQR[ZDTº(I0QR´Ìc´ @*CNS4¾{iI0]J¿0¶NQ d B½ÌcBDBDC e ÇUTU´S@NmBS-ÄkGa¶*B$m´ CNle~SBDTUBSÁ[´SBDO=³aQ l#¸U¶*B$m´ O ´mDQu¸UBSÁ@*C*e ÂÃ´ TU[ O e C*BQRTJQRTUTxQ AH¾X2nK

(33) ¿WÂÃ´ TÌc´R¸U¶^¸UTxQRCNl[e ¸¸Ue C*?·QRCNS TUBmBDe d5e C*?^QRTUTxQ AlDÄ

(34) »(C4í ï#QRCNS8í uï{M7Q^f¹»/f_ðTxQ S*QRT

(35) ³Je ¸U¶Ó³Je~SBDO Aº(lBD¼NQRTxQu¸UBS¹QRÇUBDC*CNQ l2³aQ l

(36) l/¸U@NSe BSQRCNS¹¸U¶*BiI0] ¶NQ laÌcBDBDC¹mDQRO~m@*O~Qu¸UBS^ÂÃ´ Ta¸xQRTU? B¸J¼c´le ¸Ue ´ C¹QRCNSHd BDO ´me ¸/A Ä#]Ble~SBlDM e~QRC¹n7eÞÝaÆ ¶NQ d B

(37) e C5d Bl/¸Ue ?Qu¸UBS¸U¶*Bg³aQ d BÂÃ´ TU[ ´ ¼¸Ue [e ÐQu¸Ue ´ CF´RÂkf¹»/f_ TxQ S*QRTJÌNQ lBSH´ CH¸U¶*BiI0] @Nle C*?m´ O ´mDQu¸UBSFX2nKä¸UTxQRCNl[e ¸¸Ue C*?QRCNSFTUBmBDe d5e C*?ÁQRÇUBDC*CNQ l í Dï1í ï{Ä1Ô¶*e O B

(38) QRO O*¸U¶*B2QRÌcúd BJ³´ TUµlk¶NQ d BJBP¼*O ´ e ¸UBS·¸U¶*B

(39) Q lA5[¼¸U´R¸Ue~maTUBD? e ´ C´RÂ=¸U¶*B

(40) fÓb5V8úd BDT#le ? CNQRO º¸U´Rº{C*´ e~lBaTxQu¸Ue ´ ¾b>2I2¿g¼cBDTÂÃ´ TU[ÁQRCNmB M¸U¶*B^]aQRTxQRC*µ5e CÏÌc´ @*CNSÏ¶NQ l½ÌcBDBDC8l¶*ú³JCÏ¸U´h? e d BQÓb>2IÛ¸U¶*TUBl¶*´ O~SÏQR¼*¼*TU´ Pe [ÁQu¸Ue ´ Ce C&¸U¶*B lUmBDCNQRTUe ´\´RÂkm´ O ´mDQu¸UBSHme Txm@*O~QRTJQRTUTxQ AÁm´ CÒN? @*TxQu¸Ue ´ CÓí ïWí ò ï(¾mx¶NQR¼¸UBDT 5¿Ä K0O O7¸U¶*BlB·QuÂÃ´ TUBD[BDÇUe ´ C*BSHe C5d Bl/¸Ue ?Qu¸Ue ´ CNla¶NQRCNSO B\´ C*O A¸U¶*B·mDQ lBlJ³J¶*BDTUBg¸U¶*B\@*C*µ5C*ú³JC¹¼NQRTxQR[B¸UBDTxlJQRTUB\Q lUl@*[BS ¸U´ÌcB\SB¸UBDTU[e C*e~l/¸Ue~mRÄ2b5e CNmBg¸U¶*B·l@*¼*¼c´ T¸2´RÂ1¸U¶*B½¼NQRTxQR[B¸UBDTxlJ³aQ lJC*´R¸0¸xQRµ BDC¹e ÇU´^Q mDm´ @*ÇM¸U¶*BlB½Ìc´ @*CNS*l

(41) mDQRC*C*´R¸ SBlUmTUe ÌcB¸U¶*B0úd BDTxQRO OfÓb5V&¼cBDTÂÃ´ TU[ÁQRCNmB Ä3K2l#QRCÁQRO ¸UBDTUCNQu¸Ue d B¸U´$¸U¶*B0SB¸UBDTU[e C*e~l/¸Ue~m0mDQ lB MR³BJ¼*TU´ ¼c´lB¸U´g¶NQRCNSO Ba¸U¶*B ¼*TU´ Ì*O BD[Ûe C^¸U¶*Bg]aQ A Ble~QRCÂÃTxQR[BD³´ TUµ³J¶*e~mx¶^³Je O O=¼*TUúd5e~SB$Q½¸Ue ? ¶¸aO ú³BDTÌc´ @*CNS^úd BDTQRO OçU¶*B$TxQRC*? B2´RÂ1b>2IÕQRCNSHQ ? ´5´SÁ¼*TUBSe~m¸Ue ´ C^´RÂ7¸U¶*B½b>2IÖ¸U¶*TUBl¶*´ O~S-ÄkGa¶*B$]aQ A Ble~QRCÁÌc´ @*CNS*laSBQRO=³Je ¸U¶^¸U¶*B$TxQRCNS´ [Û¼NQRTxQR[B¸UBDTaQRCNS¼*TUBl@*[B QRCÓÆJÈ=ÉÊËuÉÊk¼*TU´ ÌNQRÌ*e O e ¸/A·SBDCNle ¸/A\ÂÃ@*CNm¸Ue ´ CÓ¾Ã¼=S5Âx¿ÄWÅ0BDCNmB M ¸U¶*BDA³Je O O=QRO O ú³"@Nlk¸U´·mx¶NQRTxQ m¸UBDTUe ÐDBJQRCNS·¸U´\¼*TUBSe~m¸W[´ TUB m´ [¼*O B¸UBDO AÁf¹»/f_ÕTxQ S*QRT#¼cBDTÂÃ´ TU[ÁQRCNmB Äk>0´R¸UB0¸U¶NQu¸aC*´R¸JQRO ON¸U¶*B$]aQ A Ble~QRCÁÌc´ @*CNS*l¼*TU´ ¼c´lBSe ÇU¶*B

(42) O e ¸UBDTxQu¸U@*TUB

(43) QRTUB QRÌ*O Ba¸U´½¸xQRµ BJe ÇU´\Q mDm´ @*Çk¸U¶*B

(44) mDQ lBJ³J¶*BDCÁ¸U¶*B0¼NQRTxQR[B¸UBDTxlk´RÂ-e ÇUBDTUBl/¸QRTUB0l@*¼*¼c´lBS·¸U´½ÌcB2@*C*e ÂÃ´ TU[O ASe~l/¸UTUe Ì*@¸UBS-Ä Ga¶*BDTUBÂÃ´ TUB M=QR[´ C*?duQRTUe ´ @NlJ¸/A5¼cBl

(45) ´RÂW]aQ A Ble~QRCFÌc´ @*CNS*lDM7l@Nmx¶ÓQ l0¸U¶*BÁoce dºo=QRµuQReÌc´ @*CNS8í î ï{M=¸U¶*B·]BDO O º/b¸UBDe C5ÌcBDTU?Rº V#¼*¶*TxQRe [·º #QRC4GTUBDBl\Ìc´ @*CNS í î îï{Mk¸U¶*BF]´ Ì*TUúdlµ5Aºo=QRµuQRekÌc´ @*CNS í î òï{MWQRCNS&¸U¶*BF]aQ A Ble~QRC8K0ÌcBDOJÌc´ @*CNSÖí îó ï{MW³B m´ CNmBDÇUTxQu¸UB M e CÁ¸U¶*e~l³´ TUµ=M ´ CÁ¸U¶*B2ÔBDe~lUl/º{ÔBDe CNl/¸UBDe C^Ìc´ @*CNSh¾ÃÔÔÖ]J¿2¾lBDB·í î uï-QRCNS¸U¶*B2TUBmBDÇ³´ TUµ·´RÂI0QR¼c´ ¼c´ T¸ QRCNS4_

(46) l¶*[ÁQRC'í î ï

(47) í î ïÃ¿M ³J¶*e~mx¶4mDQRC8SBQROW³Je ¸U¶Ï¸U¶*B^@*C*e ÂÃ´ TU[O AÏSe~l/¸UTUe Ì*@¸UBSÏ¼*TUe ´ T\Q lUl@*[¼¸Ue ´ C8QRCNSÏe~l½´ C*B´RÂa¸U¶*B ¸Ue ? ¶¸UBl/¸½Ìc´ @*CNS´RÂa¸U¶*BÁÔBDe~lUl\QRCNSÔBDe CNl/¸UBDe CÏÂQR[e O A4í î Dï2í î ï{ÄFGa¶*BlUmBDCNQRTUe ´H@*CNSBDT\e C5d Bl/¸Ue ?Qu¸Ue ´ CÓe~l½Q¹f¹»/f_ TxQ S*QRT

(48) Bl/¸Ue [ÁQu¸Ue C*?F¸U¶*BSe TUBm¸Ue ´ Cº{´RÂº(QRTUTUe duQRO¾Eg_0K

(49) ¿$´RÂaQ^¸xQRTU? B¸½QRCNSÓ¸U¶*Bm´ [¼*O BPÓTxQ S*QRTº(mTU´lUl/º(lBm¸Ue ´ C&¾I0iab*¿

(50) e C ¸U¶*Bgb5³BDTUO e C*? \mDQ lB ÄW_2@*TaTUBl@*O ¸xlQRTUB$SBDTUe d BSÁe C^¸U¶*B$mDQ lB

(51) ´RÂO e C*BQRT$¾Ã¼c´lUle Ì*O AÁC*´ Cº{@*C*e ÂÃ´ TU[¿Wm´ O ´mDQu¸UBS^QRTUTxQ Al#Qu¸ BD[e~lUle ´ CFQRCNS^TUBmBD¼¸Ue ´ C7Ä Ga¶*BgTUBl/¸0´RÂ1¸U¶*e~lJ¼NQR¼cBDT0e~l0´ TU?QRC*e ÐDBS^Q laÂÃ´ O O ú³0lDöb5Bm¸Ue ´ Chò·¼*TUBlBDÇxla¸U¶*B½? BDC*BDTxQRO-¼*TU´ Ì*O BD[ lB¸U@*¼¹³Je ¸U¶hm´ O ´mDQu¸UBS O e C*BQRT#QRTUTxQ AlDÄ7»(C^b5Bm¸Ue ´ CÁó*M ³B0mDQRO~m@*O~Qu¸UBa¸U¶*B2m´ TUTUBl¼c´ CNSe C*?$ÔÔÖ]4[ÁQu¸UTUe P\ÂÃ´ Tk¸U¶*B0Eg_0KÖQRCNS\¸U¶*B2m´ [¼*O BPI0iab QRCNSF³B½¼*TUúd Bge ¸xl2Se~QR? ´ CNQROl/¸UTU@Nm¸U@*TUB ÄJ»(Cb5Bm¸Ue ´ C NM=le [\@*O~Qu¸Ue ´ CNl0QRTUB½¼*TUBlBDÇUBS^¸U´Hm´ CÒNTU[Ø¸U¶*B\? ´5´S¹QRÌ*e O e ¸/AH´RÂ ÔÔÖ]ä¸U´F¼*TUBSe~m¸$¸U¶*B[BQRCÏlUÍ@NQRTUB·BDTUTU´ T¾fÓb5V¿

(52) ´RÂW¸U¶*B[ÁQuPe [\@*[ÇÆ\ÈNË ÜÝ(ÞÉÊËuÉÊ¾fhK0j¿

(53) e CÏQRO OkTxQRC*? B·´RÂJb>2I ¾Q lA5[¼¸U´R¸Ue~m

(54) QRCNSÁ¸U¶*TUBl¶*´ O~SÁTUBD? e ´ CNlx¿Äk>0@*[BDTUe~mDQROc¼*TU´mBS@*TUB

(55) e~le ÇUTU´S@NmBSÁ¸U´QRCNQRO A5ÐDB0¸U¶*Bgf¹»/f_ÕTxQ S*QRTlAl/¸UBD[ e C^¸UBDTU[ÁlJ´RÂ1QRÇUBDC*CNQ·? BD´ [B¸UTUA ÄkÑ3e CNQRO O A MNb5Bm¸Ue ´ C ·STxQ ³0lW¸U¶*B½m´ CNmO @Nle ´ CNlDÄ ò -. ,.. /. 0. 5. %1. 32 4. 6. 7. !-. ,8. :9. ;-.

(56) /. ,0. 35. <8. =-. >/. ,6.

(57) -æ u° éN¬ ¯ë ÔBm´ CNle~SBDT^Q8le C*? O Bº¸xQRTU? B¸O ´mDQRO e ÐQu¸Ue ´ ClUmBDCNQRTUe ´Ï³Je ¸U¶ Q8f¹»/f_ÙTxQ S*QRTlAl/¸UBD[ ÂÃ´ TU[BS'³Je ¸U¶ m´ O ´mDQu¸UBS O e C*BQRT2¾Ã¼c´lUle Ì*O A·C*´ Cº{@*C*e ÂÃ´ TU[¿3¸UTxQRCNl[e ¸¸Ue C*?\QRCNSTUBmBDe d5e C*?½QRTUTxQ Al3³Je ¸U¶ QRCNS QRÇUBDC*CNQ lDM TUBl¼cBm¸Ue d BDO A ÄVWQ mx¶ ¸UTxQRCNl[e ¸¸Ue C*?·QRÇUBDC*CNQ½lBDCNS*lW´ @¸JQ\Se À=BDTUBDÇ³aQ d BÂÃ´ TU[FM ³J¶*e~mx¶e~lWµ5C*ú³JC7ÄkGa¶*B m´ [¼*O BPÁTUBmBDe d BSÁle ? CNQRONe~l ¸U¶*BDCF? e d BDC^Ì5AÓí Dï{ö M ¾/î ¿ ? A@. CBDFEHGD. I. M. N. N ×1. 5. y(t) = βb(θ)aT (θ)x(t) + n(t) = βC(θ)x(t) + n(t) t = 1 . . . T,. ³J¶*BDTUB e~l0¸U¶*B·C5@*[\ÌcBDT

(58) ´RÂlCNQR¼Nl¶*´R¸xlDÄ$b5e CNmB·Ìc´R¸U¶h¸UTxQRCNl[e ¸¸Ue C*?HQRCNS¹TUBmBDe d5e C*?^QRTUTxQ Al0QRTUB\O e C*BQRTMN¸U¶*Bl/¸UBDBDTUe C*? d Bm¸U´ Txl¶NQ d B2¸U¶*B$ÂÃ´ O O ú³Je C*?\ÂÃ´ TU[Ál ¾ò ¿ ¾ó¿ T. T 2π 2π a(θ) = exp(−j a1 sin θ), . . . , exp(−j aM sin θ) , λ λ T 2π 2π b(θ) = exp(−j b1 sin θ), . . . , exp(−j bN sin θ) , λ λ. MQRCNS M M5QRTUBJ¸U¶*B2¼c´le ¸Ue ´ CNlW´RÂ-¸U¶*B2BDO BD[BDÇxl ³J¶*BDTUB SBDC*´R¸UBlk¸U¶*B

(59) ³aQ d BDO BDC*?R¸U¶7MR³J¶*BDTUB M ¾Ã³gÄ TÄ ¸ÄkQ·TUBÂÃBDTUBDCNmB2¼c´ e Ç¿W´RÂ¸U¶*B

(60) ¸UTxQRCNl[e ¸¸Ue C*?QRCNS^TUBmBDe d5e C*?·QRTUTxQ AlDMTUBl¼cBm¸Ue d BDO A Ä1>0´R¸UB

(61) ¸U¶NQu¸aÂÃ´ T¸U¶*Bgm´ O ´mDQu¸UBS lUmBDCNQRTUe ŃM=¸U¶*BEg_0KÛe~l

(62) ¸U¶*BlUQR[BÁQ l$¸U¶*BSe TUBm¸Ue ´ Cº{´RÂº(SBD¼NQRT¸U@*TUBH¾Eg_

(63) E$¿M¶*BDCNmB e~l$¸U¶*BÁ´ C*O Ah´ C*BÁO ´mDQu¸Ue ´ C¼NQuº TxQR[B¸UBDT$´RÂ¸U¶*B¸xQRTU? B¸Äf¹´ TUBDúd BDTM e~l½Q lUl@*[BSh¸U´¹¶NQ d BQRCÖÆ·È=ÉÊËuÉÊ

(64) @*C*e ÂÃ´ TU[Se~l/¸UTUe Ì*@¸Ue ´ CÏúd BDT

(65) ¸U¶*Bl@*¼*¼c´ T¸ ÄWÑ*´ T2le [¼*O e~me ¸/Aê CH¸U¶*B\SBDTUe duQu¸Ue ´ C7M*³BgSBÒNC*B Q la¸U¶*Bg¼NQRTxQR[B¸UBDTJ´RÂ1e ÇUBDTUBl/¸0e CNl/¸UBQ SH´RÂ1³´ TUµ5e C*? Se TUBm¸UO AF³Je ¸U¶ Ä\>0´R¸UB\¸U¶NQu¸M-¸U¶*B·¼=S5ÂWÂÃ´ T e~l Ä½ÔBÁSBDC*´R¸UB Ä\Ga¶*B BDO BD[BDÇxlJ´RÂ QRTUB$? e d BDCHÌ5A Ä e~la¸U¶*Bgd Bm¸U´ TJ´RÂ1BQ mx¶^¸UTxQRCNl[e ¸¸UBS ³aQ d BÂÃ´ TU[FÄ0ÔB·l@*¼*¼c´lB½¸U¶NQu¸

(66) ¸U¶*BlB\³aQ d BÂÃ´ TU[Ál0QRTUB½e CNSBD¼cBDCNSBDÇgQRCNSh¶NQ d Bg¸U¶*B½ÂÃ´ O O ú³Je C*?^BD[¼*e TUe~mDQRO3múduQRTUe~QRCNmB [ÁQu¸UTUe P-ö M-³J¶*BDTUB Ä

(67) ÔBQRO~l´^SBÒNC*B Q l0¸U¶*B C*´ e~lB0d Bm¸U´ TxlDM ³J¶*e~mx¶HQRTUB2Q lUl@*[BS·¸U´\ÌcB

(68) e CNSBD¼cBDCNSBDÇJQRCNSÁe~SBDÇUe~mDQRO O AÁSe~l/¸UTUe Ì*@¸UBS^me Txm@*O~QRTUO A·m´ [¼*O BP^Î$QR@NlUle~QRC ³Je ¸U¶hÐDBDTU´Rº{[BQRC¹QRCNShmúduQRTUe~QRCNmB$[ÁQu¸UTUe P Ä e~l0¸U¶*B½¸xQRTU? B¸

(69) m´ [¼*O BP¹QR[¼*O e ¸U@NSB\TUBDO~Qu¸UBSF¸U´Á¸U¶*B·I0iabF´RÂ ¸U¶*Ba¸xQRTU? B¸#e C·¸U¶*B

(70) b5³BDTUO e C*? gmDQ lB M QRCNS·¶NQ l#QRChÆaÈ=ÉÊËuÉÊkme Txm@*O~QRT#m´ [¼*O BPÁÎ$QR@NlUle~QRCSe~l/¸UTUe Ì*@¸Ue ´ C³Je ¸U¶ÁÐDBDTU´$[BQRC QRCNSFduQRTUe~QRCNmB McCNQR[BDO A Ä

(71) Ga¶*B½BDO BD[BDÇxl2´RÂ1¸U¶*B\@*C*µ5C*ú³JC¹¼NQRTxQR[B¸UBDT0d Bm¸U´ T ³Je ¸U¶ MJQRTUBhm´ CNle~SBDTUBS4¸U´8ÌcBÓl/¸xQu¸Ue~l/¸Ue~mDQRO O A4e CNSBD¼cBDCNSBDÇFl@Nmx¶'¸U¶NQu¸¸U¶*B /´ e Ç^¼=S5Â Ä X0CNSBDTH¸U¶*B8Q lUl@*[¼¸Ue ´ C ´RÂ\e CNSBD¼cBDCNSBDÇh´ ÌNlBDTUduQu¸Ue ´ CNlDM0¸U¶*BÏO e µ BDO e ¶*´5´SÖ´RÂ\¸U¶*BÂÃ@*O O\lB¸F´RÂ·´ ÌNlBDTUduQu¸Ue ´ CNl e~la? e d BDCHÌ5A ¾ 5¿ ó λ. ai i = 1 . . . M. bj j = 1 . . . N. θ. θ. [0, π]. u = sin(θ). θ. u. [C(θ)]k,l =. C(θ). Rx =. 1 T. PT. t=1. √1 π 1−u2 exp(−j 2π λ (al + bk ) sin θ). C(θ) = b(θ)aT (θ) ∈ CN ×M. p(u) =. x(t)xH (t) = Diag(σs2 ). x(t). 2 T σs2 = [σ12 , . . . , σM ]. {n(t)}Tt=1. R = σn2 I β. J8. σβ2. β ∼ CN (0, σβ2 ). Θ = [u, βR , βI ]. βR = Re{β}, βI = Im{β}. T. AK. p(Θ) = p(u)p(βR )p(βI ). Y =. [y(1), y(2), . . . , y(T )]. ! T 1 1 X H p(Y; Θ) = N T N T exp − 2 (y(t) − βC(θ)x(t)) (y(t) − βC(θ)x(t)) . π σn σn t=1. L-.

(72) 7æ 5ì®® 5ì è ®D¯ 5ì è écë è ê {éN° ç °u±*ê3±° ® 1±°u± ¯ ° 5®D¯ì 8±¯ìé è Ga¶*BaÔBDe~lUl/º{ÔBDe CNl/¸UBDe CÌc´ @*CNS^í î uï{M SBDC*´R¸UBS·Ì5A MR´RÂN¸U¶*BJ@*C*µ5C*ú³JC½¸xQRTU? B¸k¼NQRTxQR[B¸UBDTxl e~lJQ [ÁQu¸UTUe PHl@Nmx¶H¸U¶NQu¸a¸U¶*B$ÂÃ´ O O ú³Je C*?TUBDO~Qu¸Ue ´ C^¶*´ O~S*l M <N. D. O N. D. ,D. SR. QP. UT. TWV. YX ZI. ;-. %EQD ,D UD. E. WWB. Θ = [u, βR , βI ]. 3×3. E. . ˆ −Θ Θ. . ˆ −Θ Θ. T . E. Gkl =. . ˆ −Θ Θ. . ˆ −Θ Θ. T . ¾ ¿ /. ≥ WWB = sup HG−1 HT , hi ,si i=1,2,3. e~lW¸U¶*B$fÓb5V4´RÂ3QRC5AÁ]aQ A Ble~QRCÁBl/¸Ue [ÁQu¸U´ T QRCNS ³J¶*BDTUB [ÁQu¸UTUe P^´RÂ¸UBl/¸º{¼c´ e ÇxlDÄ#Ga¶*B$BDO BD[BDÇxlJ´RÂ¸U¶*B [ÁQu¸UTUe P QRTUB$? e d BDC^Ì5A ˆ Θ. 3×3. T. H = [h1 , h2 , h3 ]. e~lW¸U¶*B. 3×3. G. ¾ ¿. . E [Lsk (Y; Θ + hk , Θ) − L1−sk (Y; Θ − hk , Θ)][Lsl (Y; Θ + hl , Θ) − L1−sl (Y; Θ − hl , Θ)] , E {Lsk (Y; Θ + hk , Θ)} E {Lsl (Y; Θ + hl , Θ)}. 0. ³J¶*BDTUB MQRCNS Ä½Ga¶*B·BP¼cBm¸xQu¸Ue ´ CNl$QRTUB\¸xQRµ BDCÓúd BDT0¸U¶*B /´ e Ç Ä ¼=S5Â IJe ? ´ TU´ @NlO A MR¸U¶*B$Í@NQRÇUe ¸/A [\@Nl/¸ÌcB

(73) [ÁQuPe [e ÐDBSÁ³gÄ TÄ ¸Ä QRCNS O BQ Se C*?g¸U´Q½¶*e ? ¶^m´ [¼*@¸xQu¸Ue ´ CNQRO m´ [¼*O BPe ¸/A Ä1Å0ú³BDd BDTMRe ¸a¶NQ l#ÌcBDBDC^l¶*ú³JC¹í î ï1í ò uïçU¶NQu¸amx¶*´5´le C*? MQRCNSÁQ\Se~QR? ´ CNQRO*[ÁQu¸UTUe P O BQ S*l#¸U´Q\Ìc´ @*CNSHl/¸Ue O OçUe ? ¶¸Ä#Ga¶*BDTUBÂÃ´ TUB M³B$Q lUl@*[B

(74) ¶*BDTUB0¸U¶NQu¸ M M*QRCNSÁ¸U¶NQu¸ Ä »(C^¸U¶*e~l0mDQ lB M5¸U¶*BgBDO BD[BDÇxla´RÂ3¸U¶*Bg[ÁQu¸UTUe P mDQRCFÌcBg³JTUe ¸¸UBDCFQ l ¾ R¿ L(Y; Θ + hi , Θ) =. p(Y,Θ+hi ) p(Y,Θ). [K. si ∈ [0, 1], i = 1, 2, 3. p(Y, Θ). HG−1 HT. si. hi. ,7. (8. si =. 1 2 , ∀i. = 1, 2, 3. H. H = Diag([h1 , h2 , h3 ]T ). si = 1/2 ∀i. G. Gkl =. η(hk , hl ) + η(−hk , −hl ) − η(hk , −hl ) − η(−hk , hl ) , η(hk , 0)η(hl , 0). :5. ³J¶*BDTUB$³BgSBÒNC*B η(α, γ) =. Z Z Υ. ¾ ¿ 6. 1. 1. p(Y, Θ + α) 2 p(Y, Θ + γ) 2 dYdΘ, Ω. ³J¶*BDTUB QRCNS QRTUB2¸U¶*Bg´ ÌNlBDTUduQu¸Ue ´ CHQRCNSH¼NQRTxQR[B¸UBDTJl¼NQ mBlDM5TUBl¼cBm¸Ue d BDO A Ä ]A^SBDC*´R¸Ue C*? Ω. Υ. 0. η (Θ, α, γ) =. η(α, γ). mDQRCFÌcBgTUBD³JTUe ¸¸UBDCFQ l η(α, γ) =. Z. Z. 1. ¾ ¿ 7. 1. p(Y; Θ + α) 2 p(Y; Θ + γ) 2 dY, Ω. 1. j#O @*? ? e C*?F¾ 5¿#e ÇU´F¾ ¿M³B

(75) ´ Ì¸xQRe CHQ·mO ´lBS5ºÂÃ´ TU[ðBP¼*TUBlUle ´ CÁÂÃ´ T ¾¸U¶*BgSB¸xQRe O~laQRTUB2? e d BDCe C « L-. 7. Υ. η0. \I]ID. ! T 1 X H [f (Θ + α, t) − f (Θ + γ, t)] [f (Θ + α, t) − f (Θ + γ, t)] , η (Θ, α, γ) = exp − 2 4σn t=1 0. -. ¾/î ¿ è ê3ì *Ä î ¿ ¾/î î ¿ ,8. 1. η 0 (Θ, α, γ)p(Θ + α) 2 p(Θ + γ) 2 dΘ.. L^. 0.

(76) ³J¶*BDTUB Äa>0BP5¸MNQRCHe ÇUBD? TxQu¸Ue ´ CHúd BDT¸U¶*B½l@*¼*¼c´ T¸0´RÂ¸U¶*Bg¼NQRTxQR[B¸UBDTxle~lJTUBÍ@*e TUBSê CF´ TxSBDT¸U´ SBDTUe d B Ä#>0´R¸UB

(77) ¸U¶NQu¸M*e Â1³BgQ lUl@*[B$Q·@*C*e ÂÃ´ TU[Æ

(78) È=ÉÊËuÉÊW¼=S5Âk´ C M5¸U¶*Bg¼=S5Â3ÂÃ´ T e~l ³J¶*e~mx¶FmDQR@NlBl QRCÓQRCNQRO A¸Ue~mDQRO1SBDd BDO ´ ¼*[BDÇ

(79) ´RÂ#¸U¶*B·TUBÍ@*e TUBS¹e ÇUBD? TxQu¸Ue ´ C¹@*C*´ Ì¸xQRe CNQRÌ*O B ÄgGa¶*e~l

(80) O BQ S*l0¸U´FQ^C5@*[BDTUe~mDQRO3e ÇUBD? TxQu¸Ue ´ C ¸U´mDQRO~m@*O~Qu¸UBÁ¸U¶*BFm´ CNle~SBDTUe C*?¹Ìc´ @*CNS8³J¶*e~mx¶8e~l\d BDTUAÓ¸Ue [Bº(m´ CNl@*[e C*?NÄf¹´ TUBDúd BDTM3QÓmO ´lBS5ºÂÃ´ TU[BP¼*TUBlUle ´ CÏe~l QRO ³aQ Al#e ÇUBDTUBl/¸Ue C*?\ÌcBmDQR@NlB

(81) e ¸0QRO O ú³0lQRC^BQ lAe ÇUBDTU¼*TUB¸xQu¸Ue ´ C^´RÂ7¸U¶*B$Ìc´ @*CNS-M5eÄ B Ä M³B$mDQRCHl/¸U@NSAÁ¸U¶*B

(82) ¼cBDTÂÃ´ TU[ÁQRCNmB ´RÂ#¸U¶*BÁf¹»/f_ðTxQ S*QRT$lAl/¸UBD[³gÄ TÄ ¸Ä·l´ [BSBle ? CÓ¼NQRTxQR[B¸UBDTxlDÄ·iW´ CNlBÍ@*BDÇUO A M-³Je ¸U¶Ó¸U¶*Be ÇUBDÇUe ´ CÓ´RÂ´ Ì¸xQRe C*e C*? mO ´lBS5ºÂÃ´ TU[ØBP¼*TUBlUle ´ CH´RÂkÔÔÖ]

(83) Mc³B½¼*TU´ ¼c´lBgQSe À=BDTUBDÇ

(84) QR¼*¼*TU´Q mx¶H³J¶*BDTUBg³B\Q lUl@*[B½QÁ@*C*e ÂÃ´ TU[ ÆgÈ=ÉÊËuÉÊa¼=S5Â ´ C ¾lBDB\QRO~l´Fí òîï³J¶*BDTUB$¸U¶*B\lUQR[B

(85) ¸U¶*e C*?^QR¼*¼cBQRTxlae CH¸U¶*B\m´ ÇUBP5¸0´RÂkmO~Q lUle~mDQRO-ÌcBQRTUe C*?Bl/¸Ue [ÁQu¸Ue ´ Cc¿Äf¹´ TUBDúd BDTM ³J¶NQu¸UBDd BDT2¸U¶*B¹Æ\È=ÉÊËuÉÊ0¼=S5Â´ C M-e ¸ge~l2¸U¶*BD´ TUB¸Ue~mDQRO O AF¼*TUúd BSF¸U¶NQu¸$³Bl/¸Ue O O1´ Ì¸xQRe CQHl/¸UTUe~m¸$Ìc´ @*CNSÓ´ Ch¸U¶*BfÓb5V0Ä ÔB\QRO~l´^l¶*ú³ÕÌ5AFle [\@*O~Qu¸Ue ´ C8¾Ãe Cb5Bm¸Ue ´ C NÄ ò ¿W¸U¶NQu¸2¸U¶*B½¸/³´^QR¼*¼*TU´Q mx¶*Bla? e d Bgd BDTUAFmO ´lBgÌc´ @*CNS*l

(86) QRCNSH¸U¶*B·lUQR[B ¼*TUBSe~m¸Ue ´ CH´RÂ¸U¶*B\b>2I ¸U¶*TUBl¶*´ O~S-Ä K2lUl@*[e C*?4Q&@*C*e ÂÃ´ TU[ ÆFÈ=ÉÊËuÉÊ¼=S5Â½´ C Ma¸U¶*Bl/¸UTU@Nm¸U@*TUB¹´RÂ$¸U¶*BÓ[ÁQu¸UTUe P e~l? e d BDCÖQ lÂÃ´ O O ú³0lh¾lBDB « è ê3ì *Ä ò½ÂÃ´ TJ[´ TUB$SB¸xQRe O~lx¿ö ¾/î ò ¿ f (Θ, t) = βC(θ)x(t) η. θ. u. p(u) =. √1 π 1−u2. u. u. >-. u. \I]ID. L^. WWB. 0. .  WWB = Diag sup h1. ³J¶*BDTUB

(87) ¸U¶*B\QRTUTxQ Ale ? CNQRO-¸U´ÁC*´ e~lB¾ ¼cú³BDTMQRCNS^³J¶*BDTUB e~lJ? e d BDCHÌ5A. ASN R. h21. 1 , G11 2 T ASN R +. ¿e~l2SBÒNC*BSFQ l. P N/σn2. G11. G11 = 4. ³J¶*BDTUB$³BgSBÒNC*B. 1 2 σβ. 1 , 2 T ASN R +. ³J¶*BDTUB. P =. PM. k=1. 1 2 σβ. σk2. .  ,. e~la¸U¶*Bg¸U´R¸xQRO-¸UTxQRCNl[e ¸¸UBS. [(2 − h1 )f (2, h1 ) − 2(1 − h1 )] f (1, h1 )2 , (2 − h1 )2 f (2, h1 ). M N ϕ2π 1 X X σk2 f (ϕ, h1 ) = σβ2 T 1 − cos h (a + b ) + 1. 1 k r 2 σ2 λ r=1 n k=1. ¾/îó¿ ¾/î 5¿ ;-. »(CH¸U¶*B$ÂÃ´ O O ú³Je C*?NM³B½QRCNQRO AlB

(88) ¸U¶*Bg¼*TU´ ¼c´lBS^Ìc´ @*CNSHe CF¸UBDTU[Ála´RÂ#b>2IÕ¸U¶*TUBl¶*´ O~S^¼*TUBSe~m¸Ue ´ C¹QRCNSFQRTUTxQ A? BD´ [B¸UTUA SBle ? C7Ä æ « è ± 7®ì® ÉUËUÈNÞÉÝÊÞÜ\Ë gÝ *Þ xË =â Ga¶*BDTUBgQRTUB

(89) ¸U¶*TUBDB$¼*TU´ ¼cBDT¸Ue Bla´RÂ¸U¶*BgÔÔÖ]Ö[ÁQu¸UTUe P¾/î ò ¿ö î ÄJE2@*Bk¸Uá¸U¶*BWSe~QR? ´ CNQRORl/¸UTU@Nm¸U@*TUBk´RÂ5¸U¶*BkÔÔÖ]Ó[ÁQu¸UTUe P-MD¸U¶*B#O ú³BDT-Ìc´ @*CNS*l-´ C$Bl/¸Ue [ÁQu¸Ue ´ CgBDTUTU´ Txl=QRTUBkSBm´ @*¼*O BS-Ä %BL`. _. a 4bc4Jd. e. Lf Zg 3h*i. /.

(90) ò ÄJGa¶NQRC*µl#¸U´½¸U¶*e~lSBDTUe duQu¸Ue ´ C7M ¸U¶*B2e C*e ¸Ue~QRO=[\@*O ¸Ue~Se [BDCNle ´ CNQROc´ ¼¸Ue [e ÐQu¸Ue ´ CÁ¼*TU´ Ì*O BD[Û³gÄ TÄ ¸Ä M M5QRCNS ¶NQ l ÌcBDBDCFTUBS@NmBS¸U´Q·[´ C*´Se [BDCNle ´ CNQRO=´ ¼¸Ue [e ÐQu¸Ue ´ CF¼*TU´ Ì*O BD[Ø³gÄ TÄ ¸Ä ´ C*O A Ä ó*ÄJGa¶*B\ÔÔÖ] ÂÃ´ T QRCNS QRTUB$¸U¶*B·lUQR[B\Q lJ¸U¶*BDe T$]aQ A Ble~QRChiI0] e C¹¸U¶*B·mDQ lB½³J¶*BDTUB e~l2Q lUl@*[BSH¸U´^ÌcB µ5C*ú³JCÏ¾¸U¶*Bg¼*TU´5´RÂ3e~l0l/¸UTxQRe ? ¶¸ÂÃ´ TU³aQRTxSN¿Ä "ÈNÞÉ ËuÉ ÁÆ xÞ\Ê ÁÈ=ÉUÞxâuÊ ÝÊ Ý *Þ Ë xÆ Æ =âÝ *Þ ðÝ 5ÉUÞÜ *Ë â Ga¶*B$QRe [Û´RÂ-¸U¶*e~l¼NQRT¸e~lW¸U´\BP*QR[e C*B0¸U¶*B

(91) @NlBÂÃ@*O C*BlUlW´RÂ@Nle C*?½¸U¶*B2ÔÔÖ]"¸U´·¼*TUBSe~m¸W¸U¶*B

(92) @*O ¸Ue [ÁQu¸UB

(93) ? O ´ ÌNQROcfÓb5V QRCNSÏ¸U¶*Bhb>2I ¸U¶*TUBl¶*´ O~S-Ä&Ñ*´ T\¸U¶NQu¸ÁTUBQ l´ C7M1³BFm´ CNle~SBDT½¸U¶*BFBD[¼*e TUe~mDQRO? O ´ ÌNQROfÓb5Vð´RÂ0¸U¶*BFfhK0jðBl/¸Ue [ÁQu¸U´ TM ³J¶*e~mx¶he~l2BDduQRO @NQu¸UBS¹úd BDT\î Áf¹´ ÇUBiQRTUO ´Á¸UTUe~QRO~lDÄ

(94) Ga¶*Ble [\@*O~Qu¸Ue ´ Che~l2¼cBDTÂÃ´ TU[BS¹³Je ¸U¶Ó¼NQRTxQR[B¸UBDTxl0Q l0ÂÃ´ O O ú³0lDö ¸U¶*B¸UTxQRCNl[e ¸\QRTUTxQ AFe~l½l¼NQRTxlB³Je ¸U¶ lBDCNl´ Txl$QRCNS³Je ¸U¶&e ÇUBDTº{BDO BD[BDÇ½l¼NQ me C*?&¾Ãe CÏ@*C*e ¸\´RÂa³aQ d BDO BDC*?R¸U¶Nlx¿ BÍ@NQRO~l¸U´ QRCNS^¸U¶*B½TUBmBDe d BgQRTUTxQ AÁe~l0QRC¹X2nK ³Je ¸U¶ lBDCNl´ TxlJQRCNSH³Je ¸U¶¹e ÇUBDTº{BDO BD[BDÇ2l¼NQ me C*?¹¾Ãe C¹@*C*e ¸2´RÂ ³aQ d BDO BDC*?R¸U¶Nlx¿0BÍ@NQRO~l

(95) ¸U´ *Ä Ä^b5@Nmx¶&QRTUTxQ Ahm´ CÒN? @*TxQu¸Ue ´ CÓe~lg³BDO O º{µ5C*ú³JCÓe C&f¹»/f_ TxQ S*QRTÁ¾/í ò ïW¼7Ä ¿2¸U´hmTUBQu¸UBÁQ >0AÍ@*e~l/¸kd5e T¸U@NQROQRTUTxQ A

(96) ³Je ¸U¶? ´5´S½¼cBDTÂÃ´ TU[ÁQRCNmB Ä3Ga¶*Ba@*CNm´ TUTUBDO~Qu¸UBS½f¹»/f_"TxQ S*QRT1³aQ d BÂÃ´ TU[ÁlQRTUB? BDC*BDTxQu¸UBS½@Nle C*? Å2Q S*QR[ÁQRTxSm´SBl³Je ¸U¶ lCNQR¼Nl¶*´R¸xlDÄ1Ga¶*B2¸UTxQRCNl[e ¸¸UBS^¼cú³BDTe~laQ lUl@*[BSÁ¸U´·ÌcB$@*C*e ÂÃ´ TU[O ASe~l/¸UTUe Ì*@¸UBS´ C QRO O-¸UTxQRCNl[e ¸

(97) QRÇUBDC*CNQ lDÄ e~l2mx¶*´lBDCF´ CH¸U¶*B\l@*¼*¼c´ T¸ ÄÑ3e ? @*TUBÁî½l¶*ú³0la¸U¶*B½ÔÔÖ]äQRCNS^¸U¶*B\fhK0jÖ´RÂ1¸U¶*B ¼NQRTxQR[B¸UBDTW´RÂ7e ÇUBDTUBl/¸ d BDTxl@NlWK

(98) b>2IgÄ»{¸JmDQRCÌcB$lBDBDÇU¶NQu¸¸U¶*B

(99) ÔÔÖ]'¼*TUúd5e~SBlWQ½? ´5´SÁ¼*TUBSe~m¸Ue ´ C^´RÂ-¸U¶*B$fÓb5V e CFQRO O=TUBD? e ´ CNlDÄ3»{¸0¼*TUBSe~m¸xlW¸U¶*B

(100) ¸U¶*TUBl¶*´ O~SFb>2I O ´mDQu¸Ue ´ ÇU´ÌcB ÁS*]ÌcBDO ú³¸U¶*B

(101) ¸U¶*TUBl¶*´ O~SFb>2I e CNSe~mDQu¸UBS^Ì5AÁ¸U¶*B fhK0j1Ä >0BP5¸M3Q lgQuÂÃ´ TUBD[BDÇUe ´ C*BS-M=¸U¶*BDTUBÁQRTUB\¸/³´F¼c´lUle Ì*O BÁQ lUl@*[¼¸Ue ´ CNl2ÂÃ´ T$¸U¶*BÓÆ\È=ÉÊËuÉÊ

(102) ¼=S5Âa´ C M´ C*BO BQ S*l2¸U´hQ C5@*[BDTUe~mDQROke ÇUBD? TxQu¸Ue ´ C&QRCNSÏQRC*´R¸U¶*BDT½O BQ S*l$¸U´ÓQ¹mO ´lBS5ºÂÃ´ TU[ÙBP¼*TUBlUle ´ C´RÂa¸U¶*BÌc´ @*CNS-ÄF»(C8Ñ3e ? @*TUB^òM3³Bl¶*ú³ ¸U¶NQu¸¸U¶*B

(103) ¸/³´Q lUl@*[¼¸Ue ´ CNlW? e d B

(104) d BDTUAÁmO ´lB

(105) Ìc´ @*CNS*laQRCNS¸U¶*B$lUQR[B

(106) ¼*TUBSe~m¸Ue ´ C^´RÂ¸U¶*Bgb>2IÖ¸U¶*TUBl¶*´ O~S-Äk>0´R¸UB

(107) ¸U¶NQu¸M e C¹´ @*T0le [\@*O~Qu¸Ue ´ C7M*¸Ue [B½C*BDBSBSH¸U´mDQRO~m@*O~Qu¸UB$¸U¶*B½Ìc´ @*CNSF³Je ¸U¶¹C5@*[BDTUe~mDQRO7e ÇUBD? TxQu¸Ue ´ CHe~l0úd BDTgî \¸Ue [Bl0´RÂ1¸Ue [B e C¹¸U¶*B\´R¸U¶*BDT$mDQ lB ÄgiW´ CNlBÍ@*BDÇUO A M*¸U´^¶NQ d B\QÓ¾l/¸Ue O O3¸Ue ? ¶¸¿JÌc´ @*CNSh³Je ¸U¶hB×me BDÇUO A¹m´ [¼*@¸xQu¸Ue ´ CNQRO1m´l/¸

(108) QRCNS¹BQ lA e ÇUBDTU¼*TUB¸xQu¸Ue ´ C7M¸U¶*B\ÔÔÖ]äSBDTUe d BSF³Je ¸U¶¹¸U¶*B·Q lUl@*[¼¸Ue ´ C¹´RÂk@*C*e ÂÃ´ TU[ Æ$È=ÉÊËuÉÊJ¼=S5ÂW´ C e~l0¼*TUBÂÃBDTUTUBSFBl¼cBme~QRO O A ÂÃ´ T0SBle ? C*e C*?f¹»/f_ TxQ S*QRTJlAl/¸UBD[FÄ ÆuÉÉUÆuà ÞxË ÁÞÝÉàÁÊ uÞÜÝÊ ÆuÝÊË Ga¶*e~lJ¼NQRT¸0e C5d Bl/¸Ue ?Qu¸UBl¸U¶*B½ÌcBD¶NQ d5e ´ TJ´RÂ3¸U¶*B\SBDTUe d BS^ÔÔÖ] ³gÄ TÄ ¸Ä#¸U¶*B\QRTUTxQ AÁ? BD´ [B¸UTUA´RÂ1Ìc´R¸U¶F¸U¶*B½TUBmBDe d BDT QRCNS\¸UTxQRCNl[e ¸¸UBDTÄ1>0´R¸UBa¸U¶NQu¸#¸U¶*BJ¼*TU´ ¼c´lBS·Ìc´ @*CNS·¶NQ lkÌcBDBDCSBDTUe d BS·e C·¸U¶*B2m´ ÇUBP5¸#´RÂ=O e C*BQRTWQRTUTxQ Al3Ì*@¸W¼c´lUle Ì*O A C*´ Cº{@*C*e ÂÃ´ TU[FÄkiW´ CNlBÍ@*BDÇUO A Mue ¸kTUBD[ÁQRe CNl1Q

(109) SBD? TUBDB´RÂNÂÃTUBDBS´ [ä¸U´

(110) BDC*¶NQRCNmB¸U¶*Ba¼cBDTÂÃ´ TU[ÁQRCNmB ÄÔBÒNTxl/¸kQRCNQRO A5ÐDBW¸U¶*B e [¼NQ m¸W´RÂ-¸U¶*B2TUBmBDe d BDTWQRTUTxQ A½? BD´ [B¸UTUA·³J¶*e O B0¸U¶*B0¸UTxQRCNl[e ¸¸UBDTWe~lW[ÁQ SBJÂÃTU´ [ÛQ\mO~Q lUle~mDQRONX2nK½Ä5]´R¸U¶Á¸UTxQRCNl[e ¸¸UBDT QRCNSÓTUBmBDe d BDTgQRTUB·[ÁQ SB³Je ¸U¶ lBDCNl´ TxlDÄGa¶*B·¸UTxQRCNl[e ¸¸UBDTge~l½Q^Ò*PBSX2nK¾Ã¶NQRO Âº{³aQ d BDO BDC*?R¸U¶Óe ÇUBDTº BDO BD[BDÇWl¼NQ me C*?5¿3QRTUTxQ A ÄGa¶*BJTUBmBDe d BDT1¶NQ l#Se À=BDTUBDÇ#O e C*BQRT#QRTUTxQ A$? BD´ [B¸UTUe Bl1µ BDBD¼*e C*?

(111) ¸U¶*B0QRTUTxQ A½QR¼cBDT¸U@*TUB ¾Ãe C@*C*e ¸a´RÂ ¿kÒ*PBSh¾ÃeÄ B Ä3¸U¶*B

(112) ¼c´le ¸Ue ´ CNl#´RÂ-¸U¶*B$ògBP5¸UTUBD[B

(113) lBDCNl´ Txlk´RÂ-¸U¶*B

(114) O e C*BQRTQRTUTxQ A\QRTUB0Ò*PBSN¿Ä1»(C´R¸U¶*BDTW³´ TxS*lDM ¸U¶*BDe Tge~l½ò ò^¼c´le ¸Ue ´ CNlgQ duQRe O~QRÌ*O B\ÂÃ´ T$¸U¶*B H´R¸U¶*BDT½lBDCNl´ Txl·¾ *îó^¼c´lUle Ì*e O e ¸Ue Blx¿ÄÁGa¶*BÁC5@*[\ÌcBDTg´RÂa´ ÌNlBDTUduQu¸Ue ´ CNl2e~l h1 h2. h3. h1. βR. a 4 jF4Yklk\m. Le. n. 3ipo. βI. i. u. o. iFq. Lf [q(2 cg 32rts%u. 3i. Lf s*vw. Lf. xf 32. ,8c8c8. M =8. 4. N =8. >8. /. 530. T = 64 h1. [−1, 1]. u. -. u. ,8c8. u. a 4zy{4$rU|r~}. Zq. 3n. i. q. 3i. M =N =8. D = 23. λ 2. \0. 0. :5

(115) -{0.

(116)

(117) Ä ÔB\¶NQ d B½? BDC*BDTxQu¸UBS¹QRO O¸U¶*BQRTUTxQ AHm´ CÒN? @*TxQu¸Ue ´ CNl0QRCNShm´ [¼*@¸UBS¹¸U¶*BQ lUl´me~Qu¸UBSHÔÔÖ]

(118) Ä=»(CÑ3e ? @*TUB\ó*M QR[´ C*?¸U¶*BlB *îóÁÌc´ @*CNS*l

(119) ³B¶NQ d B\¼*O ´R¸¸UBSh¸U¶*BÌc´ @*CNSÓ³J¶*e~mx¶ÓO BQ S*l2¸U´^¸U¶*BO ú³Bl/¸½b>2I ¸U¶*TUBl¶*´ O~S4¾SBDC*´R¸UBS Ìc´ @*CNSÓK

(120) ¿

(121) QRCNS¹¸U¶*BÌc´ @*CNSh³J¶*e~mx¶Ó¶NQ l2¸U¶*BO ú³Bl/¸$fÓb5VÕe Ch¸U¶*BÁQ lA5[¼¸U´R¸Ue~m\QRTUBQÓ¾SBDC*´R¸UBSÓÌc´ @*CNSÓ]J¿Ä-ÔB·¶NQ d B C*´R¸Ue~mBSh¸U¶NQu¸\Ìc´R¸U¶Ï? BD´ [B¸UTUe Bl

(122) O BQ Se C*?F¸U´FÌc´ @*CNS&K QRCNSÏ]ÚQRTUB·C*´R¸½³Je ¸U¶&[e C*e [\@*[ÙTUBS@*CNS*QRCNmAh´ Tg[e C*e [\@*[ ?QR¼7Ä8»(CNSBDBS-M#ÂÃ´ T·¸U¶*e~lÁm´ CÒN? @*TxQu¸Ue ´ C7Mke ¸mDQRC4ÌcBhlBDBDC8¸U¶NQu¸Áe ¸ÁBPe~l/¸xl\ÂÃ´ @*T[e C*e [\@*[·º{TUBS@*CNS*QRCNmA8QRTUTxQ AlDÄ&ÔB ¶NQ d B½m´ [¼*@¸UBSH¸U¶*B\Ìc´ @*CNS*l2Q lUl´me~Qu¸UBS^¸U´Á¸U¶*BlB½ÂÃ´ @*T0¼c´lUle Ì*e O e ¸Ue Bl

(123) QRCNSF¼*O ´R¸¸UBSH¸U¶*B\ÌcBl/¸

(124) ´ C*B½e CF¸UBDTU[Ál0´RÂ#fÓb5V ÂÃ´ T$m´ [¼NQRTUe~l´ Ch¼*@*TU¼c´lB ÄgGa¶*BÌc´ @*CNSÓK¾ÃeÄ B Äg¸U¶*B·Ìc´ @*CNSÓ³Je ¸U¶h¸U¶*BO ú³Bl/¸gb>2I ¸U¶*TUBl¶*´ O~SN¿0l¶*ú³0l0¸U¶NQu¸gÌcB¸¸UBDT Q lA5[¼¸U´R¸Ue~m¹¼cBDTÂÃ´ TU[ÁQRCNmBmDQRCÖÌcB´ Ì¸xQRe C*BS-Äð»{¸FBPe~l/¸xlHQ8? BD´ [B¸UTUA'm´ CÒN? @*TxQu¸Ue ´ CO BQ Se C*?8¸U´"lO e ? ¶¸UO A'ÌcB¸¸UBDT Q lA5[¼¸U´R¸Ue~mJ¼cBDTÂÃ´ TU[ÁQRCNmB\¾ÃÌc´ @*CNS^]J¿#Ì*@¸a³Je ¸U¶^Q½³´ Txl/¸ab>2I¸U¶*TUBl¶*´ O~S-Äk>0´R¸UB0¸U¶NQu¸M5³B2¶NQ d B0´ Ì¸xQRe C*BS¸U¶*B$lUQR[B TUBl@*O ¸xlDMeÄ B ÄklUQR[B2Ìc´ @*CNS*lQ lUl´me~Qu¸UBS·¸U´\¸U¶*B

(125) lUQR[B2´ ¼¸Ue [ÁQROc? BD´ [B¸UTUe BlDMe CÁ¸U¶*B

(126) ´ ¼*¼c´le ¸UB

(127) mDQ lB0³J¶*BDTUB0¸U¶*B

(128) TUBmBDe d BDT e~l

(129) QÁÒ*PBSÓX2nKðQRCNS¹³J¶*BDTUB½¸U¶*B\¸UTxQRCNl[e ¸¸UBDT$QRTUTxQ A^? BD´ [B¸UTUAHe~l

(130) ´ ¼¸Ue [e ÐDBSFÂÃ´ O O ú³Je C*?¸U¶*BlUQR[B\QuÂÃ´ TUBD[BDÇUe ´ C*BS ³aQ A Ä -æ é è ² ë3®ìé è ® »(Cg¸U¶*e~l1m´ TUTUBl¼c´ CNSBDCNmB MD³BW¶NQ d BWSBDTUe d BSgQ2mO ´lBS5ºÂÃ´ TU[ BP¼*TUBlUle ´ Cg´RÂ¸U¶*BWÔBDe~lUl/º{ÔBDe CNl/¸UBDe C½Ìc´ @*CNSgÂÃ´ T3f¹»/f_ I0Q S*QRTJ³Je ¸U¶Óm´ O ´mDQu¸UBSHO e C*BQRT2QRTUTxQ AlDÄ#»{¸

(131) ¶NQ lJÌcBDBDChlBDBDCF¸U¶NQu¸0¸U¶*B\ÔBDe~lUl/º{ÔBDe CNl/¸UBDe C¹Ìc´ @*CNSHÂÃ´ TJ¸U¶*B\¼NQRTxQR[B¸UBDTJ´RÂ e ÇUBDTUBl/¸

(132) ¼*TUúd5e~SBl2Q^? ´5´S¹¼*TUBSe~m¸Ue ´ Ch´RÂ#¸U¶*BfÓb5VÕe CQRO O3TUBD? e ´ CNlDÄ

(133) »{¸gQRO~l´^¼*TUBSe~m¸xl0¸U¶*B\¸U¶*TUBl¶*´ O~SÓb>2IÚO ´mDQu¸Ue ´ C C*BQRT0¸U¶*B\¸U¶*TUBl¶*´ O~Shb>2I e CNSe~mDQu¸UBShÌ5AH¸U¶*BfhK0j1Ä=ÔB·¶NQ d B\QRO~lé C5d Bl/¸Ue ?Qu¸UBSH¸U¶*B·e CõN@*BDCNmB\´RÂQRTUTxQ A? BD´ [B¸UTUA ´ C^¸U¶*BgÌcBD¶NQ d5e ´ Ta´RÂ3¸U¶*B$ÔBDe~lUl/º{ÔBDe CNl/¸UBDe CFÌc´ @*CNS-Ä 7æ « è ê3ì ·ÞÉÊ uÆuÝÊË ÏË >0´R¸UB$¸U¶NQu¸ ¾/î ¿ ³J¶*BDTUB. T = 64. U5

(134) -{0. . U. cB. \I]ID. 4bc4 . L^. . 3i. e. η 0 (Θ, α, γ). ! T X 1 p(Y; Θ + α) p(Y; Θ + γ) = N T N T exp − ζ(Θ, α, γ, t) , π σn t=1 1 2. 1 2.

(135) /. 1 1 [y − f (Θ + α, t)]H [y − f (Θ + α, t)] + 2 [y − f (Θ + γ, t)]H [y − f (Θ + γ, t)] 2 2σn 2σn 1 1 1 1 1 = 2 yH y − yH f (Θ + α, t) − yH f (Θ + γ, t) − f (Θ + α, t)H y − f (Θ + γ, t)H y σn 2 2 2 2 1 1 H H + f (Θ + α, t) f (Θ + α, t) + f (Θ + γ, t) f (Θ + γ, t) . 2 2. ζ(Θ, α, γ, t) =. E2BÒNC*B. z = y − 12 f (Θ + α, t) − 21 f (Θ + γ, t). M*³B$¶NQ d B 5. ¾/î ¿ ,0.

(136) ζ(Θ, α, γ, t) =. ¾/î R¿. 1 H 1 z z + ζ 0 (Θ, α, γ, t), 2 σn 4. 35. ³J¶*BDTUB S´5BlaC*´R¸0SBD¼cBDCNSF´ C MNQRCNS ζ 0 (Θ, α, γ, t) =. [f (Θ + α, t) − f (Θ + γ, t)]H [f (Θ + α, t) − f (Θ + γ, t)]. 1 2 σn. ÄkiW´ CNlBÍ@*BDÇUO A M. ζ 0 (Θ, α, γ, t). Y. ! T X 1 0 1 H exp − ( 2 z z + ζ (Θ, α, γ, t)) dY σn 4 Ω t=1 ! T X1 ζ 0 (Θ, α, γ, t)) . = exp − 4 t=1. 1 η (Θ, α, γ) = N T N T π σn 0. Z. ·ÞÉÊ uÆuÝÊË ÏË ÁÆuÝÉÊ Ñ*TU´ [Ù¾/î ¿M¸U¶*BglB¸0´RÂ3ÂÃ@*CNm¸Ue ´ CNlae C5d ´ O d BSe C. 4 jF4 . . 3i. eJn. . WWB. ,6.   η 0 (Θ, h1 , h1 ) = 1,       η 0 (Θ, −h1 , −h1 ) = 1, 0. G11. ¾/î ¿ ,7. n. 2 − 21 (βR. βI2 )T. σk2 2 r=1 σn. PM PN. σk2 2 σn. . . 1 − cos. 4π λ h1 (ak. η0. u. Θ.  1  η(h1 , h1 ) = 2−h  2 ,     2−h1    η(−h1 , −h1 ) = 2 , 1−h1 η(h1 , −h1 ) = , 2 PM PN σk  4π 2 1T  σβ k=1 r=1 σ2 [1−cos( λ h1 (ak +br ))]+1  2  n   2−h1 1   .  η(h1 , 0) = 2 2 1 PM PN σk2 1−cos σβ 2 T [ ( 2π k=1 r=1 σ2 λ h1 (ak +br ))]+1 n. Ga¶*B½lB¸J´RÂ3ÂÃ@*CNm¸Ue ´ CNlJe C5d ´ O d BSe C. Gkk , k = 2, 3. o. + br ) , o 1 − cos 2π . λ h1 (ak + br ). Ga¶*BDC7MNle CNmBgQRO O-¸U¶*B

(137) ÂÃ@*CNm¸Ue ´ CNl S´ÁC*´R¸0SBD¼cBDCNSF´ C MÌ5Ae ÇUBD? TxQu¸Ue C*?·³gÄ TÄ ¸Ä M*³B$´ Ì¸xQRe C. ¾ò ¿ (8. QRTUB$? e d BDCHÌ5A.   η 0 (Θ, hk , hk ) = 1,       η 0 (Θ, −hk , −hk ) = 1,. Ga¶*BDC7M. ,6. QRTUB$? e d BDCHÌ5A.  η (Θ, h1 , −h1 ) = exp +  k=1   n    η 0 (Θ, h , 0) = exp − 1 (β 2 + β 2 )2T PM PN 1 R I k=1 r=1 2.       . ¾/î ¿. PM σk2 η (Θ, hk , −hk ) = exp −h2k N T k=1 , 2 σn PM 2 σk η 0 (Θ, hk , 0) = exp − 41 h2k N T k=1 . 2 σn 0. .   η(hk , hk ) = 1,       η(−hk , −hk ) = 1,. 2 PM σk2 h  η(hk , −hk ) = exp −h2k N T k=1 exp − σk2 ,  2 σn  β  P  2  h2k  η(h , 0) = exp − 1 h2 N T M k=1 σk exp − . k 4 k σ2 4σ 2 6. n. β. ¾òî ¿. ¾ò ò ¿.

(138) Ga¶*B½lB¸J´RÂ3ÂÃ@*CNm¸Ue ´ CNlJe C5d ´ O d BSe C                                                                         . G12. QRTUB

(139) ? e d BDCHÌ5A. (. M X N X σk2 2π 2π 2 β 1 − cos h (a + b ) + β h 1 − cos h (a + b ) + 1 k r R 2 1 k r σ2 R λ λ k=1 r=1 n ) 2π 2π 1 2 2 h1 (ak + br ) − βI h2 sin h1 (ak + br ) + h2 , βI 1 − cos λ λ 2 ( M X N X σk2 2π 2π 2 η 0 (Θ, −h1 , −h2 ) = exp 2T β 1 − cos h (a + b ) − β h 1 − cos h (a + b ) + 1 k r R 2 1 k r σ2 R λ λ k=1 r=1 n ) 2π 2π 1 2 2 βI 1 − cos h1 (ak + br ) + βI h2 sin h1 (ak + br ) + h2 exp , λ λ 2 ( M X N 2 X σk 2π 2π 0 2 η (Θ, −h1 , h2 ) = exp 2T β 1 − cos h1 (ak + br ) + βR h2 1 − cos h1 (ak + br ) + σ2 R λ λ k=1 r=1 n ) 2π 2π 1 βI2 1 − cos h1 (ak + br ) + βI h2 sin h1 (ak + br ) + h22 exp , λ λ 2 ( M N 2 XX σ 2π 2π 2 0 k η (Θ, −h1 , −h2 ) = exp 2T β 1 − cos h1 (ak + br ) − βR h2 1 − cos h1 (ak + br ) + σ2 R λ λ k=1 r=1 n ) 2π 1 2 2π 2 βI 1 − cos h1 (ak + br ) − βI h2 sin h1 (ak + br ) + h2 exp . λ λ 2 0. η (Θ, h1 , h2 ) = exp 2T. ¾òRó¿. iW´ CNlBÍ@*BDÇUO A M η(h1 , h2 ) = η(−h1 , h2 ) =. ¾ò 5¿. 2 − h1 t1 t2 , 2. 3-. ³J¶*BDTUB #) M N T X X σk2 2π 1 exp 1 − cos h1 (ak + br ) + 2 2 σ2 λ σβ k=1 r=1 n √ πσβ × n h io 12 , 2 2 P P σβ σk M N T 2π 1 4 2 2 k=1 r=1 σ2 1 − cos λ h1 (ak + br ) + σ2 n β   h 2 i2 P P σk M N   T 2π   h2 2 k=1 r=1 σ2 sin λ h1 (ak + br ) n h P i t2 = exp 2 P σk N   4 T2 M  1 − cos 2π + σ12  2 k=1 r=1 σn λ h1 (ak + br ) β √ πσβ × n h io 12 . 2 σβ T PM PN σk2 2π 1 4 2 2 k=1 r=1 σ2 1 − cos λ h1 (ak + br ) + σ2 h2 t1 = exp − 22 2σβ. !. (. h22 4. ". b5e [e O~QRTUO A Mke ¸e~ll/¸UTxQRe ? ¶¸ÂÃ´ TU³aQRTxSÓ¸U´lBDB^¸U¶NQu¸ ÄgÔBÁQRO~l´^¶NQ d B M M n. G12 = 0. ¾ò ¿ c/. ¾ò ¿ (0. ³J¶*e~mx¶7Mk¸U´ ? B¸U¶*BDT·³Je ¸U¶Õ¾ò 5¿M#O BQ SÏ¸U´ M7QRCNS Ä\Ga¶*BDTUBÂÃ´ TUB MçU¶*B·[ÁQu¸UTUe P e~l β. M. η(h1 , −h2 ) = η(−h1 , −h2 ). G21 = 0 G13 = 0 G31 = 0 G23 = 0. 7. G32 = 0. 3-. G.

(140) Se~QR? ´ CNQRO=³Je ¸U¶FBDO BD[BDÇxla? e d BDCHÌ5A 2−h1 2. G11 = 2 .  2−h1. G22. G33. 2. −. σ2 β 1 2 2T σn. σ2 β 1 2 2T σn. PM. PN. k=1. PM. k=1. ¾ò R¿. 1−h1. 4π 2 r=1 σk [1−cos( λ h1 (ak +br ))]+1. 1. PN. r=1. σk2 [1−cos( 2π λ h1 (ak +br ))]+1. 2 PM σk2 h 1 − exp −h22 N T k=1 exp − σ22 2 σn β , PM =2 2 h22 1 2 k=1 σk exp − 2 h2 N T σ2 exp − 2σ2 n β 2 PM 2 σ h k 1 − exp −h23 N T k=1 exp − σ23 2 σn β , PM =2 2 h23 1 2 k=1 σk exp − 2 h3 N T σ2 exp − 2σ2 n. {5. 2 , . ¾ò ¿ (6. ¾ò ¿ (7. β. aG ¶NQRC*µl#¸U´½¸U¶*B$SBÒNC*e ¸Ue ´ C^´RÂ e C¾/î 5¿M mDQRCÌcB

(141) TUBD³JTUe ¸¸UBDC^Q lWe C¾/îó¿ÄkX2le C*?·QRO OçU¶*B$Q lUl@*[¼¸Ue ´ CNlWQRCNS TUBl@*O ¸xlJQRÌcúd B M¸U¶*B½mO ´lBS5ºÂÃ´ TU[ BP¼*TUBlUle ´ C^´RÂ¸U¶*Bg[ÁQu¸UTUe P mDQRCHÌcBg³JTUe ¸¸UBDC¹Q l ¾ó ¿ ;-. f (ϕ, h1 ). G11. WWB. h2 h2 h2 WWB = Diag sup 1 , sup 2 , sup 3 , h1 G11 h2 G22 h3 G33. c8. ³Je ¸U¶ ? e d BDCHe C&¾/îó¿M7¾ò ¿M*QRCNSÏ¾ò ¿MTUBl¼cBm¸Ue d BDO A Ä >0´R¸UB\¸U¶NQu¸½e CÓ¸U¶*B´ TUe ? e CNQROÂÃ´ TU[,´RÂW¸U¶*BÔBDe~lUl/º{ÔBDe CNl/¸UBDe CÌc´ @*CNS"¾ ¿M7³B¶NQ d B½¸U´F´ ¼¸Ue [e ÐDB·¸U¶*B[ÁQu¸UTUe P ³gÄ TÄ ¸Ä QRCNS Ägb5e CNmB SBD¼cBDCNS*l

(142) ´ C*O AH´ C MçU¶*B\´ ¼¸Ue [e ÐQu¸Ue ´ C¹¸xQ lµHe~l2TUBS@NmBSF¸U´ ³J¶*e~mx¶ e~la[´ TUB

(143) ¸UTxQ m¸xQRÌ*O B Äk>0BP5¸M*³B$? e d BgQmO ´lBS5ºÂÃ´ TU[ BP¼*TUBlUle ´ CÂÃ´ T M M ¾ó*î ¿ (6. G11 , G22 , G33. (7. /. h1 , h2 ,. h3. WWB. h2k , ∀k sup Gkk hk. hi. Gii. h2. k sup Gkk k = 2, 3. hk. WWBkk = sup. x exp(−ax) , a > 0, − exp(−2ax)]. x≥0 2[1. ³J¶*BDTUB. x = h2k. QRCNS. a = NT. PM. 2 k=1 σk 2 2σn. +. 1 2 2σβ. Äk»{Â1³BgSBDC*´R¸UB. g(x) =. x exp(−ax) 2[1−exp(−2ax)]. M¸U¶*BDC. dg(x) exp(−ax) [1 − ax − exp(−2ax) − ax exp(−2ax)] = , dx 4[1 − exp(−2ax)]2. ³J¶*e~mx¶&e~l·QRO ³aQ Al$C*BD?Qu¸Ue d B ÄiW´ CNlBÍ@*BDÇUO A M [ÁQuPe [\@*[ Qu¸ ÄkÑ3e CNQRO O A M*³B$´ Ì¸xQRe C7ö ∀x > 0. g(x). e~l·Q¹[´ C*´R¸U´ C*e~mDQRO O ASBmTUBQ le C*?FÂÃ@*CNm¸Ue ´ C4QRCNS&¶NQ l½e ¸xl. x=0. sup hk. ³J¶*e~mx¶Fm´ CNmO @NSBl¸U¶*Bg¼*TU´5´RÂ/Ä. 1 h2k PM = σk2 Gkk 2 N T k=1 + σ2. î. n. ,8. ¾óò ¿. 1 2 σβ. , k = 2, 3,. ¾ó ó¿.