Tài liệu Introduction to Communication Systems, The Theory of Probability ppt

14 582 0
Tài liệu Introduction to Communication Systems, The Theory of Probability ppt

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

Thông tin tài liệu

Lqwurgxfwlrq wr Frppxqlfdwlrq V|vwhpv Wkh Wkhru| ri Suredelolw| Suredelolw| Wkhru| Zk| Suredelolw| Suredelolw| wkhru| lv dssolhg wr vlwxdwlrqv zkhuh xqfhuwdlqw| h{lvwv1 Wkhvh vlwxdwlrqv lqfoxgh 41 Wkh fkdudfwhul}dwlrq ri wud!f dw wkh lqwhuvhfwlrq XV 793 dqg Shsshuv Ihuu| Urdg +l1h1/ wkh qxpehu ri fduv wkdw furvv wkh lqwhuvhfwlrq dv d ixqfwlrq ri wlph, 51 Wkh suhglfwlrq ri wkh zhdwkhu lq Eodfnvexuj 61 Wkh qxpehu ri vwxghqwv wudyhuvlqj wkh Guloo Ilhog ehwzhhq ;=83 dqg &> N > D j duh dovr hyhqwv1 &# @ 7=8368 S , zkhuh lv wkh vdpsoh vsdfh/ zklfk lv wkh vhw ri doo srvvleoh rxwfrphv ( lv wkh vljpd dojheud ru vljpd hoC/ zklfk lv wkh vhw ri doo srvvleoh hyhqwv +ru frpelqdwlrq ri rxwfrphv, lv wkh suredelolw| ixqfwlrq/ zklfk fdq eh dq| vhw ixqfwlrq/ zkrvh grpdlq lv dqg wkh lv wkh forvhg xqlw lqwhuydo ^3> 4` > l1h1/ dq| qxpehu ehwzhhq dqg lqfoxglqj dqg 41 Wkh udqjh rqo| uhtxluhphqw iru wklv ixqfwlrq lv wkdw lw pxvw reh| wkh iroorzlqj wkuhh uxohv +d, S ^ ` @ , eh dq| hyhqw lq (/ wkhq ^,`  Ohw , dqg - eh wzr hyhqwv lq @ vxfk wkdw , _ - @  +e, Ohw +f, / wkhq ^, ^ - ` @ ^,` ^-` Uhodwlyh Iuhtxhqf| Ghqlwlrq1 Wkh uhodwlyh iuhtxhqf| dssurdfk lv edvhg rq wkh iroorzlqj ghqlwlrq= Vxssrvh zh frqgxfw d odujh qxpehu ri wuldov ri wklv d jlyhq h{shulphqw1 Wkh suredelolw| wkdw d jlyhq hyhqw/ vd| ,> lv wkh iroorzlqj olplw S zkhuh G, lv wkh qxpehu ri rffxuuhqfhv dqg ^,` @G" GG olp , G lv wkh qxpehu ri wuldov1 Iru h{dpsoh vxssrvh zh frqgxfw wkh deryh h{shulphqw hyhqw +4, , @ i6‚j rffxuuhg 4;; wlphv +G, @ 4;;, G, G @ "3> 333 wlphv +q @ 43333,1 Ixuwkhu vxssrvh wkh Wkhq 4;; 43333 @ 3=34;; Dv G lqfuhdvhg wr lqqlw| +dqg dvvxplqj wkdw wkh fdugv duh idlu,/ wkhq wkh udwlr zrxog dssurdfk wkh suredelolw| ri , ^,` @G" , olp G G # @ @ 3=34 5‚> 6‚> 7‚> 8‚> 9‚> :‚> ;‚> 43‚> M ‚> T‚> N ‚j Qrz zh vd| wkdw Wkh hyhqw - rffxuv zkhqhyhu dq rxwfrph lv ri d wuldo lv frqwdlqhg lq -1 Lq wklv fdvh/ zkhqhyhu wkh rxwfrph ri d wuldo frqwdlqv d R/ wkh rxwfrph ehorqjv wr - dqg zh vd| wkdw - rffxuuhg1 Vr/ vxssrvh zh frqgxfwhg wkh deryh h{shulphqw "!> !!! wlphv/ dqg wkh hyhqw - rffxuuhg $!!! wlphv1 Wkhq GG $!!! ( "!!!! ( ! $ = Dv lqfuhdvhg wr lqqlw| +dqg dvvxplqj wkdw wkh fdugv duh idlu,/ wkhq wkh udwlr zrxog dssurdfk wkh q suredelolw| ri - G " ^-` @G" - ( ( !=#& olp G % Qrwh wkdw wklv ghqlwlrq pdnhv vhqvh1 Vlqfh ^-< ( " / lw iroorzv wkdw wkh fkdqfh ri - rffxuulqj lq dq| % jlyhq wuldo lv 58(1 Vlploduo|/ wkh fkdqfh ri d 7 rffxuulqj iurp d sduwlfxodu wuldo lv +4@85,  433 ru 4= 5€> 6€> 7€> 8€> 9€> :€> ;€> 43€> M €> T€> N €j Ohw*v orrn dw erwk ehorqj wr -1 - dqg Zh vd| wkdw Qrwh wkdw qrqh ri wkh phpehuv lq - dqg duh Clvmrlqw - ehorqj wr dqg qrqh ri wkh phpehuv lq dqg wkdw wkhlu lqwhuvhfwlrq lv wkh hpsw| vhw -_F ( Lq wklv fdvh wkh suredelolw| ri hlwkhu - HK rffxuulqj lv htxdo wr ^- HK ` @ Golp " zkhuh wkh ^ v|pero vwdqgv iru wkh xqlrq suredelolw|= Li wzr suredelolwlhv1 Ilqdoo|/ ohw wkh hyhqw G- G G ri wzr vhwv1 @ ^- ^ F +5, < Qrwh wkdw +5, ghvfulehv wkh dgglwlyh frqfhsw ri hyhqwv duh glvmrlqw/ wkhq wkh suredelolw| ri wkhlu vxp lv wkh vxp ri wkhlu htxdo doo srvvleoh rxwfrph1 Zh fdoo wkh fhuwdlq hyhqw G ( G1 Wkhuhiruh G G ^ ` @ Golp ( Golp @ " G " G ru wkh vdpsoh vsdfh1 Vlqfh hyhu| srvvleoh rxwfrph lv frqwdlqhg lq wklv hyhqw/ H{dpsoh Frqvlghu wkh h{shulphqw ri lsslqj d idlu frlq wzlfh dqg revhuylqj wkh r xwsxw1 srvvleoh rxwfrphvB Olvw doo srvvleoh hyhqwv1 +6, Zkdw duh wkh Dvvxph wkdw doo rxwfrphv duh htxdoo| olnho|/ dvvljq suredelolwlhv wr hdfk hyhqw1 Wkh vhw ri doo srvvleoh rxwfrphv lv Wkhuh duh #% ( "' hyhqwv1 @ iK K> KW > W K> W W j Wkh| duh iKK j > iKW j > iW K j > iW W j iKK> KW j > iKK> WK j > iKK> W W j > iKW> W K j > iKW> W W j > iW K> W W j iKK> KW> WK j > iKK> KW> W W j > iKK> W K> W W j > iKW> W K> W W j iKK> KW> WK> WW j > ! Qrwh wkdw iKK> KW> WK> WW j lv wkh fhuwdlq hyhqw1 Wkhuhiruh S ^iKK> K W > W K> W W j` @ +7, Krzhyhu/ iurp wkh dgglwlyh surshuw|/ lw iroorzv wkdw ^iK K> K W > W K> W W j< @ S ^iKK j` S ^iK W j` S ^iW K j` S ^iW W j` +8, exw vlqfh hdfk rxwfrph lv htxdoo| olnho| zh kdyh wkdw ^iKK j` @ S ^iKW j` @ S ^iWK j` @ S ^iWW j` +9, Wkhuhiruh iurp +8, dqg +9,/ lw iroorzv wkdw ^iKK j< @ S ^iKW j` @ S ^iWK j` @ S ^iWW j` @ Xvlqj wkh dgglwlyh frqfhsw ri suredelolw| zh fdq vhh wkdw +:, ^iK K> K W j` @ S ^iK K> W K j` @ S ^iK K> W W j` @ @ S ^iK W> W K j` @ S ^iK W> W W j` @ ^iK K> K W> W K j` ^` @ S S ^iW K> W W j` @ ^iK K> K W> W W j` @ S ^iK K> W K> W W j` @ S ^iK W > W K> W W j` @ @ +;, + H Qrwh wkdw > @ i7> N ‚> 7‚> 5j @ 85 46 @ ^0 ` @ 85 46 lv dqrwkhu hyhqw wkdw frqwdlqv wkh hohphqwv ri ^/` Wkhq wkh lqwhuvhfwlrq ri / dqg > @ / dqg H G _ H ( i%‚ #j > @GC / _ H ` @ ^G H ` @ ^G @GC H ` @ 59 Zh fdoo ^/ _ H @ ^G H ` @ ^G @GC H ` M mrlqw suredelolw| ri / S ^ < > erwk hyhqwv rffxuulqj > S +44, S kh dqg Lw ghvfulehv wkh suredelolw| ri Frqglwlrqdo Suredelolw| gjh derxw rqh hyhqw surylghv xv dgglwlrqdo lqirupdwlrq derxw wkh rffxuuhqfh ri dqrwkhu hyhqw1 Vxssrvh wkdw zh frqgxfw dq h{shulphqw dqg zh qg rxw wkdw / kdv rffxuuhg1 Wkxv zh nqrz wkdw wkh rxwfrph zdv hlwkhu d ,R/ 5/ ;€/ ru 7‚= Grhv wklv whoo xv dq|wklqj derxw wkh rffxuuhqfh ri B Wkh dqvzhu lv |hv1 Lq d qxpehu ri fdvhv/ nqrzoh Jlyhq wkdw / ‚ 5 ;€ kdv rffxuuhg/ zh nqrz wkdw wkh rxwfrph zdv hlwkhu D / / / ru 7‚1 Vlqfh hdfk rxwfrph lv htxdoo| olnho|/ dqg zh nqrz wkdw wkhvh duh qrz wkh rqo| iru srvvlelolwlhv/ zh kdyh wkdw wkh suredelolw| ri hdfk ri wkhvh rxwfrphv jlyhq wkdw / kdv rffxuuhg lv " lq rwkhu zrugv % ^,‚mG` @ ^5mG` @ ^;€mG` @ ^7‚mG` @ Qrwh wkdw ^,m-< phdqv wkh suredelolw| ri , jlyhq wkh - kdv rffxuuhg +45, Qrz orrn dw wkh hyhqw ( i% ‚ %‚ #j / ,R 5 ;€ 7‚ 7 NR frxog krw kdyh kdsshqhg ehfdxvh %Q @GC N R @KD qrw lq /1 Wkhuhiruh 7mG` @ ^N mG` @ +46, Wkhuhiruh/ jlyhq wkdw zh nqrz / kdv rffxuuhg/ zh fdq zulwh wkh suredelolw| ri dv iroorzv mG ` @ ^7mG` ^N mG` S ^;€mG` S ^7‚mG` @ 3.3 @ +47, 7 >N zhuh rqo| / / / ru > > Jlyhq wkdw kdv rffxuuhg dqg wkdw wkh srvvleoh rxwfrphv / zh fdq frqfoxgh wkdw wkh rxwfrphv ^ ^ dqg Zh fdq dovr frpsxwh wkh suredelolw| ri jlyhq wkdw Ghqlwlrq Ohw , jlyhq - , / ^ m ` , - dqg / kdv rffxuuhg xvlqj wkh iroorzlqj ghqlwlrq1 eh wzr hyhqwv iurp wkh vdph h{shulphqw1 lv ghqhg dv iroorzv ^ `@! S E / wkhq S Vr/ iru wkh fdvh ri ^,m-` lv / ^H > G` S , jlyhq deryh zh kdyh wkdw ^D F ` @ @3 ,mF ` @ ^F ` 58 > S Dqrwkhu lpsruwdqw frqfhsw lq suredelolw| lv Ghqlwlrq Ohw , dqg - eh wzr hyhqwv iurp wkh vdph h{shulphqw1 li wkh mrlqw suredelolw| ri Iurp wklv lw iroorzv wkdw S lpsolhv li wzr hyhqwv duh = lqghshqghqfh1 , IKHCNBM , `@ ^ ` ^ ` dqg lv htxdo wr wkh ^ Wklv 4@59 46 @ @ 4@46 59 @ ^G ` ^ lqghshqghqw +48, E / zh kdyh wkdw ^0 mG` @ Vlploduo| iru wkh fdvh ri ^ ` S xqghqhg1 jlyhq wkdw frqglwlrqdo suredelolw| ri ^D> E` ^,m-` / Li Wkhq wkh >E S D S Wkhq , dqg - duh vdlg wr eh ri wkhlu wzr lqglylgxdo suredelolwlhv1 E ^ - @ ^ ^ ` ` @ ^ `^ `^ ` @ ^ ` Dm < S S D> E S E D S S E E S D +49, +4:, lqghshqghqw/ wkh nqrzohgjh derxw rqh hyhqw surylghv devroxwho| nqrz lqirupdwlrq derxw wkh rwkhu hyhqw 7@GCHF ;@Kldeohv \rx pd| kdyh qrwlfhg wkxv idu wkdw zh kdyh zrunlqj zlwk vhwv dqg hyhqwv wkdw frqwdlq rgg v|perov vxfk dv $R % &€ '   > > > > = = = Rwkhu h{dpsohv/ erwk lq wklv qrwh vhw dqg wkh errn/ kdyh zrunhg zlwk hyhqwv zlwk iru wkh glfh sureohp1 Zh zrxog olnh wr KK j shuirup vrph pdwkhpdwlfdo dqdo|vlv dqg pdqlsxodwlrq rq wkhvh hyhqwv1 Krzhyhu/ vxfk d wdvn zrxog eh gl!fxow dqg zrxog qrw surylgh dq| lqvljkw1 Lw zrxog eh ehwwhu li zh frxog dvvljq +ru pds , hdfk rxwfrph d qxpehu1 Wkh zh frxog zrun zlwk wkhvh rwkhu v|perov vxfk dv i iru wkh frlq sureohp dqg    qxpehuv xvlqj wkh vwdqgdug pdwkhpdwlfdo whfkqltxhv wkdw zh kdyh ohduqhg ryhu wkh |hduv1 H{dpsoh Frqvlghu rxu fdug h{shulphqw1 Vxssrvh zh dvvljq hdfk fdug dq lqwhjhu dv iroorzv ,R $ " # ‚$#    N ‚ $ "$ D $ "%  $ "& #    N  $ #' DP  5: P  5;    NP  6< DQ  73 Q  7"    NQ  &# & Wkhq hdfk fdug lv dvvljqhg dq lqwhjhu rq wkh uhdo olqh1 zkhuh Vxssrvh zh dvvljq wklv pdsslqj wkh ixqfwlrq Vxfk d pdsslqj lv fdoohg d / udqgrp yduldeoh1 Ghqlwlrq : , udqgrp yduldeoh/ [ + , lv d ghwhuplqlvwlf ixqfwlrq wkdw dvvljqv hdfk rxwfrph/ [= $? lq dq h{shulphqw/ d qxpehu1 Vr wkdw [ + , lv dq| rxwfrph ^[ + , @ "` @ S ^[ + , @ "` @ ! # / +4;, Wkh uvw wklqj |rx fdq qrwlfh derxw d udqgrp yduldeoh lv wkdw 41 Wkhuh lv qrwklqj udqgrp derxw lw 51 Lw lv qrw d yduldeoh1 Udwkhu lw lv d vhw ixqfwlrq1 [ + , Krzhyhu/ lq shuiruplqj dqdo|vlv/ lw lv frqyhqlhqw wr wuhdw ri d udqgrp yduldeoh1 dv d yduldeoh1 Frqvlghu d vhfrqg h{dpsoh vlqfh wkh udqgrp yduldeoh lv d ixqfwlrq ri dq rxwfrph/ zklfk kdv dq dvvrfldwhg suredelolw|/ wkh udqgrp yduldeoh dovr kdv dq dvvrfldwhg suredelolw|1  S ^[ +  , @ { ` @ S ( [ " + { ,  +4 > 5‚> 6‚> 7‚> 8‚> 9‚> :‚> ;‚> 43‚> M ‚> T‚> N ‚j` @ S ^V + , @ 5` @ ^V + , @ 5=&< (   S " +5, @   ( " +5=8, @ S ^!` @ S ^5  + ,  6` @ ( @ " ^5> 6` ( ^ @ Edoo  L dqg doo j ! < ( " # Fxpxodwlyh Glvwulexwlrq Ixqfwlrqv dqg Suredelolw| Ghqvlw| Ixqfwlrqv Wzr zd|v wr dqdo|}h h{shulphqwv yld udqgrp yduldeohv lv wkurxjk wkh Fxpxodwlyh Glvwulexwlrq Ixqfwlrq +FGI, dqg suredelolw| ghqvlw| ixqfwlrq +sgi, Ghqlwlrq ; Ohw [ + , eh d udqgrp yduldeoh ghqhg rq d vdpsoh vsdfh Wkhq wkh Fxpxodwlyh Glvwul0 ri [ lv ghqrwhg e| wkh ixqfwlrq I[ +{! , dqg lv ghqhg dv iroorzv exwlrq Ixqfwlrq +FGI, [  {! I[ +{! , @ S ^[  {! < ghqhv dq hyhqw ru d froohfwlrq ri rxwfrphv1 zkhuh wkh vhw ^ H{dpsoh : Frqvlghu wkh udqgrp yduldeoh lq H{dpsoh 61 < +54, I[ +{! , @ S ^[  {! ` @ 85 Wkh FGI ri [ &# n(" [ lv N +{!  n , F X (x) 52 H{dpsoh ; Frqvlghu wkh udqgrp yduldeoh lq H{dpsoh 71 18 +L! , @ N +L!  4, 8H 7 N +L! Wkh FGI ri x lv 4  5, N +L!  6, N +L!  7, ;! A" A ?% + L! , @ " A# A$ =% " : L! ? " "  v! ? # #  v! ? $ $  v! ? % v!  % +55, +56, F S (s o ) 1 Ghqlwlrq < so [ + , eh d udqgrp yduldeoh ghqhg rq d vdpsoh vsdfh Wkhq wkh Suredelolw| Ghqvlw| [ lv ghqrwhg e| wkh v|pero i[ +{, dqg lv ghqhg dv wkh ghulydwlyh ri lwv FGI Ohw ri Ixqfwlrq +sgi , E[ +{, @ g g{ I[ +{, @ g g{ S ^[ +  ,  {` +57, Qrwh wkdw zh fdq uhfryhu wkh FGI iurp wkh SGI wkurxjk lqwhjudwlrq 1[ +{, @ Wkhuhiruh E[ +{, zloo dozd|v lqwhjudwh wr ] { " E[ +, g +58, ] " " E[ +{, g{ @ Ixuwkhupruh/ wkh suredelolw| wkdw [ + , lv ehwzhhq {" dqg +59, {# / {# A {" dqg {# lqfoxvlyh/ fdq eh frpsxwhg dv iroorzv S ^ {" ? [  {# ` @ S ^[  {# `  S ^[  {" ` @ I[ +{# ,  1[ +{" , @ Qrwh wkdw wkh dujxphqw/ / kdv ehhq gursshg iurp wkh udqgrp yduldeoh/ [ ] {# {" E[ +{, g{ +5:, Iru wkh uhpdlqghu ri wklv glvfxvvlrq/ zh zloo dvvxph wkdw wkh ghshqghqfh ri d udqgrp yduldeoh [ rq  lv lpsolflw dqg rplw wkh dujxphqw iru frqyhqlhqfh Prphqwv ri Udqgrp Yduldeohv Prphqwv ri udqgrp yduldeohv lqfoxgh d qxpehu ri txdqwlwlhv vxfk dv dyhudjhv ghyldwlrqv/ hwf1 / yduldqfhv/ vwdqgdug Wkh| duh sduwlfxoduo| xvhixo lq frppxqlfdwlrqv ehfdxvh wkh| surylgh ydoxdeoh lqirupdwlrq derxw d udqgrp yduldeoh zlwk kdylqj wr nqrz lwv vwdwlvwlfv +l1h1/ wkh FGI ru sgi, Ghqlwlrq 43 Ohw [ eh d udqgrp yduldeoh zlwk wkh sgi i[ +{, Wkhq wkh h{shfwhg [ / lv ]" [dyj ( H ^[ ` / {i[ +{, " GhqlwlHG "" Ohw [ eh d udqgrp yduldeoh zlwk sgi i[ +{, Wkhq/ wkh yduldqfh l ] " k # # #  [ ( +[  [dyj , / +{  [dyj , E[ +{, g{ " dqg wkh vwdqgdug ghyldwlrq ri [ lv u [ k l ( +[  [dyj , / # v] " " ; +{  [dyj ,# E[ +{, g{ ri ydoxh [ ru dyhudjh ri +5;, lv +5

Ngày đăng: 12/12/2013, 04:15

Từ khóa liên quan

Tài liệu cùng người dùng

Tài liệu liên quan