Adaptive Deconvolution and Cross Equalization

Transcription

1 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges Adpve Decovoluo d Cross Equlzo By: Dr. M. urh ury er.er@rocsoldges.co roduco: Augus 998 Adpve lerg hve bee roduced by Wdro, hch ler led o he develope o eurl eors. g lo o lberes by borrog y coceps roduced by Wdro or he purpose o reroducg he coceps o dpo o decovoluo. recoed redg o he boo by Wdro d Sers or urher udersdg d proos o he coceps dscussed belo. hve lso suded he ehod preseed by Grhs e l 977, hch s soeh slr o Wdro's pproch d dpo ehod used he ProMx Adpve Decovoluo odules. roduco: Mos o he ehods descrbed belo re bsed o he updg he operors he dreco o seepes desce ords he opu soluo. hve oud he "Del Rule" used he eurl eor rg copuos gves he sples explo o he operor updg, bsed o he les e error squres copuo. hs ehod s horoughly covered he boo by Wdro d Sers. Geerlzed Del Rule: Oe o he erles ors o eurl eors goes bc o Wdro's roduco o Percepro. hs cossed o sgle euro, hch could be red o respod dpve er. Wdro 96 shoed h he Percepro could be red o peror ler lerg or dscro. e X {x,x,...x} represe he pu rg per he or o colu vecors. We sh o solve or colu vecor o M elees such h; X b here he elees o b re he oupu vlues speced by he rg se. bry clssco b specy he correspodg X se belog o he clss A d b specy he correspodg se X belogs o he clss B. Sce X rx hs uber o colus d M uber o ros d M <, he e c solve hs se o equos by he clsscl les e error squres ehod. We c ob he orl equo squre rx by preulplyg boh sdes by he rspose o he X rx; X. X. X b Pge

2 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges d solve or ; X. X. X b 3 hs expresso c be spled o ; Xˆ. b 4 here Xˆ s he pseudoverse o he orgl recgulr X. X. X rx, ˆ X X. X. X. 5 hs verse heoreclly c be copued drecly. Hoever e y ge soe osescl vlues o represeg he rel suo. hs s copued by erve procedure clled he ler percepro lgorh hch leds us o he Del rule. We sh o copue sgle se o eghs o yeld correc se o oupus b or ll pu pers xp. We sr h rbrry se o vlues o he upde by he ollog rule; j j ρ b j j x j]. x j 6 hs updg coues ul ll o he pers re clssed correcly, hch e he egve o grde b j j x j] becoes zero or very sll. prcce hs co be reched os o he cses, hece he ero ould be sopped he su o he squres o errors reches belo soe prescrbed hreshold vlue. We c re he expresso 6 he or; W η. δ.x 7 here δ s he derece beee he desred oupu d he copued cul oupu produced by he percepro. hs s he delrule expresso, hch ses h he chge he egh vecor should be proporol o he del he error d o he pu per. We c lso sho expresso 6 ore lr or s; j j h. d. X 8 Wdro shos h he produc o d X s he ovee he seepes desce dreco. Geerl Opzo d MS Mehod o Wdro: hs s he ehod descrbed by Wdro d Sers d Grh e l. he ehod s bsed o he c h he orulo o les e error squre soluo lso represe he egve o he grde he l soluo coordes. o deosre hs, e ll reve he develope o predcve decovoluo operor copuo. We sh o desg se o operors, he covolved h he d se ll predc s uure vlues soe predco dsce. he les e error squres ehod help desg hese operors h zes he su o squres o dereces beee he cul d he predced rce. e he rce be gve by d he predcor operors gve by ; Pge

3 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges Pge 3 } { ε, 9 here predced rce s he resul o covoluo o d h operors;. By subsug equo bc o equo, e ll ge; ]} { ε Fgure. Reloshp beee operors boo sesc rce op d predced poso. oe h covoluo operors re reverse order h respec o sesc rce. coveol copuo, e deere he operors by seg prl dervves o he equo h respec o he uo operors equl o zero. ]} ]..{ / e Equo s he expresso o he grde o he les e error squres surce he l ese o. By seg hese expressos o zero e deere he u coordes o he error surce, hch gves us he desred se o operors, h zes he error les e error squres er. / e 3 hereore, e hve l ese o he operor, s correco ll be ords he u ollog he egve o he grde dreco; ]} ]..{ h 4 Here, e go bc o he orgl orulo o he equos, s sho o equos 9 d, d re he equo 4 ore copc or; ]}.{ h 5 hs s he geerl zo soluo. oe h hs upde s bsed o zo over e do. Wdro suggesed he use o seous vlues sed o doed verges. he error e s;

4 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges ε { ]} 6 Slrly he grde ll be ; e /.{. ]} 7 hereore, correcos he seepes desce dreco ll be he or o;,, h. ] } hs Equo 8 orulo s ered by Wdro s he MS lgorh, hch s pleeed he ProMx odule. Wdro's sudes dce h o cheve covergece, h, he lerg re should be ep < h < /l x, here l x s he lrges egevlue o he uocorrelo rx o he orl equos. s eresg h, oe could rech he se cocluso s equo 8 by heursc pproch. We could sy h e ould le o djus operors grdully ech oupu sequece so he derece beee he desred d cul oupu s zed. We ry o eep he djuse ous sloly vryg h e. Hoever, s dcul heursclly prove h hs pproch ll coverge. ProMx pleeo: hs s he sples pplco o he dpo o predcve decovoluo. s he pleeo o Wdro's MS ehod, hch s ecoocl y o dpg he cul oupu o he desred oupu sple by sple couous er. hs ehod s cluded h he ProMx odules. he ehod s cpble o upde he operors by coprg he predced oupu h he cul desrble oupu d. hs coprso could be de sple by sple, s pleeed he ProMx odule, or by esurg he RMS error vlue over soe ovg prese e do. he help le gves he ollog explo: " A rce sple s predced ro subse o ps rce vlues, d he error s he cul rce vlue us hs predced vlue. he legh o he subse s probbly equl o he legh o he operor copued oe o severl ys vlble he odule. he error s zero, he oupu sple s zero, d he uchged predco ler s le o predc he ex sple. he error s o zero, ech ler coece s cresed by ou equl o he error es he correspodg rce vlue es he re o dpo. he ler s o redy o predc he ex rce vlue." hs procedure, essece, slr o Wdro's pproch, operors re correced h respec o error ou ech sple. By eepg he lerg re s sll uber, he ou o odco s ep uder corol. Boh d or operor desg lgorhs d Burg lgorh re oered s decovoluo operor desg procedures. We c express hs pproch s; 8 j j h b j j x j]. x j 9 Adpve Subrco or Cross Equlzo: Predcve decovoluo eses uure vlues o he sesc rce ro s ps vlues. he cse o cross equlzo e predc vlues o oe rce ro oher rce. hereore he rce ll be replced by he predcor rce g o he covoluo sho equo. We ssue h he rce o be predced s he su o sgl s d soe or o ose h hch could be predced ro he secod rce chel g. s h Pge 4

5 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges e he cul oupu be; h g Ad he error uco be gve s;. ε { s h g }. Prl dervves o he error uco ll gve he grde s; ε /.{ g s h g ]} Assug h correlo beee g d s s eglgble, he he grde orul ll sply o; ε /.{ g h h } 3 hereore he updg procedure ll be;,, η. h h ] g } 4 hs s slr o he updg procedure o predcve decovoluo. Excep, here hve used o sded ler order o ccoode vele shpe dereces beee h d g. Copuol Procedure: Sce he updg lgorh ollos he seepes desce roue, operors ll eveully pproch he les e error squres soluo or sory e seres. hereore, e c sr h y sple ler eghs s e sh. eurl eor rg ll o he eghs re chose s rdo vrbles. predcve decovoluo or he cross equlzo cse, e c sr h resoble se o operors eghs. We c ssue h lly he e seres re sory d sr h he coveol Weer lers copued over sucely log dos. We o h hese lers re opu or he e do, bu hey y o be opu or osory d ses. Sce e re srg h overll sscs o he d se, hese ler eghs ll be resobly close o he locl opu soluo, hus hey c be used s he srg vlues o he ler se. ProMx odule uses severl dere ler desgs s he srg vlues, s poed bove. Cocluso: hve preseed o bsc ehods or dpve decovoluo. Oe s bsed o zo over soe e do d he secod s bsed o seous vlues o he grde uco. Boh o he ehod re vble predcve decovoluo d cross equlzo dpve subrco pplcos. hve oud he boo by Wdro d Sers very orve d gves very sysec explo o ll o he heores volved he dpve procedures. hs provded e good bcgroud o udersd he pper by Grhs e l. hve lso dscussed he "Del" rule ulzed eurl eor copuo. sly, hve dscussed he dpve cross equlzo dpve subrco, hch s developed he slr les s he predcve decovoluo. Pge 5

6 Adpve Decovoluo Dr. M. urh ury er Roc Sold ges Reereces: Grhs,. J., Sol, F. R., d rebly,. D., 977, Adpve decovoluo: e echque or processg evryg sesc d; Geophyscs,4, o. 4, ProMx Help Fle, Adpve Decovoluo; Advce Geophyscl Co. Wdro, B. d Sers S. D.,985, Adpve sgl processg; PreceHll Boo Co. Wdro, B., 96. Geerlzo d oro sorge eors o Adle "euros". M.C. Yovs, G.. Jcob, d G. D. Goldse Eds., Sel Orgzg Syses, pp 43546, Spr Boos, Wshgo, D.C. Pge 6