3 Error Equations for Blind Equalization Schemes

Transcription

1 3 Error Equatios or Blid Equalizatio Schms I this sctio dirt rror quatios or blid qualizatio will b aalzd. Basd o this aalsis a suitabl rror quatio will b suggstd aimd at providig bttr prormac. Th modl o th qualizr usd or th rror aalsis is giv i Figur.1. For covic that sam igur is also show i Figur 3.1. s mtiod i th prvious chaptr, i a blid qualizatio schm, istad o usig a traiig squc, som statistical proprt o th sigal is usd or th dtrmiatio o th istataous rror,, which is th usd or updatig th qualizr coicit vctor. Filtr iput daptiv Filtr Filtr output daptatio Tchiqu Proprt Masurmt Figur 3.1: daptiv Filtr Modl or Blid Equalizatio. Grall, th costat modularit o th output sigal is th dsird proprt, which ds to b rstord. Th adaptiv algorithm that works mail to rstor th costat modulus proprt o th output sigal is kow as th Costat Modulus lgorithm CM, ad th sam algorithm will b usd to aalz th proprtis o dirt rror quatios. Error quatios usd i th proprt masurmt block o Figur 3.1 masur th dviatio o th output sigal amplitud rom th dsird amplitud lvl ad orc th adaptiv algorithm to chag th qualizr coicits to compsat or th prst rror. How much th adaptiv algorithm will compsat or a spciic dviatio o th output sigal amplitud rom th dsird amplitud lvl is compltl dpdt o th rror quatio. Thror th rror quatio usd i a adaptiv sstm rciss sigiicat cotrol ovr th ovrall 5

2 sstm prormac. I th ollowig sctios our dirt tps o rror quatio will b did ad aalzd. 3.1 Error Sigals Out o th ma possibl rror quatios th ollowig ar slctd or aalsis. Th irst thr quatios wr suggstd arlir [8]. Tp1: Tp : Tp3 : Tp4 : l,, i i > < 3.1 I 3.1, is th dsird amplitud lvl. ll th slctd rror quatios ar oliar ad this o-liarit o th rror quatio causs th blid qualizatio schm to b o-liar. ll o th rror quatios bcom zro wh th amplitud o has th propr valu. Th also provid sig iormatio to idicat whthr th absolut valu o is lss tha or gratr tha th dsird valu. This sig iormatio dtrmis th dirctio o adaptatio, i.. whthr th prst valus o th iltr coicits d to b icrasd or dcrasd. Figur 3. shows how th rror varis with th absolut valu o th iltr output or th dirt rror tps. dsird output amplitud o 1 o was usd i plottig th rror curvs. Tp1 is th most commol usd rror quatio with CM basd blid qualizatio. Thror th blid qualizatio schm that uss th Tp1 rror quatio with a ractioall spacd qualizr FSE will b amd th Covtioal Blid Equalizr ad this schm will b usd throughout th rport or compariso purposs. 6

3 Figur 3.: Dirt Tps o Error Fuctios. Th adaptiv algorithm with a Tp1 rror quatio will compsat mor wh th amplitud lvl o th output sigal is highr tha th dsird amplitud lvl. This kid o compsatio will rsult i a bttr covrgc rat at th bgiig o th adaptatio procss sic, i this situatio grall, th rror sigal has a larg positiv valu. Howvr, ar th covrgc poit, or th sam positiv ad gativ dviatios o th amplitud lvl rom its dsird valu, th Tp1 rror dos ot provid th sam compsatio. Du to this o-smmtric bhavior o th rror quatio, with Tp1 rror, th covrgc rat adjact to th miimum ma ot b as good as ar th bgiig o th adaptatio procss. Rcall that, or Tp1 rror, th valu o th rror icrass potiall with th amplitud dviatio. Thror, ois spiks ma caus th qualizr to dviat rom a stabl solutio. Th rror quatio o Tp has a ai rlatio with th amplitud o th qualizr output ad is smmtric about zro rror. Du to th smmtric atur, ulik or a Tp1 rror, a Tp rror rsults i th sam amout o compsatio or th sam amout o dviatio o th output amplitud rom th dsird valu, irrspctiv o th sig o th dviatio. Th sig ol dtrmis th dirctio o th compsatio. For this raso with this rror quatio tp, th stad stat prormac is pctd to b bttr tha with Tp1. Th covrgc rat with a Tp rror quatio ma ot b as good as or Tp1 at 7

4 th bgiig o th adaptatio procss i th vlop o th qualizr output has a much highr valu tha th dsird amplitud lvl. Th adaptatio procss with a Tp3 rror quatio is pctd to bhav th sam as or Tp adjact to miima o th cost surac, du to th smmtric atur o th cost surac ar th global miimum. Howvr, a Tp3 rror ma caus a wors covrgc rat at th bgiig o th adaptatio procss. gai, this tp o rror is vr ssitiv to ois, lik a Tp1 rror, but i th opposit dirctio, i.., wh th qualizr output amplitud bcoms too small. Morovr, this is th ol tp o rror Sctio 3..3, or which th output sigal appars i th domiator o th wight updat quatio. Thror, i th output bcoms zro or som raso prhaps du to ois th qualizr ma bgi to divrg. This problm ca b avoidd b ot allowig th output to go blow a crtai prdid lvl. Th Tp4 rror quatio is dsigd to avoid th ssitivit to ois ad also to provid a bttr covrgc rat at th bgiig o th adaptatio procss. Th Tp4 rror is also smmtric about zro adjact to th miimum poit o th cost surac. Howvr, ar th covrgc poit, th Tp4 rror rsults i a highr valu tha a o th othr rror tps. For this raso, th stad stat ois loor or th Tp4 rror is pctd to b highr tha that or a othr rror tp. Sic th mthod o calculatig th rror sigal dtrmis th quatios or updatig th adaptiv iltr coicits or tap wights, th wight updat quatios or dirt tps o rror will b drivd i th t sctio. gai, or a o th abov tps o rror, th sstm bcoms o-liar ad aalticall itractabl. Thror simulatio rsults will b usd to compar th adaptatio rat or th dirt tps o rror quatio. Th updat quatios, drivd i th t sctio, will b usd i th simulatio. 8

5 3. Wight Updat Equatios Updatig th coicits o th iltr tap wights or liar trasvrsal iltr is aimd at optimizatio o th cost uctio. Th cost uctio is grall a uctio o th rror sigal ad dotd as J[]. Th cost uctio J[] should hav th ollowig charactristics [3]: a b 0 J i 0 [ ] J[ ] 1, J[ 1 ] J[ ] 3. Th most commol usd cost uctio is th pctd valu o th squar o th rror sigal. Thus th objctiv o th adaptiv algorithm is to miimiz th ma squar rror. For simplicit o calculatio, th sam cost uctio will b usd to driv th wight updat quatios. [ ] E{ } J 3.3 E{ } i 3.3 is th statistical pctatio oprator. Thus J[] is th ma squar o th rror ssd i masurig adhrc to th costat vlop proprt. Th costat modulus algorithm CM mplos th approimat gradit dsct mthod to miimiz th cost uctio giv i 3.3, ad th updat quatio or th iltr coicits bcoms, + 1 µ J 3.4 I 3.4, [0 1. N-1] T is th iltr tap wight vctor at istat, ad µ is th adaptatio stp siz, which is a small positiv umbr. Th tru gradit i 3.4, at istat, is approimatd b its istataous valu ad th otatio is chagd to ˆ. ˆ J { } 3.5 9

6 30 Now th wight updat quatios 3.4 will b drivd or th dirt tps o rror giv i Tp1 3.6 Hr is th cojugat o th output o th iltr. To valuat 3.5 or th Tp1 rror, ds to b prssd i trms o. compl sstm will b assumd i th drivatio., 1 0 l l N l H 3.7 I 3.7, H is th compl cojugat traspos o th iltr coicit vctor tap wight vctor ad [ ] T N L is th iput vctor to th qualizr adaptiv iltr at istat. N is th iltr lgth. Now, { } { } { } { } ˆ J H H H 3.8 Hr, H. Thror, 4 4 ˆ J 3.9

7 31 Substitutig J i 3.4 b J ˆ, th wight updat quatio or Tp1 rror ca b writt as: 1 µ I 3.10 all th costats ar lumpd togthr ito th sigl costat µ, th stp siz o th adaptatio procss. Th sam will b do or th othr tps o rror quatio. 3.. Tp 3.11 Th gradit o th cost uctio, { } ˆ J 3.1 Th wight updat quatio ca b obtaid rom 3.4 b rplacig th valu o J ˆ. Thror, 1 µ Tp3 l 1 l 3.14

8 3 Th gradit o th cost uctio, { } { } 1 l l ˆ J 3.15 Th wight updat quatio rom 3.4, 1 µ Tp4 < > i, i, 3.17 Cas 1: > i, { } 1 ˆ J 3.18

9 33 Cas : < i, { } 1 ˆ J 3.19 I gral, th gradit ca b rprstd as: ˆ sig J. 3.0 Thror th wight updat quatio or th Tp4 rror is: 1 sig + µ 3.1 To mak it similar to th Normalizd Last Ma Squar lgorithm NLMS, th wight updat quatios or th dirt tps o rror sigals wr rormulatd ad th w ormalizd updat quatios ar show i Tabl 3.1. Ths quatios wr usd i th simulatios to updat th coicits o th Fractioall Spacd Equalizr FSE. Tabl 3.1: Wight Updat Equatios or Dirt Error Tps. Tp1: 1 µ + 3. Tp: 1 µ Tp3: 1 µ Tp4: 1 sig + µ 3.5

10 3.3 Simulatio Rsults Simulatios wr prormd to dtrmi th covrgc rat o th sstm or th dirt tps o rror sigals. Fractioall Spacd Equalizr FSE with a ovrsamplig actor o 3 M 3 was usd i th simulatio. For simplicit, BPSK modulatio was usd i th simulatio. Th block diagram o th simulatd sstm is show i Figur 3.3. Th ractioall spacd qualizr FSE o Figur 3.3 is a adaptiv qualizr ad its uctioal diagram is show i Figur 3.1. Th ol dirc btw th implmtd qualizr ad th o i Figur 3.1 is that, i th simulatio, th qualizr coicits wr updatd or vr Mth M is th ovr-samplig actor o th qualizr output sampl. This is du to th act that th FSE rquirs M sampls to mak a dcisio or a sigl output. For this raso, i Figur 3.3, th FSE ad th dow samplr ar show i th sam block. Whit Gaussia Nois Discrt Mmorlss Data Sourc DMS Chal T s /M Fractioall Spacd Equalizr Dow Samplr D M Output T s Smbol Priod M Ovr-samplig Factor Figur 3.3: Block Diagram o Simulatd Sstm Usd or alsis o Dirt Error Sigals. Sic th mai objctiv hr is to aalz th covrgc proprtis ad th stad stat prormac o th blid qualizatio schms or dirt rror tps, all simulatios wr prormd with tral ois. Th sigal-to-ois ratio SNR usd i th simulatio was 40 db or all primts. To obsrv how ach o th rror quatios acts th covrgc rat o th qualizr udr dirt chal coditios, dirt tps o chals wr usd i th simulatio. ll th implmtd chals ar o th sam lgth 3 i baud spac but hav dirt rquc rsposs. Rcall that i Sctio.1.1 it has b 34

11 show that th whol sstm with FSE ca b dcomposd ito multipl chals, whr th umbr o sub-chals is th sam as th ovr-samplig actor M. It is also show i Sctio.1. that to hav a prct qualizr, th sub-chals baud spacd o all o th paths should ot hav commo zros. For this purpos, a tra zro was isrtd at tim id k6 i ractioal spac or all th chals whil covrtig th baud spac chal to th ractioal spac chal. This tra zro maks th chal lgth qual to 10 i ractioal spac. I Sctio.1.3 it has b show that, i th lgth o th chal ractioal spac ad th qualizr ar ML ad MN, th or prct sourc rcovr, N should satis th ollowig coditio: L 1 N 3.6 M 1 I 3.6, M is th ovr-samplig actor. Thror, or a chal o lgth 10 i ractioal spac, ad a ovr-samplig actor o 3, th lgth o th FSE should b at last Th lgth o th qualizr FSE usd i th simulatio was 1. Th rsults o th simulatio will b prstd sparatl or dirt chals Miimum Phas Chal with Two Impulss Th chal impuls rsposs, both i baud spac ad i ractioal spac, ar show i Figur 3.4. For simplicit, rctagular puls shapig was usd i th simulatio. Th sam puls shapig will b usd or all th chals ulss spciicall mtiod othrwis. Sic th dla associatd with th total sstm ds to b spciid whil iitializig th blid qualizr, th group dla o th sstm is st to 1 or this spciic chal. Th Ma Squard Error MSE curvs or th sstm with dirt rror quatios, whil adaptig th qualizr or th miimum phas chal with two impulss, ar show i Figur 3.5. Figur 3.6, which ocuss o th iitial adaptatio procss, is a zoomd vrsio o Figur

12 Figur 3.4: Impuls Rspos o th Miimum Phas Chal with Two Impulss. Figur 3.5: MSE Curvs or Dirt Error Equatios or th Chal Show i Figur 3.4. Figur 3.5 shows that, v though th sstm has a SNR o 40 db, th MSE curvs go blow 40 db or th Tp ad Tp3 rror quatios. This is bcaus, i ths cass, th qualizr ot ol rmovs th ISI rom th rcivd smbols but also provids a spctral shapig o th whit tral ois. Du to this spctral shapig ct, th rg o th ois at th output o th qualizr bcoms lss tha 40 db compard to th sigal rg 0 db. Howvr, this kid o spctral shapig o th ois sigal causs biasd stimatio o th ivrs o th chal. Biasdss o th stimatio ca b ovrcom b imposig som 36

13 rstrictios o th adaptatio quatios. Sic th objctiv hr is to compar dirt tps o rror quatios, w will ot coctrat o this aspct ow, ad lav it or utur stud. Figur 3.6: Zoomd Vrsio o Figur 3.5. From Figur 3.6 it is clar that, at th bgiig o th adaptatio procss, th covrgc rat is much astr or th qualizr with a Tp1 rror quatio. But atr ~50 sampls, wh th rror is aroud 10 db, th covrgc rat with Tp1 bcoms slowr tha or th qualizr with a Tp rror. Th raso or this is th o-smmtric bhavior o a Tp1 quatio aroud zro rror Figur 3.. Figur 3. shows that both Tp ad Tp3 hibit almost th sam kid o smmtr aroud zro rror. Ths two rror quatios provid almost th sam valu o rror wh 0.8 < < 1.. Cosqutl, th stad stat MSE lvls or ths two Tps ar almost th sam Figur 3.5. Equalizatio with a Tp3 rror ilds a slowr covrgc rat at th bgiig, as pctd rom Sctio 3.1. Th qualizr with a Tp 4 rror, v though it has a modratl good covrgc rat at th bgiig, producs a much highr stad stat ois lvl. This rsult is also as pctd rom Sctio Epotiall Dcaig Chal Th chal impuls rsposs, both i baud spac ad i ractioal spac, ar show i Figur 3.7. Th dca tim costat o th chal is o. Th chal was ormalizd or uit rg. Othr tha th chal, th othr simulatio paramtrs ar th sam as thos 37

14 usd i Sctio dla o o smbol was usd whil iitializig th qualizr coicits. Th MSE curvs or th potiall dcaig chal or dirt tps o rror quatio ar show i Figur 3.8. Figur 3.7: Impuls Rspos o th Epotiall Dcaig Chal. Figur 3.8: MSE Curvs o th Equalizr with Dirt Error Equatios or th Epotiall Dcaig Chal. Figur 3.8 shows almost th sam cts o th dirt rrors o th covrgc rat o th qualizr as show i Figur

15 3.3.3 Smmtric Chal Th impuls rspos o th chal is dscribd b a cosi uctio [7] as: 1 π 1 + cos h W 0 1, 0,1, ls 3.7 Th paramtr W cotrols th amout o amplitud distortio producd b th chal, with th distortio icrasig with W. Evtuall W cotrols th igvalu sprad o th corrlatio matri o th tap iputs to th qualizr, with th igvalu sprad icrasig with W. W 3.1 is usd i th simulatio. I th simulatio, h was ormalizd to hav uit rg so that, h Figur 3.9: Impuls Rspos o th Smmtric Chal. Th impuls rspos o th chal, i baud spac ad i ractioal spac, is giv i Figur 3.9. Du to th smmtr proprt, th chal itroducs a group dla o o smbol. Thror, ulik with th prvious two chals, istad o usig a o smbol sstm dla, a two-smbol sstm dla was cosidrd durig th iitializatio o th qualizr coicits. Th othr paramtrs i th simulatio wr th sam as thos i th 39

16 prvious two cass. For th smmtric chal th MSE curvs o th qualizr adaptatio procss or dirt tps o rror ar show i Figur Figur 3.10: MSE Curvs o th Equalizr with Dirt Error Equatios or th Smmtric Chal. From Figur 3.5, Figur 3.8, ad Figur 3.10 w coclud that th Tp rror quatio provids th optimal solutio or th blid qualizr. This is bcaus th sstm with a Tp rror quatio ilds a astr covrgc rat tha th sstm with a Tp3 or Tp4 rror quatio. gai Tp rsults i a lowr stad stat rror or a spciic tral ois. Th problm with a Tp rror ad also with a Tp3 rror is that, or som chals, th stimat o th qualizr ivrs o th chal ma b biasd. To mak th solutio ubiasd som kid o costraits d to b usd whil miimizig th cost uctio. 3.4 Ect o daptatio Stp Siz I Sctio 3.3 it has b show that th stad stat rror o CM with Tp1 rror is highr tha th stad stat rror with Tp or Tp3. Howvr th stad stat rror is a uctio o th adaptatio stp siz µ ad th sam µ will ot act similarl or dirt rror tps bcaus o th uqual lumpd costats o quatios 3.10, 3.13, 3.16 ad 3.1. Sic all th rsults or dirt rror quatios show i Sctio 3.3 wr gratd or th 40

17 sam adaptatio costat µ, cousio ma aris as to whthr th dirt stad stat rrors ar th rsult o usig th sam µ or dirt rror tps or th rsult o th rror tps. To rmov this cousio aothr simulatio was prormd with Tp1 ad Tp rror. I this simulatio, dirt valus o th adaptatio stp siz wr chos or dirt rror tps so as to gt th sam covrgc rat i ach cas. Th stp siz usd with Tp1 rror was 0.05 ad that usd with Tp rror was 0.1. Th smmtric chal o Figur 3.9 was usd or this simulatio. ll othr paramtrs o th simulatio wr th sam as thos usd to grat Figur Th simulatio rsults ar show i Figur Figur 3.11: MSE Curv or Tp1 ad Tp Error with Sam Covrgc Rat. Figur 3.11 shows that v though both rror tps provid almost th sam iitial covrgc rat, th Tp rror rsults i bttr stad stat prormac. s dscribd arlir, th smmtric atur o th Tp rror about zro rror causs this rror quatio to prorm bttr tha Tp1, which is ot smmtric about zro rror. Figur 3.1 shows th MSE curvs o th adaptatio procss with Tp rror or dirt stp sizs µ. From Figur 3.1 it is clar that whil th stp siz dtrmis th covrgc rat o th adaptatio procss, it dos ot hav a sigiicat ct o th stad stat bhavior. Thror, it ca b cocludd that th stad stat rror o CM mail dpds o th tp o rror quatio usd i th adaptatio procss ad o th tral ois lvl o th sstm. 41

18 Figur 3.1: MSE Curv with Tp Error or Dirt Stp Siz. 3.5 Ect o Smmtr bout Zro Error Wh a rror quatio is smmtric about th zro rror or smmtric i th rgio o th global miima, th adaptiv algorithm with this rror quatio corrcts th qualizr coicits, up or dow quall, dpdig o th amplitud o th dviatio. I this cas th sig o th dviatio dtrmis ol th dirctio, ot th magitud, o th corrctio. I th rror is ot smmtric about th zro rror, it will rsult i dirt amouts o corrctio or th sam amout o amplitud dviatio dpdig o th dirctio sig o th dviatio. For this raso th smmtric rror tps ar pctd to hibit bttr stadstat prormac. simulatio was prormd to show th ct o th smmtric proprt o th rror quatio o th stad stat prormac o th adaptiv sstm. Tp1, Tp ad Tp3 rror quatios with corrctio actors ar usd i th simulatio. Th corrctd quatios ar giv blow. 1 Tp1c : Tpc : Tp3c : l 3.9 Th corrctios wr mad to mak all th rror quatios hav th sam slop at th zro rror poit. Sic th corrctios will chag th lumpd costats o th wight updat 4

19 quatios, quatios 3., 3.3 ad 3.4 wr usd with th corrctd rror quatios without a chag. Th smmtric chal Figur 3.9 was usd i th simulatio. Th simulatio rsult or th adaptatio stp siz o 0.1 µ 0.1 is show i Figur ll othr paramtrs ar th sam as thos usd or gratig Figur Th sigal to ois ratio usd i th simulatio is 40 db. 500 dirt ralizatios wr prormd with idpdt iput squcs ad th rsults o thos primts wr avragd to grat th curvs o Figur Th sam simulatio was also prormd with th dirt adaptatio stp siz µ 0. ad th rsults o this primt ar show i Figur Fit dirt ralizatios wr usd to plot th curvs o Figur Figur 3.13: MSE Curvs or Dirt Error with Sam Slop at Zro Error m 0.1. Figur 3.13 shows that th stad stat MSE lvls o th adaptatio procss with dirt rror quatios ar almost th sam whil Figur 3.14 shows a dirt stad stat rror lvl or dirt rrors. Figur 3.14 also shows that Tp1, v atr corrctio or th slop, prorms worst i trms o stad stat MSE, amog th thr rror tps usd i th simulatio. Th raso or this poor prormac is th o-smmtric bhavior o th Tp1 rror. For a small adaptatio stp siz, th o-smmtric bhavior o th rror quatios bcoms isigiicat ad thus Figur 3.13 dos ot show a dirc i stad stat bhavior o th adaptatio procss with dirt rror tps. 43

20 Figur 3.14: MSE Curvs or Dirt Error with Sam Slop at Zro Error m Ect o Iitializatio I th prvious chaptr Sctio.3, it was mtiod that i a blid adaptatio schm, th iitializatio o th qualizr coicits dtrmis whthr th qualizr will covrg to a global miimum or to a local miimum. I this sctio w will show that th covrgc to a global or a local miimum ot ol dpds o th iitializatio but also o th tp o rror quatio that is usd i th adaptatio procss. To show th ct o th iitializatio w will simulat th sam sstm as usd i Sctio.3 but with dirt tps o rror quatio. Th simulatd sstm is show i Figur.4. Th sstm has a sigl pol chal with a trasr uctio giv b, 1 C z z baud spacd FIR qualizr o lgth will b usd to ulli th ct o th chal o th rcivd sigal. It is clar rom 3.30 that, to covrg to th corrct miimum poit, th qualizr coicit vctor should hav th valu o [1 0.7]. I a dirtial codig schm is usd, [-1-0.7] would also b a accptabl solutio. Thror, i th qualizr spac -spac, th positios o th accptabl or global miima ar ±[1 0.7]. I th adaptatio procss causs th qualizr to covrg to som othr poits 44

21 i -spac, thos poits will b cosidrd to b local miima sic thos valus o qualizr coicits do ot provid a corrct or accptabl solutio to th problm. Similar to Sctio.3, th ct o th iitializatio o th covrgc bhavior will b obsrvd b iitializig th qualizr coicits with 0 dirt poits i -spac. Ths 0 poits ar slctd such that th all ar quall spacd ad lig o a circl o radius. Th sam iput squc o lgth 4000 will b usd or all th dirt iitializatio situatios. adaptatio stp siz o µ was usd i th simulatios. Th simulatio rsults or dirt tps o rror ar show blow. Figur 3.15: Ect o Iitializatio o Blid Equalizatio with Tp1 Error. Figur 3.16: Ect o Iitializatio o Blid Equalizatio with Tp Error. 45

22 Figur 3.17: Ect o Iitializatio o Blid Equalizatio with Tp3 Error. Figur 3.18: Ect o Iitializatio o Blid Equalizatio with Tp4 Error. Figur 3.15, Figur 3.16, Figur 3.17, ad Figur 3.18 show th ct o iitializatio o th covrgc bhavior o a blid qualizr or rror Tp 1,, 3, ad 4 rspctivl. Ths igurs show that or all th rror tps, cpt or th Tp3 cas, covrgc to global miima dpds o th iitializatio. Ths igurs also sm to show that th probabilit o covrgig to local miima is highr or a sstm with a Tp4 rror quatio. I Figur 3.17 th circls o idicat th iitial poits ad th astrisks idicat th ial valu o th qualizr coicits atr 4000 itratios. From Figur 3.17 it is clar that Tp3 rror hibits a compltl dirt proprt tha th othr rror tps. Figur 46

23 3.17 shows that with a Tp3 rror ad a dirtiall codd sourc, th qualizr will alwas covrg to th global miima, irrspctiv o th iitializatio o th qualizr. Howvr, th Tp3 simulatio rsults also show problms with covrgc o th qualizr cass a, b, c, ad d i Figur Th divrgig bhavior o th qualizr with Tp3 rror ca b plaid rom th rror quatio ad th wight updat quatio, which ar giv blow. l + 1 µ 3.31 From 3.31 it is clar that i or a raso th output bcoms gligibl, this will rdr th rror vr larg - or 0. I th wight updat quatio appars i th domiator. Thror, i bcoms too small, th qualizr coicit vctor will b updatd b a larg corrctio. s a cosquc o ths two acts, th qualizr will start to divrg. To avoid divrgc o th qualizr, th Tp3 rror quatio will b modiid as ollows: 1. Us a hard-limitr at th output o th qualizr. Th hard-limitr will ot allow th output amplitud o th qualizr to go blow a crtai lvl. I th simulatio, sic th dsird amplitud lvl o th output sigal was 1, this lvl was st to To avoid a vr high valu o rror or a small valu o output amplitud, ad also to mak th rror quatio smmtric about zro rror, th rror quatio will b modiid to: l, or 3.3 l, or < 47

24 Th modiid Tp3 rror quatio ad th origial Tp3 rror quatio ar show i Figur Th ct o iitializatio with th modiid Tp3 rror quatio is show i Figur 3.0. Th sam simulatio paramtrs wr usd to grat th plots o Figur 3.0 as wr usd or gratig Figur Th sam wight updat quatio 3.31 was usd or th modiid Tp3 rror quatio. Figur 3.19: Modiid Tp 3 Error Fuctio. Figur 3.0: Ect o Iitializatio o Blid Equalizatio with Modiid Tp 3 Error. Similar to Figur 3.17, i Figur 3.0 th o s idicat th iitial ad th s idicat th ial valus o th qualizr coicits. Th arrows i Figur 3.0 show th positios o th global miima. It is clar rom Figur 3.0 that, atr th modiicatios, a Tp3 rror quatio will alwas lad th qualizr to covrg to a global miimum ad thrb 48

25 limiat th dpdc o th covrgc bhavior o th iitializatio. Sic, irrspctiv o th iitializatio, th qualizr alwas covrgs to th global miima, it ca b pctd that th modiid Tp3 rror quatio provids a cost surac with o local miimum. Figur 3.1: Cost Surac o CM with Modiid Tp3 Error. Figur 3.: Cotour Plot o th Cost Surac o Figur 3.1. Th cost surac ad th cotour plot o th cost surac o CM with Modiid Tp3 rror ar show i Figur 3.1 ad Figur 3. rspctivl. Both rprstatios o th cost surac Figur 3.1 ad Figur 3. coirm th absc o local miima i th cost surac o CM with Modiid Tp3 rror. 49

26 Figur 3.3: MSE Curvs o th FSE with Tp 3 Error ad Modiid Tp3 Error or th Miimum Phas Chal Figur 3.4; m0.1. To s how th modiicatio o th Tp 3 rror acts th covrgc rat o th qualizr, a FSE o lgth 1 was adaptd usig th modiid rror quatio or th miimum phas chal show i Figur 3.4. Th valu o th adaptatio stp siz usd i th simulatio was 0.1. Th sam primt was prormd i Sctio with th origial Tp3 rror. Both MSE curvs ar show i Figur 3.3. Figur 3.3 shows that th modiicatios mad to th Tp3 rror quatio do ot sm to hav a ct o th covrgc rat or th stad stat MSE o th qualizr. Basd o th simulatio rsults w coclud that th qualizr with a Tp rror quatio hibits bttr covrgc rat tha with a othr tp o rror. Tp3 rror ilds bttr rsults tha th othr rror tps i th ss o alwas covrgig to th global miima. Ev though w hav bttr prormac with th Tp or modiid Tp3 rror quatio, a Tp1 rror quatio will b usd i th t chaptr to dsig a w blid qualizr. This is do ol to acilitat as o compariso, bcaus th Tp1 rror is commol usd i covtioal blid qualizrs. 50