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Transcription

1 Kathlieke Uiversiteit Leuve Departemet Elektrtechiek ESA-SISA/R - A imprved ptimizati algrithm fr time dmai equalizer desig i ADSL mdems Katlee Va Acker, Geert Leus, Marc Me, Sta Claes, livier va de Wiel ctber Published i the Prceedigs f the Iteratial Wrkshp Cpper Wire Access Systems `ridgig the last cpper drp', udapest, ugary, ctber - his reprt is available by aymus ftp frm ftpesatkuleuveacbe i the directry pub/sisa/vaacker/reprts/-psgz KULeuve, Dept f Electrical Egieerig (ESA), Research grup SISA, Kardiaal Mercierlaa, Leuve, elgium, el //, Fax //, WWW: katleevaacker@esatkuleuveacbe Marc Me is a Research Assciate with the FW (Fud fr Scietic Research- Fladers) Geert Leus is a Research Assistat supprted by the FW Fladers his research wrk was carried ut at the ESA labratry f the Kathlieke Uiversiteit Leuve, i the frame f the elgia State, Prime Miister's ce - Federal ce fr Scietic, echical ad Cultural Aairs - Iteruiversity Ples f Attracti Prgramme - IUAP P- (-): Mdelig, Ideticati, Simulati ad Ctrl f Cmplex Systems ad the Ccerted Research Acti MIPS (`Mdel-based Ifrmati Prcessig Systems') f the Flemish Gvermet he scietic respsibility is assumed by its authrs xdsl echlgies, radbad Divisi, ALCAEL elecm, Atwerpe, elgium Access t Netwrks, Crprate Research Cetre, ALCAEL elecm, Atwerpe, elgium

2 AN IMPRVED PIMIZAIN ALGRIM FR IME DMAIN EQUALIZER DESIGN IN ADSL MDEMS Katlee Va Acker, Geert Leus, Marc Me, Sta Claes, livier va de Wiel Kathlieke Uiversiteit Leuve - ESA Kardiaal Mercierlaa, everlee - elgium tel //, fax // katleevaacker@esatkuleuveacbe geertleus@esatkuleuveacbe marcme@esatkuleuveacbe xdsl echlgies radbad Divisi A L C A E L elecm Atwerpe, elgium Access t Netwrks Crprate Research Cetre A L C A E L elecm Atwerpe, elgium Abstract I this paper, a imprved ptimizati algrithm fr the time dmai equalizer is develped i case f Frequecy Divisi Multiplexig - based ADSL I ctrast with existig algrithms, ur algrithm explicitly disregards the uused tes i the ptimizati fucti as well as i the -triviality cstrait Sme simulati results are preseted fr the upstream chael It is shw that the achievable bitrate is higher tha with a classical algrithm which impses a uit rm cstrait the full target impulse respse Itrducti Asymmetric Digital Subscriber Lie (ADSL) prvides a high bitrate dwstream chael ad a lwer bitrate upstream chael ver twisted pair cpper wire he trasmissi methd is Discrete Multite (DM) It divides the used badwidth i parallel subchaels r tes he icg bitstream is split betwee all available tes i rder t QAM-mdulate them At the emissi, a cyclic prex extesi is added t each time symbl, cmputed by a iverse Furier rasfrm his techique eables the recepti t be perfrmed by a FF, fllwed by a sigle cmplex tap equalizer per te t cmpesate fr the chael amplitude ad phase eects, prvided that the chael impulse respse is shrter tha the cyclic extesi I rder t avid t lg cyclic extesi, e ca advatageusly make use f a time dmai equalizer (EQ), which rle is t shrte the chael impulse respse Imperfectly shrteed chael impulse respses give rise t iterblck iterferece (II) betwee tw successive symbls, ad itercarrier iterferece (ICI) betwee dieret carriers May algrithms have bee develped t iitialize this EQ Falcer et al [] shwed hw a target shrteed impulse respse ca be cmputed as the eigevectr crrespdig t the smallest eigevalue f a chael depedet matrix his is used i the ctext f maximum likelihd sequece estimati A uit rm cstrait is put this target impulse respse t avid trivial slutis I [], Chw et al describe algrithms suited fr multicarrier mdulati which use adaptati i the frequecy dmai ad widwig i the time dmai Va ladel et al [] csider EQ deterati by meas f a MMSE criteri, agai resultig i a eigevalue prblem A fast cmputati algrithm fr the EQ is discussed i [] by Lee et al where a uit tap cstrait the target impulse respse is used Al-Dhahir et al argue i [] that the MSE itself is t the ptimum equalizati criteri i cjucti with DM A ew criteri is derived, called gemetric SNR Melsa et al [] derive a ptimal shrteig algrithm with respect t the shrteig SNR Als i [], Va Kerckhve et

3 W; E y >< y? : : : y?(?) y + y y >: +N? : : : y +N? {z } Y w w w? {z } W? x x N : : : x N?(?) x x x N : : : x N? {z } X b b >= b {z } >; () al demstrate usatisfactry perfrmace f the MSE algrithm A uit rm cstrait is used fr the EQ t avid the trivial all-zer sluti but the it is shw that a ther uwated EQ ca appear i case Frequecy Divisi Multiplexig is used fr up- ad dwstream: the EQ-eergy mves t the carriers that are t used his is slved by ijectig virtual ise i the stpbad ere, we fcus the Frequecy Divisi Multiplexig case, ad aim at imprvig up the results i [] Ulike the apprach i [], ur algrithm explicitly disregards the uused tes he -triviality cstrait is impsed ly the used tes such that trivial slutis i the passbad are avided his is achieved by itrducig weight matrices ad is shw t imprve the resultig capacity f the system his paper is rgaized as fllws he secd secti explais sme aspects f capacity ptimizati, the classical apprach ad ur prpsed imprvemets he third secti describes the algrithm Secti dees the cstrait ad gives a iterpretati t it Simulati results are preseted i secti Chael shrteig versus capacity ptimizati he ptimal EQ shuld maximize the trasmissi capacity Sice this ptimizati prblem is highly -liear, we fcus urselves t a apprach based the DM sigal prperties x i h delay i yi IR EQ W Figure : Mdel fr chael shrteig A algrithmic apprach prve t be eciet i literature [] - [] is depicted i gure he ei mea square errr f e i (i the time dmai r E i i the frequecy dmai) is imized ver a rage f pssible delays subject t the cstrait that kwk = r kk = t avid trivial slutis, where W ad are the ceciet vectrs f the EQ ad target impulse respse (IR) respectively here is hwever a fudametal prblem with this mea square errr criteri: there is direct relati betwee imum mea square errr ad maximum capacity i fucti f the delay A smth capacity fucti f the delay is desirable t limit the umber f tested delays ere we try t achieve a higher capacity by isertig weightig matrices i the ptimizati fucti i rder t imize the mea square errr ly ver the used tes ad mdify the cstrait t a mre meaigful e Mdel ad algrithm ur startig pit is a frmula which explicitly takes it accut the ature f the iput ad utput samples It ctais a system f N equatis where N is the FF size, ie the legth f a symbl withut prex I ur mdel, is the legth f the cyclic prex he errr f time dmai shrteig f the chael is imized ver the cmplete symbl legth, as shw i equati () ere x i ad y i are respectively iput ad utput data samples f the chael, W = w :? is the time dmai equalizer ad = b : the target impulse respse is the delay itrduced by the chael It is ted that the data matrices have a eplitz structure Nw we mdify this cst fucti by imizig ly ver the used tes herefre we trasfrm ur system t the frequecy dmai, leadig t W; E ks FF (Y W? X )k () ere, FF is the N N FF-matrix, S NN is a selectr matrix, ie a diagal matrix with es the psitis crrespdig t the used tes ad zers therwise y writig the utput samples y i as a fucti f x i ad the additive white chael

4 W; E >< >: S FF X ext N I (?)(+) I (+)(+)?W >= >; () ise i, e btais equati (), with (N?+ ) = N = X ext N(N?+ ) = X ext h h h ()? : : :?(?) + +N? : : : +N? () x : : : x +N? x x xn x N () x is a eplitz matrix, ad a `extesi' f X N is a eplitz matrix with ise samples h N ctais the chael impulse respse slve this system i `square rt' frm, we eed the (square rts f) crrelati matrices f ise ad iput, amely ~R = E ~N ~N = E N IFF S (S FF N ) () he S matrices select the apprpriate rws ad clums i the FF respectively IFF matrix he prduct IFFS SFF gives a real, symmetric circulat weight matrix, see () ~R the becmes: ~R = E = >< >: N a b c : : : c b b a b c b N b c : : : a NN >= >; N a (N?)b : : : (N? +)f (N?) b N a (N? +)f : : : N a () E k () I expressi (), the elemets f the real, symmetric eplitz matrix are frmed by takig the sum f the diagals f the weight matrix f equati () his ca be see as fllws, take e elemet f ~R : ~R ij = E N (i; :) IFFSSFF N (:; j) () Suppse i < j (the same ca be shw fr i j) he the N? (j? i) rst elemets f the ith clum f N are the same as the N? (j? i) last elemets f the jth clum because N is eplitz Whe develpig (), ly terms with a prduct f white ise samples with the same idex have t be csidered hese are the ly terms left after takig the expected value Nw suppse the j?i+ clum f the weight matrix IFFS SFF starts with the elemet g the ~ R = E ij ( N (i; :) N (:; j) ) a b : : : g : : : b b g g g b : : : = (N? (j? i)) g E k a NN () I calculatig ~R ext xx, we make sme assumptis such that we ca frm it similar t R ~ We eglect the prex structure i X ext ad assume the

5 iput samples x i are white his is a apprximati i that it is based the assumpti that all tes are set As already metied, the ptimizati itself is made ver the tes that are eectively used his leads t ~R ext xx E ~X ext E X ext M E x k ~X ext IFF S S FF X ext () M is a eplitz matrix It is frmed i the same way as the matrix i equati () wever its dimesi is larger, amely (N? + ) (N? + ) herefre the? uter -diagals csist f zers: M= N a (N?)b : : : b : : : (N?)b b b b : : : N a () N-triviality cstrait As a -triviality cstrait, we take E his ca be writte as ~X = () E X IFF S FF X = E (N??) FF X circ X circ IFF S = (N??) () with X circ NN the circular extesi f X with prex take i accut X circ = x x N : : : x x x x N : : : x () After iserti f a IFFFF pair betwee the circulat matrix ad the target impulse respse vectr, e btais E( : : : N? his meas X X N? X X N? ) N? S = () P ifused tesg j ij = () uder the assumpti that E jx i j = S, as far as the cstrait is ccered, the prex is take i accut Practically, the cstrait is calculated as fllws : : : N? (N??) IFFS FF S N? = = (N??) R Q Q R = () Nw call ~ = R he resultig ptimizati fucti the becmes equati (), subject t ~ ~ = his cstrait is meaigful because f the fllwig reas: the sluti f the cstraied MSE prblem equals the sluti f the ucstraied MSE prblem W;? W ~Rxx ~R xy ~R yx ~R yy ~R xx? W () where the ~R-matrices are crrelatis f ~Y = S FF Y ad ~ X = S FF X his frmula ca be see as =SNR fr the subbad f the used tes Simulati results I this secti we preset sme simulati results fr the prpsed ptimizati algrithm It is shw that there is a imprvemet fr the upstream case where a -pit FF is used f the tes, te up t te are used I the dwstream case, based a -pit FF, ly tes up t f the carriers are used S relatively speakig mre tes are used i the dwstream case Prbably fr this reas, the weight factrs f ur algrithm have t much iuece the resultig capacity f the dwstream system

6 W; ~ "chl( ~R ext xx ) chl( R ~ ) # I (?)(+) I (+)(+) I R??W ~ () I the simulatis, we assume additive white chael ise i the EQ as well as i the capacity calculatis Fr upstream we have: N =, =, sample frequecy F s = : Mz ad t calculate the capacity: SNR gap? = : d, ise margi m = d, cdig gai c = d he amut f bits b assiged t te i is calculated as fllws: b i = r lg + SNR i???m+c () he `r' peratr is iserted t rud b i t the earest smaller iteger he rate is calculated with the frmula! X F s rate = b i () N + i=used te he trasfer characteristics f the chaels are pltted i gures ad Figures t shw the simulati results he full ad dashed lies i d es Figure : rasfer characteristic f a m upstream chael Rate i bits/s x Delay d Figure : Capacity i fucti f the delay fr a m upstream chael with additive white chael ise ad EQ legth = es Figure : rasfer characteristic f a m upstream chael these gures are respectively the results f the algrithm with weight matrix crrespdig t the used tes ad f the classical algrithm with kk = he target impulse respses are i each case calculated by the sigular value decmpsiti after qr-factrizati f the apprpriate system f equatis Figure shws the capacity i a delay rage [- :] f a upstream chael f m he EQ legth is, the pwer spectral desities are psd x = - dm/z ad psd = - dm/z he zer referece delay is take as the maximum f the abslute value f the chael impulse respse e sees that the rate is less depedet f the delay SNR i d es Figure : SNR fr the best EQ fr the tw methds shw i gure Figure shws the SNR fr the best EQ's f the tw methds I gures ad the same is pltted fr = he dierece betwee the reachable

7 capacities is higher as is als clear i the SNR plt Figures ad are made fr a upstream chael f m with = Agai, the capacity btaied with weightig matrix based apprach is higher x SNR i d Rate i bits/s es Figure : SNR fr the best EQ fr the tw methds shw i gure Delay Figure : Capacity i fucti f the delay fr a m upstream chael with additive white chael ise ad EQ legth = Cclusis I this paper we have develped a algrithm fr the EQ i FDM-ADSL mdems A imprved perfrmace with respect t kk = is shw fr the upstream chael he depedece is lwered which results i less cmplexity ad is als iversely prprtial t the umber f EQ-taps SNR i d es Figure : SNR fr the best EQ fr the tw methds shw i gure x Ackwledgmets Marc Me is a Research Assciate with the FW (Fud fr Scietic Research- Fladers) Geert Leus is a Research Assistat supprted by the FW Fladers his research wrk f Katlee Va Acker, Geert Leus ad Marc Me was carried ut at the ESA labratry f the Kathlieke Uiversiteit Leuve, i the frame f the elgia State, Prime Miister's ce - Federal ce fr Scietic, echical ad Cultural Aairs - Iteruiversity Ples f Attracti Prgramme - IUAP P- (-): Mdelig, Ideticati, Simulati ad Ctrl f Cmplex Systems ad the Ccerted Research Acti MIPS (`Mdel-based Ifrmati Prcessig Systems') f the Flemish Gvermet he scietic respsibility is assumed by its authrs Rate i bits/s Delay Figure : Capacity i fucti f the delay fr a m upstream chael with additive white chael ise ad EQ legth = Refereces [] N Al-Dhahir, JM Ci ptimum Fiite-Legth Equalizati fr Multicarrier rasceivers, IEEE rasactis Cmmuicatis, Vl, N, pp-, Ja [] JAC igham Multicarrier Mdulati fr Data rasmissi: A Idea Whse ime as Cme, IEEE Cmmuicatis Magazie, Vl, N, pp, May [] JS Chw, JM Ci, JAC igham Equalizer raiig Algrithms fr Multicarrier

8 Mdulati Systems, Prceedigs f the Iteratial Cmmuicatis Cferece, ICC ', pp- [] DD Falcer, FR Magee Adaptive Chael Memry rucati fr Maximum Likelihd Sequece Estimati, ell System echical Jural, Vl, N, pp-, Ja [] I Lee, JS Chw, JM Ci Perfrmace Evaluati f a Fast Cmputati Algrithm fr the DM i igh-speed Subscriber Lp, IEEE Jural Selected Areas i Cmmuicatis, Vl, N, pp-, Dec [] PJW Melsa, RC Yuce, CE Rhrs Impulse Respse Shrteig fr Discrete Multite rasceivers, IEEE rasactis Cmmuicatis, Vl, N, pp-, Dec [] M Va ladel, M Meeclaey ime-dmai Equalizati fr Multicarrier Cmmuicati, Prceedigs f the IEEE Glbal elecmmuicatis Cferece, Glbecm ', pp- [] JF Va Kerckhve, P Spruyt Adapted ptimizati Criteri fr FDM-based DM- ADSL Equalizati, Prceedigs f the Iteratial Cmmuicatis Cferece, ICC ', pp-