Coded Adaptive Linear Precoded Discrete Multitone Over PLC Channel

Transcription

1 Coded Adaptve Lnear Preoded Dsrete Multtone Over PLC Channel Fahad Syed Muhammad, Jean-Yves Baudas, Jean-Franços Hélard, Mattheu Crussère To te ths verson: Fahad Syed Muhammad, Jean-Yves Baudas, Jean-Franços Hélard, Mattheu Crussère. Coded Adaptve Lnear Preoded Dsrete Multtone Over PLC Channel. Internatonal Symposum on Power- Lne Communatons and Its Applatons, Apr 2008, Jeju Island, South Korea. pp , 2008, < /ISPLC >. <hal > HAL Id: hal Submtted on 30 Sep 2008 HAL s a mult-dsplnary open aess arhve for the depost and dssemnaton of sentf researh douments, whether they are publshed or not. The douments may ome from teahng and researh nsttutons n Frane or abroad, or from publ or prvate researh enters. L arhve ouverte plurdsplnare HAL, est destnée au dépôt et à la dffuson de douments sentfques de nveau reherhe, publés ou non, émanant des établssements d ensegnement et de reherhe franças ou étrangers, des laboratores publs ou prvés.

2 Coded Adaptve Lnear Preoded Dsrete Multtone Over PLC Channel Fahad Syed Muhammmad, Jean-Yves Baudas, Jean-Franços Hélard, and Mattheu Crussère Insttute of Eletrons and Teleommunatons of Rennes Rennes Cedex, Frane Emal: Abstrat Dsrete multtone modulaton (DMT) systems explot the apabltes of orthogonal subarrers to ope effently wth narrowband nterferene, hgh frequeny attenuatons and multpath fadngs wth the help of smple equalzaton flters. Adaptve lnear preoded dsrete multtone (LP-DMT) system s based on lassal DMT, ombned wth a lnear preodng omponent. In ths paper, we nvestgate the bt and energy alloaton algorthm of an adaptve LP-DMT system takng nto aount the hannel odng sheme. A oded adaptve LP- DMT system s presented n the power lne ommunaton (PLC) ontext wth a loadng algorthm whh aommodates the hannel odng gans n bt and energy alulatons. The performane of a onatenated hannel odng sheme, onsstng of an nner We s 4-dmensonal 16-states trells ode and an outer Reed-Solomon ode, n ombnaton wth the proposed algorthm s analyzed. Theoretal odng gans are derved and smulaton results are presented for a fxed target bt error rate n a multarrer senaro under power spetral densty onstrant. Usng a multpath model of PLC hannel, t s shown that the proposed oded adaptve LP-DMT system performs better than oded DMT and an aheve hgher throughput for PLC applatons. Index Terms Channel odng, lnear preoded dsrete multtone (LP-DMT), multaess ommunatons, power lne ommunatons (PLC), resoure management. I. INTRODUCTION The thrvng growth of ndoor and outdoor networks s drvng an ever nreasng demand for hgh speed data transmsson. Among many ompetng tehnologes, power lne ommunaton (PLC) has ts unque plae due to already avalable power supply grds n both ndoor and outdoor envronments. Harsh hannel haratersts wth deep fades aused by multpaths, frequeny-dependent able losses, and hostle nose ondtons are some of the bggest hurdles n the way of PLC system desgners. To ope wth them, one requres robust and effent modulaton and hannel odng tehnques. Combnatons of multarrer (MC) and lnear preodng (LP) have proved ther sgnfane n the dgtal subsrber lne (DSL) ontext [1]. The dea of usng lnear preodng to mprove performane over fadng hannels s related to that of [2], [3] and [4], where a real orthogonal preoder s appled to maxmze the hannel utoff rate [4], or maxmze the mnmum produt dstane [2], [3]. Unoded lnear preoded dsrete multtone (LP-DMT) has already been suggested for PLC networks n [5] and [6] wth a loadng algorthm that handles the subarrer, ode, bt, and energy resoure dstrbuton among the atve users but wthout takng nto aount the hannel odng sheme. Assumng perfet hannel state nformaton (CSI) at the transmttng sde, energy and bts are effently dstrbuted among the preodng sequenes by the loadng algorthm to aheve ether hgh throughput or hgh robustness. In ths paper, we examne the performane of an LP-DMT system explotng a resoure alloaton algorthm whh takes nto aount the hannel odng sheme. The loadng algorthm, proposed n [5] and [6], s modfed to aommodate the odng gans assoated to the hannel odng sheme. The proposed bt and energy alloaton algorthm an be used n ombnaton wth any hannel odng sheme, no matter t has onstant or varable odng gans for dfferent modulaton orders, provded the obtaned odng gans are known for all the modulaton orders. Gven an adaptve LP-DMT system, the sutable odng sheme should have large odng gans, reasonable mplementaton omplexty and some measure of burst mmunty. Seleted on these bases, the proposed onatenated hannel odng sheme onssts of an nner We s 4-dmensonal (4D) 16-states trells ode [7] and an outer Reed-Solomon (RS) ode. Ths ombnaton has already proved ts sgnfane n xdsl systems and has been nluded n many standards [8]. The effent performane of We s 4D 16-states trells ode over gaussan hannel has also been demonstrated n [9] and [10]. Here, we analyze the performane of the proposed onatenated hannel odng sheme for an LP-DMT system usng a multpath referene model of power lne hannel [11]. The rest of the paper s organzed as follows. In Seton II, an unoded LP-DMT system s presented along wth a bref desrpton of the hosen odng sheme and the struture of a oded LP-DMT system. The modfed bt loadng algorthm s desrbed takng nto aount the odng gans obtaned from the hannel odng sheme. Seton III presents the theoretal dervatons related to the odng system and gves the expresson of the obtaned odng gan along wth the loss nurred due to redundanes. In Seton IV, smulaton senaros are dsussed and results are presented for the lassal dsrete multtone (DMT) system and the proposed adaptve LP-DMT system usng a multpath PLC hannel model [11] for both oded and unoded mplementatons. Energy dstrbutons for oded DMT and oded LP-DMT are also ompared. LP-DMT and DMT smulatons, for both oded and unoded senaros,

3 B. Appled Channel Codng Shemes The suggested odng sheme for adaptve LP-DMT system onssts of an nner We s 4D 16-states trells ode and an outer RS ode. Fg. 1. Unoded LP-DMT transmtter struture are run for varous hannel lengths, whle usng length profles of the power lne hannel model suggested n [11]. It s shown that the proposed oded LP-DMT system wth the modfed bt loadng algorthm performs better than oded DMT and aheves hgher throughput for PLC applatons. Fnally, Seton V onludes ths paper. A. Adaptve LP-DMT II. SYSTEM MODEL The struture of the onsdered adaptve LP-DMT system s shown n Fg. 1 as suggested n [5]. The entre bandwdth s dvded nto N parallel subarrers whh are splt up nto N k sets S k of L subarrers. The preodng funton s then appled blok-wse by mean of preodng sequenes of length L. In fat, the preodng funton an be vewed as a spreadng omponent arred out n the frequeny doman as n multarrer ode-dvson-multple-aess (MC-CDMA). Fator L s suh that L N, whh mples that N k = N L. Note that the subsets n a gven set are not neessarly adjaent. Eah user u of the network s beng assgned a set B u of subsets S k. We emphasze that u, B u are mutually exlusve subsets. Consequently, multple aess between the N u users s managed followng a frequeny dvson multple aess (FDMA) approah, nstead of a ode dvson multple aess (CDMA) approah. It s worthy to menton here that we are gong to onsder the only ase of a sngle user multple blok system whh an be easly extended for a mult user multple blok senaro. The number of preodng sequenes used to spread nformaton symbols on one subset S k s denoted by N (k), wth 0 N (k) L sne we assume orthogonal sequenes. A ertan amount of energy e (k) wll be assgned to eah preoded sequene assoated to a gven modulaton symbol of b (k) bts, where 1 N (k). Fnally, n the followng, H = {1,..., N} wll be the set of the useful subarrers of the multarrer spetrum. 1) We s 4D 16-States Trells Code: Trells oded modulaton ombned wth nterleavng enables a better trade off between performane and bandwdth effeny, whle enjoyng low-omplexty Vterb deodng. Trells oded modulaton systems aheve sgnfant dstane gans whh are dretly related to the number of states. However, the odng gan saturates upon approahng a ertan number of states and the onstellatons must be hanged to aheve hgher gans. Multdmensonal onstellaton then gves a potental soluton. An nherent ost of 2D oded shemes s that the sze of the onstellaton s doubled over unoded shemes. Ths s due to the fat that a redundant bt s added to every sgnalng nterval. Wthout that ost, the odng gan of those oded shemes would be 3 db greater. Usng a m m+1 multdmensonal onstellaton wth a trells ode of rate an redue that ost beause fewer redundant bts are added for eah 2D sgnalng nterval. For example, that ost s redued to about 1.5 db f four-dmensonal onstellatons are used, whh s the ase n the suggested odng sheme. The trells ode onsdered here s a 4D 16-states ode developed by We [7]. Ths ode provdes a fundamental odng gan of γ f,db = 4.5 db, omputed as a 6.0 db nrease n the mnmum squared dstane between allowable sgnal sequenes, less a 1.5 db penalty nurred for a normalzed redundany of 0.5 bts per 2D symbol [10]. 2) RS Codes: The bnary data at the nput s frst fed to an outer nterleaved RS ode wth ode length n and nformaton length k. To orret t random errors n a blok of n symbols, n k = 2t party hek symbols are requred for an RS ode. The RS ode used here s based on a fnte feld (also known as Galos Feld) GF(2 8 ), and an have 256 dfferent values between 0 and 255. It s a shortened RS ode RS(240,224), supported n many standards [12], and an orret up to 8 erroneous bytes. C. Coded Adaptve LP-DMT The struture of the suggested system, nludng the proposed hannel odng sheme, s shown n Fg. 2. We s trells ode operates on the bts alloated to the preodng sequenes and produes two 2D ponts at ts output. In Fg. 2, only a sngle output s shown for We s trells enoder, beause both 2D outputs are alloated to the same ode. Also multple opes of We s enoder s shown for the purpose of llustraton whereas, n prate a sngle enoder s used to enode aross the preodng sequenes as dsussed n [9]-[10], where a sngle enoder s used to enode aross the subarrers. It wll be shown n Seton IV that smlar to ndependent and memoryless subhannels n a DMT senaro, preodng sequenes are also ndependent and memoryless n an LP- DMT senaro. The gan obtaned from the applaton of

4 Fg. 2. Coded LP-DMT transmtter struture trells ode wll therefore be the same as that obtaned n an ntersymbol nterferene (ISI) free envronment. On the other hand, the RS ode operates on the bnary stream at the nput of the system, before the bt alloaton blok as shown n Fg. 2. A onvolutonal nterleaver s used to spread the errors over a number of RS odewords. D. The Loadng Algorthm The bt loadng algorthm presented here s a smplfed form of that presented n [5] beause sngle user multple blok senaro s onsdered. The modfed algorthm takes nto aount the hannel odng sheme and s modfed to aommodate the odng gans obtaned from the odng sheme. Strtly speakng, all the subarrers are alloated to the only user n ths ase. The ahevable data rate on a gven subset S k at a gven target symbol error rate (SER) an be wrtten as R k = L log Γ L E s 1 h n N 2 0 n S k (1) where h n 2 s the gan of subhannel n, Γ s the normalzed sgnal to nose rato (also known as SNR gap), E s s obtaned from a gven power spetral densty (PSD) onstrant and N 0 s the addtve bakground whte Gaussan nose level. SNR gap, Γ, has a onstant value for all the modulaton orders of unoded quadrature ampltude modulaton (QAM) for a fxed target SER. However, (1) does not provde any pratal throughput beause t assumes nfnte granularty. 1 The task here s therefore to fnd an approprate bt dstrbuton for L avalable odes of eah subset S k, whh maxmzes the data rate. In the followng, we dsuss the alloaton poly that handles the fnte granularty problem. Gven an SNR gap Γ, a preodng fator L and a transmsson level E s, the rate aheved by an adaptve LP-DMT system usng dsrete modulaton s maxmzed f, on eah 1 nfnte granularty: non-nteger modulaton order subset S k, b (k) bts are alloated to odes and b (k) s gven as b (k) R k /L + 1 (1 n (k) ) = (2) R k /L < N (k) ) where n (k) (n (k) = L (2 R k/l R k /L 1). Then, pratally ahevable data rate R k, onsderng fnte granularty, on eah subset S k s gven as R k = L (2 R k/l R k /L 1) ( R k /L + 1) + (L L (2 R k/l R k /L 1) ) R k /L (3) Now we an optmally assgn a partular modulaton order to eah ode on dfferent subsets. The energy, e (k), assgned to these odes s gven as e (k) = (2 b(k) e (k) 1) Γ 1 L 2 N 0 h n 2 (4) n S k whh satsfes < E s. Ths energy alloaton onsders null nose margn as eah ode reeves the exat amount of energy to transmt the number of bts determned by the algorthm for a gven Γ. As dsussed above, Γ s defned for a gven target SER wth unoded QAM and have a onstant value for all the modulaton orders. In ths paper we are gong to deal wth fxed target bt error rate (BER) nstead of SER. Then takng nto aount the odng gans and fxed BERs, Γ s no more onstant for all the modulaton orders. The above algorthm s modfed to aommodate varable Γ for dfferent modulaton orders. The exat values of the SNR gaps for all the modulaton orders are stored n a predefned table and are. These values are alulated on the bass of the seleted hannel odng sheme and the requred system margn, whh we are gong to dsuss n Seton III. For a gven subset S k, ntally, we an take any value for Γ, denoted by Γ (k) say Γ ntal(k) = 1. The loser the ntal value to the exat value, the more effent s the algorthm. R k s alulated from (1) usng Γ ntal(k) whle b (k) and n (k) from (2). Now the exat value of SNR gap, Γ (k), s taken from the table dependng upon the bt vetor b (k), and e (k) s alulated from (4) usng Γ (k). Gradually bts are added n the bt vetor, b (k), tll e (k) exeeds the PSD lmt, E s, and subsequently bts are removed to respet the PSD lmt. We an summarze the modfed approah as follows: 1) Calulate R k, n (k) 2) Take Γ (k) 3) Calulate e (k) and b (k), dependng upon b (k) for Γ (k) 4) Start a( ounter, say ount ) = 1 5) Whle e (k) < E s a) b (k) n +ount + = 1 b) ount + = 1 for Γ ntal(k)

5 ) update e (k) ( ) 6) Whle e (k) > E s a) ount = 1 b) b (k) n = 1 +ount ) update e (k) III. THEORETICAL CODING EFFECTS ON SYSTEM PERFORMANCE In ths seton, we onsder the theoretal odng gan promsed by the proposed onatenated hannel odng sheme. In ths analyss we need to deal wth 2D error rates, BERs, and RS SERs, dependng upon what part of the system s beng onsdered. We use the assumptons gven n [9]-[10] for the sake of smplty. Contrary to [9]-[10], all alulatons are made dealng only wth BERs. In these assumptons, these quanttes are related by onstant fators, and the 2D error rate s used as a ommon bass. In partular, 2D SERs are onverted to BERs by multplyng by one-half. Smlarly, 2D SERs are onverted to RS SERs by multplyng by a onstant, where represents the average number of preodng sequenes ontrbutng bts to eah RS symbol [9]. P bt denotes the requred BER at the output of the overall system. From [13], the probablty of 2D symbol error n quadrature ampltude modulaton s losely approxmated by [ ] dmn P 2D 4Q (5) 2σ where d mn s the mnmum dstane between QAM onstellaton ponts at the hannel output, σ s the nose varane, and Q[.] represents the well-known Q-funton. By usng the frst assumpton, as dsussed above, the SNR gap Γ for a target BER of 10 7 s gven as Γ = γ m γ (db) (6) where γ m s the desred margn n the system and γ, the odng gan for the proposed onatenated hannel odng sheme, s gven as γ = γ t,db + γ rs,db γ loss,db (db) (7) where γ t,db and γ rs,db are the gans provded by the trells ode and the RS ode respetvely and γ loss,db s the loss nurred for nreasng the data rate. γ rs,db : Whle assumng effent nterleavng to have random errors at the nput of RS deoder and assumng that RS deoder does not attempt to orret the odeword f greater than t errors are deteted, we may relate the output RS SER, P rs, to the nput RS SER, P s, by P rs = n =t+1 ( n 1 1 ) P s(1 P s ) n. (8) Gven P bt and knowng that P 2D = 2P bt and P rs = P 2D, we an say that P rs = 2P bt and by teratvely solvng (8) for P s, the orrespondng BER at the nput of RS deoder s gven by P b = P s (9) 2 and P b s the BER at the output of the demodulator, therefore an SNR gap to obtan P b, Γ rs, an be wrtten as ( Γ rs = 1 [ ] ) 2 Q 1 Pb (10) 3 2 Γ 0,Pbt s defned as an SNR gap requred by an unoded system to aheve P bt, and s gven as ( Γ 0,Pbt = 1 [ ] ) 2 Q 1 Pbt (11) 3 2 From (11) and (10), γ rs an be gven as γ rs = Γ 0,Pbt Γ rs (db) (12) γ t,db : As P b s the requred BER at the nput to the RS deoder and Γ 0,Pb and Γ t,pb are the SNR gaps requred by an unoded and a We s 4D 16-states trells oded system respetvely to aheve P b. Then the odng gan of a We s 4D 16-states trells ode an be gven by γ t = Γ 0,Pb Γ t,pb (db) (13) γ loss,db : If P tot(b) s the mnmum amount of power requred to aheve the data rate b as defned n [9], the loss for the nreased data rate assoated wth the RS ode, γ loss,db, an be gven as γ loss,db = Ptot,dB (nb k ) P tot,db (b) (14) IV. SIMULATION RESULTS In ths seton, smulaton results are presented for the proposed onatenated hannel odng sheme ombned wth the adaptve LP-DMT system. The performane of the oded adaptve LP-DMT system s ompared wth that of the oded DMT. The generated LP-DMT sgnal s omposed of N =1024 subarrers transmtted n the band [500;20,000] khz and the preodng fator, L =16. The subarrer spang s khz. It s assumed that the synhronzaton and hannel estmaton tasks have been suessfully performed. We use the multpath model for the power lne hannel as proposed n [11] and shown n Fg. 3. The onsdered referene model s 110 m lnk 15-paths model whose frequeny response s gven by H(f) = N g e (a0+a1fk ) d e j2πfτ (15) =1 a result whh has been wdely proved n prate. The parameters of the 15-path model are lsted n Table I, and τ s the delay of path. A bakground nose level of -110 dbm/hz s assumed and the sgnal s transmtted wth respet to a flat PSD of -40 dbm/hz. Super-onstellaton sze for We s 4D 16-states trells odng s 1024 ponts. Results are gven for a fxed target BER of 10 7.

6 H(f) n db H(f) n db m 300m 100m 150m 200m Frequeny n MHz Frequeny n MHz Fg paths referene hannel model for PLC [11] Fg. 4. Length profles of the attenuaton of power lne lnks TABLE I PARAMETERS OF THE 15-PATH MODEL TABLE II ATTENUATION PARAMETERS CORRESPONDING TO THE LENGTH PROFILES attenuaton parameters k = 1 a 0 = 0 a 1 = path-parameters g d (m) g d (m) lass g 1 a 0 [m 1 ] a 1 [s/m] k 100m m m m m Coded LP DMT Coded DMT Unoded LP DMT Unoded DMT Length profle of the attenuaton of power lne lnks,.e. negletng the mpats of nothes, as proposed n [11], are shown n Fg. 4 and the orrespondng parameters are lsted n Table II. These profles are used to ompare the performane of LP-DMT wth DMT at varous dstanes for both oded and unoded mplementatons. The DMT system an be obtaned by takng L = 1 n the LP-DMT system. The smulatons are run for a sngle user multple blok senaro. The proposed adaptve oded LP-DMT system an easly be extended to a mult user multple blok senaro. The numbers of useful bts n eah LP-DMT and DMT symbol are shown n Fg. 5 for both oded and unoded mplementatons at varous lnk dstanes, whle usng the bt and energy alloaton algorthm as dsussed n Seton II. The reason for the better performane of the oded LP-DMT system s explaned n Fg. 6, where energy dstrbuton of the oded LP-DMT s ompared wth that of the oded DMT. It s lear that the oded DMT s not fully explotng the avalable energy on eah subarrer due to fnte granularty and PSD onstrants, whle the preodng omponent of the oded LP- DMT system aumulates the energes of a gven subset of Bt/Symbol Dstanes n metre Fg. 5. Aheved throughputs at varous lnk lengths subarrers to transmt addtonal bts. Both systems respet the PSD onstrant of -40 dbm/hz as defned earler. The oded adaptve LP-DMT system utlze more effently ths PSD lmt n omparson wth the oded DMT system. Fg. 6 gves the mnmal requred energy allowng the transmsson of the maxmum data rate. Aordng to the PSD mask, the resdual avalable energy would not lead to any nrease n the

7 PSD (dbm/hz) Fg Coded LP DMT: bt/sym Coded DMT: bt/sym Subarrers Energy dstrbuton omparson of oded LP-DMT wth oded DMT TABLE III THROUGHPUT COMPARISON USING 15-PATH REFERENCE MODEL OF PLC CHANNEL system useful bt/symbol oded LP-DMT oded DMT unoded LP-DMT 5016 unoded DMT 4636 data rate. The spke-shaped urve of the oded DMT shows the transtons of the modulaton orders (.e. dereasng the onstellaton szes) when no more energy s avalable to sustan the fxed target BER. Fnally, the smulatons for oded and unoded DMT and LP-DMT systems are run usng the referene model of 110 m lnk wth 15 multpaths. The results of these smulatons are summarzed n Table III. As t s shown n Fg. 5 and Table III that oded LP-DMT system has the hghest throughput and more effent energy utlzaton when ompared to oded DMT (see Fg. 6). Table III shows that there s an mprovement n the throughput of approxmately 18% and 5% when we ompare our oded LP-DMT system wth unoded LP-DMT and oded DMT respetvely usng 15-path referene model for power lne ommunaton for a fxed target BER of It s worthy to menton here that 5% s the mnmum mprovement aheved, due to the suboptmal subarrer sharng method used heren, whh an further be mproved by a subarrer swappng approah. V. CONCLUSION In ths paper, we have nvestgated the resoure alloaton problem of an adaptve LP-DMT system takng nto aount the hannel odng sheme. The proposed loadng algorthm aommodates odng gans of the hannel odng sheme n bt and energy alulatons and an handle dfferent values of the SNR gaps for dfferent modulaton orders. Ths bt and energy alloaton algorthm an be used n ombnaton wth any hannel odng sheme provded the SNR gaps of that sheme are known for all the modulaton orders. A onatenated hannel odng sheme, onsstng of an nner We s 4D 16-states trells ode and an outer RS ode, s proposed for the adaptve LP-DMT system n order to aheve hgh performane transevers for power lne ommunaton applatons. It s shown that by usng a powerful but low-omplexty odng sheme wth the proposed algorthm, throughput of LP-DMT system an be nreased sgnfantly, further the ombnaton of odng and preodng elements n a multarrer power lne senaro aheves hgher throughput. REFERENCES [1] S. Maller, F. Nouvel, J.-Y. Baudas, D. Gardan, and A. Zeddam, Multarrer CDMA over opper lnes - omparson of performanes wth the ADSL system, n Pro IEEE Int. Workshop Eletron. Desgn, Test, Appl., Jan. 2002, pp [2] J. Boutros, and E. Vterbo, Sgnal spae dversty: a power- and bandwdth-effent dversty tehnque for the Raylegh fadng hannel, IEEE Trans. Inform. Theory, vol. 44, pp , Jul [3] A. O. Hero, and T. L. Marzetta, Cutoff rate and sgnal desgn for the quas-stat Raylegh fadng spaetme hannel, IEEE Trans. Inform. Theory, vol. 47, pp , Sep [4] D. Ransh, Dversty transform for fadng hannels, IEEE Trans. Commun., vol. 44, pp , De [5] M. Crussère, J.-Y. Baudas, and J.-F. Hélard, Robust and hgh-bt rate ommunatons over PLC hannels: a bt-loadng mult-arrer spreadspetrum soluton, n Pro. IEEE Int. Symp. Power. Lne. Commun., Apr 2005, pp [6] M. Crussère, J.-Y. Baudas, and J.-F. Hélard, Adaptve spread-spetrum multarrer multple-aess over wrelnes, IEEE Journal on Seleted Areas n Communatons, vol. 24, no. 7, pp , [7] L.F. We, Trells-oded modulaton wth multdmensonal onstellatons IEEE Trans. Inform. Theory, vol. 33, no. 4, pp , Jul [8] Internatonal Teleommunaton Unon-Teleommunaton Reommendaton, Very Hgh Speed Dgtal Subsrber Lne Transevers 2 (VDSL2), ITU-T Re. G Feb [9] T.N. Zogaks, J.T. Aslans, and J.M. Coff, A Coded and shaped dsrete multtone system, IEEE Trans. Commun., vol. 43, no. 12, pp , De [10] T.N. Zogaks, J.T. Aslans, and J.M. Coff, Analyss of a onatenated odng sheme for a dsrete multtone modulaton system, n 1994 IEEE Mltary Communatons Conf., pp [11] M. Zmmermann and K. Dostert, A multpath model for the powerlne hannel, IEEE Trans. Commun., vol. 50, no. 4, pp. 553, Apr [12] European Teleommunatons Standards Insttute, Very Hgh Speed Dgtal Subsrber Lne (VDSL), ETSI TS , Jul [13] J. Coff, A multarrer prmer ANSI T1E1.4/91-157, 1991, Commttee ontrbuton, Teh. Rep. [14] W. Rhee, and J.M. Coff, Inrease n apaty of multuser OFDM system usng dynam subhannel alloaton, n Pro. IEEE Vehular Tehnology Conferene (VTC-Sprng 00), vol. 2, Tokyo, Japan, May. 2000, pp [15] Z. Lu, Y. Xn, and G.B. Gannaks, Lnear onstellaton preodng for OFDM wth maxmum multpath dversty and odng gans IEEE Trans. Commun., vol. 51, no. 3, pp , Mar [16] E. Bgler, Codng and modulaton for a horrble hannel, IEEE Commun. Mag., vol. 41, no. 5, pp , May