UI - Secteur de la ormalisatio des télécommuicatios IU - elecommuicatio Stadardizatio Sector UI - Sector de ormalizació de las elecomuicacioes Study Period 997-000 Commissio d'études ;Study Group;Comisió de Estudio } 5 Cotributio tardive;delayed Cotributio ;Cotribució tardía } D.xxx Geeva, Jue July, 999 exte dispoible seulemet e ;ext available oly i;exto dispoible solamete e } E Questio(s): 4/5 SOURCE*: ILE: IBM Europe Filtered Multitoe Modulatio ABSRAC his cotributio describes the basic priciples of filtered multitoe (FM) modulatio ad presets a aalysis of the performace ad complexity of this techique for VDSL trasmissio. FM desigates a form of polyphase filter-ba modulatio where a high level of spectral cotaimet is achieved for subchael spectral shapig. It borrows features from CAP/QAM ad DM modulatio techiques ad offers advatages that go beyod those of each of these two modulatio schemes tae idividually. he paper is tutorial i ature ad does ot aim at specifically addressig all VDSL system requiremets. hese aspects will be cosidered i a future cotributio. * Cotact: Giovai Cherubii (cbi@zurich.ibm.com) Evagelos Eleftheriou (ele@zurich.ibm.com) Sedat Ölçer (oel@zurich.ibm.com) +4--74 85 8 +4--74 85 83 +4--74 84 80 IBM Zurich Research Laboratory Säumerstrasse 4 8803 Rüschlio Switzerlad
. Itroductio his cotributio describes filtered multitoe modulatio (FM) ad its applicatio to VDSL trasmissio. FM modulatio desigates a form of polyphase filter-ba modulatio techique where a high level of subchael sigal spectral cotaimet is achieved by spectral shapig. he advatages of special subchael spectral shapig will be show i the cotext of a VDSL trasmissio system employig frequecy divisio duplexig (FDD) where echo ad crosstal sigals represet the mai sources of disturbace. I Sectio, we itroduce some geeral cocepts from the theory of filter-ba systems, which fid their roots i multirate digital sigal processig. I Sectio 3, we describe FM modulatio. Efficiet realizatios of trasmit ad receive filters that achieve a high level of subchael spectral cotaimet are discussed ad several examples are itroduced. I Sectios 4 ad 5, we idetify the advatages of the described techique with respect to CAP/QAM ad sychroous DM modulatios ad ivestigate the performace of FM-based trasmissio for a VDSL system employig FDD. Implemetatio complexity ad system latecy are two importat implemetatio aspects that are addressed i Sectio 6. Fially, i Sectio 7, we provide some cocludig remars. he purpose of this cotributio is to preset FM modulatio as a geeric techology. herefore a aalysis that addresses all the system requiremets set forth for VDSL withi various stadardizatio groups goes beyod the scope of the preset study.. Filter-ba modulatio Figure shows the bloc-diagram of a commuicatio system employig filter-ba modulatio ad (i) demodulatio cocepts. A group of M modulatio symbols A, i = 0,, M-, is iput i parallel at the rate of / ito a set of M filters with trasfer fuctios H 0 (f), H (f),, H M- (f). he otatio idicates upsamplig by a factor, i.e., isertio of - zero sigals betwee two cosecutive iput sigals. his set of M filters represets a so-called sythesis filter-ba. he chael iput sigal is geerated at the trasmissio rate of /. At the receiver, demodulatio is achieved by a aalysis filter ba that icludes M filters G 0 (f), G (f),, G M- (f) followed by dowsamplers deoted by. Whe M =, a critically sampled filter-ba structure is obtaied. (0) A () A H 0 (f) H (f) + Chael C(f) G 0 (f) G (f) (0) Â () Â (M-) A H M- (f) G M- (f) (M-) Â Figure : Geeral represetatio of a commuicatio system usig filter bas
Selectio of filter characteristics plays a cetral role i the desig of a commuicatio system employig filter-ba modulatio. he filters H i (f) ad G i (f), i = 0,, M-, typically exhibit a badpass characteristic. Figure shows a example of filter characteristics for a critically sampled filter ba where the i th filter trasfer fuctio is cetered o the frequecy i/. he ecessity of oversamplig each subchael sigal by a factor M ca be immediately deduced from this figure sice the spectral characteristic of each subchael is periodic with period M/. H 0 (f) H (f) H M- (f) H 0 (f) / M/ f I practice, filter-ba modulatio systems are almost ever implemeted directly as show i Figure. he reaso for this is that, i this cofiguratio, filters must operate at a rate that is times faster tha the symbol rate /. If the filters are appropriately selected, it is possible to achieve very efficiet realizatios. For example i the critically sampled case, if the M trasmit [receive] filters are selected as the frequecy-shifted versios of a basebad filter H(f) [G(f)], the so-called prototype filter, the system of Figure ca be redraw as show i Figure 3(a), which i tur is equivalet, for the case M = ad assumig the matched filter pair coditio g (i) = h (i)* - Figure : ypical subchael characterisitics of a critically sampled M-bad filter ba, i = 0,,, M - is satisfied, to the system depicted i Figure 3(b), as show i the Appedix []. he discrete-time modulatios with the complex expoetials have bee absorbed ito the iverse discrete Fourier trasformatios (IDFs) ad the filterig operatios o the M braches are performed by filters that correspod to the M polyphase compoets of the prototype filter. I other words, the H (i) (f), i = 0,, M-, represet the trasfer fuctios of the M polyphase compoets of H(f), ad liewise at the receiver. his M-brach polyphase filter ba structure is attractive because the filterig operatios are performed at the rate of / istead of M/. It also has the advatage that oly suitable prototype filters must be determied, ot the complete set of aalysis ad sythesis filters. Efficiet polyphase filter ba realizatios ca liewise be foud i the ocritically sampled case. he decompositio of a filter characteristic ito its polyphase compoets is explaied i the Appedix. he reader will readily otice similarities betwee the structures show i Figure ad 3 ad those of CAP/QAM as well as DM modulatio systems. A passbad sigle carrier system is obtaied whe oly oe brach i ( i > 0) of the filter ba i Figure is retaied. A DM modulatio system is obtaied whe all the polyphase filters i Figure 3(b) are reduced to uity gai. Filter bas have bee extesively studied i the area of discrete-time multirate sigal processig []. I particular, filter ba structures for which the matched filter pair coditio holds, have received cosiderable attetio. I the vast majority of these applicatios, the followig coditios are further imposed for the desig of the filter ba: he asteris deotes complex cojugatio
(0) A () A H(f) H(f) j f 0 / e π j f / e π + x Chael C(f) -j f 0 / e π -j f / e π G(f) G(f) (0) Â () Â (M-) A H(f) j f / e π M- -j f / e π M- G(f) (M-) Â (a) Filter ba modulatio system employig basebad prototype filters (0) A () A IDF H (0) (f) H () (f) P/S x M Chael C(f) S/P H (0)* (f) H ()* (f) DF (0) Â () Â (M-) A H (M-) (f) H (M-)* (f) (M-) Â (b) Efficiet implemetatio of the filter ba show i (a) for M= Figure 3: Efficiet filter ba implemetatio usig polyphase filter compoets (i) (i)* h h = δ, i = 0,,, M - - (i) ( j)* h h - = 0, i j. For a ideal trasmissio chael that does ot itroduce sigal distortio ad oise, the above coditios guaratee perfect recostructio for all the braches of the sythesis/aalysis structure. his meas that the symbols received at the output of each subchael are free of iterchael as well as itersymbol iterferece, i.e., Â = A (i) (i)
up to a time delay itroduced by the overall chael. I practice, perfect recostructio requires that substatial spectral overlap of adjacet subchaels be allowed. he above criterio appears to be meaigful for may applicatios, typically for subbad codig of audio ad video sigals. Also i discrete wavelet multitoe (DWM) modulatio [3], a techique that has bee ivestigated for DSL trasmissio, the prototype filter is desiged such that perfect recostructio is possible at the demodulator output uder the assumptio of a ideal chael without oise ad distortio. he perfect recostructio costrait may, however, be overly restrictive or eve ot suitable for filter desig i a data commuicatios cotext, because it eforces zero itersymbol ad iterchael iterferece but igores oise ad distortio that are always preset i practical commuicatio chaels. By relaxig the above cotraits ad itroducig sigal equalizatio capabilities at the receiver, prototype filters leadig to better overall system performace ad commuicatios efficiecy ca be foud. 3. Filtered multitoe (FM) modulatio: geeral priciples We desigate by filtered multitoe (FM) modulatio a polyphase filter ba modulatio techique where the protoype filter is desiged to achieve a high level of subchael spectral cotaimet. I other terms, the prototype filter is desiged such that at the output of each subchael the level of iterchael iterferece may be cosidered egligible as compared to the level of other oise sigals. I some cases, high spectral cotaimet will be more easily achieved by relaxig the zero itersymbol costrait. I some other cases, our geeral polyphase filter-ba modulatio approach will also eable alterative solutios to reach high spectral isolatio betwee subchaels, as illustrated below. At the receiver of a FM system, per-subchael sigal equalizatio is employed. Accordig to the particular applicatio ad/or the desired system performace level, usually specified i terms of the mea-square error at the receiver decisio poit or of bit error probability, a symbolspaced or fractioally-spaced liear equalizer ca be used. Alteratively, symbol-spaced or fractioally-spaced decisio-feedbac equalizatio ca be implemeted, possibly i the form of precodig [4]. High level of subchael spectral cotaimet is a desirable property for may applicatios. For example, because leaage of sigal eergy betwee subchaels may be cosidered egligible, echo cacellatio is ot eeded i frequecy-divisio duplexig (FDD) trasmissio systems where subchaels are closely placed to each other. Also, i Zipper-lie FDD [5], where all trasmissios withi the same cable bider adopt the same upstream/dowstream frequecy bad allocatios, self- EX is completely avoided. We ote that ow multicarrier modulatio techiques lead to oegligible spectral overlap betwee the subchaels. For DM modulatio, the spectra of adjacet subchaels approximately cross at the 3 db poit ad the first sidelobe for a subchael is as high as 3 db. For DWM modulatio, although the first sidelobe is as low as 45 db or less, the spectra of adjacet subchaels still cross at 3 db. Hece, a sigificat amout of subchael sigal eergy couples ito the eighborig subchaels. For FM modulatio, the trasmit polyphase filters are usually obtaied from a liear-phase FIR prototype filter of legth γm, i.e., h = 0 for < 0 ad γ M. Hece each of the M polyphase filter compoets has legth γ. I geeral, larger values of γallow better approximatio of prototype filter trasfer fuctios with sharp spectral roll-offs but lead to a icrease i system latecy. Clearly, the choice of the prototype filter allows various tradeoffs betwee umber of subchaels, level of
spectral cotaimet, sigal latecy ad trasmissio efficiecy. I this sectio, two examples will be treated to illustrate prototype filter selectio. As a first example, we cosider a critically sampled FM system that employs a prototype filter with frequecy respose providig a close approximatio to the characteristic H ~ (f + e e + ρ ) = 0 - jπf - jπf if f / otherwise, where the parameter 0 ρ cotrols the roll-off towards the spectral ulls at badedge frequecies ad f deotes frequecy ormalized by the symbol rate. his zero excess badwidth characteristic leads to a FM system ideally free of iterchael iterferece but with itersymbol iterferece withi a subchael. Figure 4 shows the subchael characteristics that are obtaied by usig a prototype FIR filter that approximates the characteristic H ~ (f) i the case where M = 56, c = 0, ad ρ = 0.. For ρ, the frequecy characteristic of a FM subchael is characterized by steep roll-off towards the badedge frequecies, suggestig that per-subchael decisiofeedbac equalizatio be performed to recover the trasmitted symbols. Figure 4: Subchael frequecy resposes for f χ [0, 0.05 M/] i a FM system with M=56 ad prototype filter desiged for =0. ad c=0.
I the secod example, we employ for the prototype filter a yquist filter with ozero excess badwidth ad allow ocritical samplig > M. A square-root raised cosie filter with excess badwidth α ad trasfer fuctio [6] H ~ (f ) = π - si ( f α ) f -α ( f α)/ + α where f is ormalized frequecy, is a well-ow example of a yquist filter. he, for same carrier spacig as i the critically sampled case, if the symbol rate is decreased by a factor /M a yquist filter with a excess badwidth of α = M - ca be used while esurig that the spectral eergy of a subchael is still etirely cotaied withi the bad of that subchael. By lettig M, the pealty i badwidth efficiecy ca be made vaishigly small at the price of a icrease i implemetatio complexity sice filters with icreasigly sharper spectral roll-offs must the be realized. ote that excess badwidth modulatio ca also be achieved i the first example if ocritical samplig > M is employed. 4. FM-based frequecy divisio duplexig for VDSL We focus i this cotributio o a FM system for VDSL that employs frequecy divisio duplexig (FDD). I order to mitigate the effects of disturbace by self-ex, we assume that each of the M subchaels is used either for upstream or for dowstream trasmissio but ot both. All trasmissios withi the same cable bider follow the same subchael frequecy allocatio. his implies a Zipper-lie FDD techique. I the ext sectio, performace ad complexity issues for FM-based FDD will be described for selected system parameters. Here we provide a discussio o the advatages of the proposed scheme by comparig it to the CAP/QAM ad DM-based techiques curretly proposed for VDSL. he followig remars ca be made with respect to CAP/QAM: FM modulatio permits efficiet implemetatio of a CAP/QAM scheme with more tha two bads as presetly proposed. Implemetatio efficiecy stems from a all-digital geeratio of a multibad CAP/QAM-lie sigal with miimum aalog filterig requiremets. Digital filters with fairly sharp spectral roll-offs ca be employed i FM modulatio, thus allowig subchaels to be placed close to each other. he guard bads eeded for subchael separatio i CAP/QAM lead to a waste of useful spectrum. Similar to CAP/QAM systems, the receiver of a FM system icorporates adaptive liear or decisio-feedbac equalizatio, possibly i fractioally spaced form. Although per-subchael equalizatio adds to system complexity, it is very importat to eep i mid that the equalizers
operate at the FM symbol rate ad ot at the high trasmissio rate lie i CAP/QAM receivers. Providig more tha two subchaels for trasmissio, typically 6, 3, 64, or 8 for VDSL applicatios, allows brigig all advatages of a multicarrier scheme (log symbol duratio, flexibility i upstream/dowstream subchael assigmet, mitigatio of arrowbad iterferece, ease of meetig spectral compatibility requiremets, etc.) ito a CAP/QAM-lie system. he followig remars ca be made with respect to DM-based Zipper: DM-Zipper requires itroducig a cyclic prefix ad a cyclic suffix to combat the effects of itersymbol iterferece itroduced by the chael ad to maitai sigal orthogoality betwee subchaels i the presece of echo sigals. hese cyclic extesios lead to a loss i trasmissio efficiecy. ote that to provide robustess agaist self-ex disturbace via the widowig ad shapig techique the cyclic extesios must be further expaded. o cyclic extesio is eeded for FM modulatio due to the high level of subchael isolatio. Eve i the case where widowig-ad-shapig is used, DM-Zipper requires that VU-O ad VU-R trasmissios be pairwise frame sychroized accordig to the timig advace techique. here are o frame sychroizatio requiremets for FM trasmissio, either at the bider or at a VU-O/VU-R pair level. I DM-Zipper, widowig-ad-shapig does ot lead to perfect rejectio of self-ex disturbace. For this reaso, there is strog motivatio to group together as much as possible the subchaels used for upstream trasmissio ad similarly for the dowstream subchaels. I FM trasmissio, the subchaels ca be arbitrarily assiged to upstream ad dowstream trasmissio without icurrig a performace pealty due to self-ex. For FM-based trasmissio a low umber of subchaels ca be employed as compared to DM-based trasmissio. he reaso for this is that FM does ot suffer from a loss of trasmissio efficiecy (o cyclic extesios are used) as the umber of subchaels is decreased, which is ot the case for DM-based trasmissio. FM modulatio thus borrows features from CAP/QAM ad DM modulatio techiques ad offers advatages that go beyod those of each of these two modulatio schemes tae idividually. 5. Performace of FM-based FDD for VDSL I this sectio, we preset simulatio results to illustrate the performace achieved by FM-based FDD systems ad also compare it to that of DM-based FDD systems. For both cases, a Zipper-lie FDD scheme is assumed. We cosider the followig cable trasfer fuctio, which correspods to UP-3 worst-case characteristics [7]: C(f, l) = e -6-3.85 0 (+ j) fl. I this trasfer fuctio, f represets frequecy i Hz, l the cable legth i meter ad costat propagatio delay has bee igored. For the results preseted i this sectio, we cosider ear-ed ad far-ed crosstal (EX ad FEX, respectively) to be the mai sources of iterferece for VDSL trasmissio. he power spectral desities of EX ad FEX sigals arisig from disturbers are modeled as
ad PSD EX (f) = PSD disturber (f ) 49 0.6 0 3 f 3/, respectively. PSD FEX (f) = PSD disturber (f ) C(f, l) 49 0.6 3 0 We measure the performace of a FM-based system i terms of achievable bit rate for give chael characteristics. he umber of bits per modulatio iterval that ca be loaded o the -th subchael is give by [8] b = log SR γ Γ γ margi code, + where SR is the sigal-to-oise ratio o the -th subchael, γ code is the codig gai, Γ deotes the SR gap betwee the miimum SR required for reliable trasmissio of L bits per modulatio iterval ad the SR required by L -ary QAM modulatio to achieve a bit error probability of 0-7, L >>, ad γ margi deotes the required additioal margi. he achievable bit rate for dowstream or upstream trasmissio is therefore obtaied by summig the values give by b for dowstream or upstream trasmissio. umerical results will be give for the case where the prototype filter is a liear-phase FIR filter approximatio to H ~ (f) defied i Sectio 3. For all cases, we use a overlap factor γ = 0 ad rolloff ρ = 0.. he various cases addressed will be referred to as idicated i able. 9 l f, Case r. of subchaels (M/) Excess badwidth a Symbol rate a 8 0 % 86.5 Hz b 8 6.5 % 8.8 Hz c 8.5 % 76.67 Hz a 3 0 % 345 Hz b 3 6.5 % 34.7 Hz c 3.5 % 306.67 Hz a 6 0 % 690 Hz 3 b 6 6.5 % 649.4 Hz c 6.5 % 63.33 Hz able : Defiitio of the cases cosidered for the simulatios he power of the trasmitted sigal is 0 dbm. rasmissio over UP-3 cable i the presece of 49 EX disturbers, 49 FEX disturbers, AWG with power spectral desity equal to 40 dbm/hz, ad a echo sigal egligible as compared to the other disturbaces is assumed. Achievable bit rates for symmetric trasmissio over the frequecy bad of 0 Hz to.04 MHz are show i Figures 5 to 6. For all cases, per-subchael equalizatio is performed by employig a b omliso-harashima precoder with e taps at the trasmitter ad a liear equalizer with f e taps
at the receiver. For the cases with zero excess badwidth, liear symbol-spaced equalizatio is used, while for the ozero excess badwidth liear equalizatio with half symbol spacig is employed. he coefficiets of the liear equalizer ad of the precoder are equivalet to the coefficiets of the forward sectio ad of the feedbac sectio of a MMSE DFE, respectively, ad are computed assumig perfect owledge of the subchael characteristics. I the figures, the otatio f e b e FM(, ) is used to idicate the umber of equalizer taps used to derive system performace. For compariso, the rates achieved by a sychroous DM-based Zipper system with M = 4096, cyclic prefix of 50 samples, cyclic suffix of 75 samples, o time-domai equalizer ad oe-tap frequecy equalizers are also illustrated i Figure 5a. Perfect owledge of the overall chael characteristics has bee assumed, as well as perfect sychroizatio of all trasmissios over the 50- pair cable. Hece, the cosidered system achieves ideal suppressio of EX iterferece. 40 35 30 Achievable rate (Mbit/s 5 0 5 0 5 DM FM(48,) 0 00 00 300 400 500 600 700 800 000 00 400 Cable legth Figure 5a: Performace of FM systems (M=56) for differet equalizer legths ad 0% excess badwidth (case a). Performace of sychroous DM (M=4096) is also show. he results of Figure 5a idicate that FM ad DM-based Zipper systems exhibit essetially idetical performace for cable legths up to 700 m. For loger cables, where cable-depedet sigal distortio becomes more sigificat, the FM system allows higher data rates to be achieved due to its more powerful equalizatio capability. ote that i all cases the FM-based scheme does ot require ay sychroizatio of trasmissios.
40 35 Achievable rate (Mbit/s 30 5 0 5 0 FM(6,6) FM(36,6) FM(48,) FM(48,)* 5 0 00 00 300 400 500 600 700 800 Cable legth Figure 6a: Performace of FM systems (M=64) for differet equalizer legths ad 6.5 % excess badwidth (case b). he asteris idicates the zero excess-badwidth case (case a). 40 35 Achievable rate (Mbit/s 30 5 0 5 0 FM(6,6) FM(36,6) FM(48,) FM(48,)* 5 0 00 00 300 400 500 600 700 800 Cable legth Figure 6b: Performace of FM systems (M=64) for differet equalizer legths ad.5 % excess badwidth (case c). he asteris idicates the zero excess-badwidth case (case a).
6. Implemetatio complexity ad latecy I this sectio, we first ivestigate the implemetatio complexity of FM trasmissio for the differet cases studied i the Sectio 5. Each etry i able represets, i giga (0 9 ) operatios (real multiply-adds) per secod, the total complexity of digital filterig ad fast Fourier trasformatio (FF) at the trasmitter ad at the receiver. he umber of operatios for a FF is computed as Mlog M, although implemetatios with sigificatly lower complexities could be used. Filter complexity is computed as the umber of filter taps times the output rate, which is equivalet to the umber of multiply-ad-adds per secod. 3 Case f b f b f b ( e, e ) = (48,) ( e, e ) = (36,6) ( e, e ) = (6,6) a..7.8 b.0.70.6 c.08.68.4 a.03.63.9 b.0.6.8 c.00.60.6 a.99.59.5 b.97.58.3 c.96.56. able : Implemetatio complexity i giga operatios per secod for the cases cosidered i Sectio 5. Aother importat system desig parameter is latecy. he values give i able 3 represet sigal latecy i ls from the trasmit IFF output to the receive FF iput, excludig propagatio delay through the chael. 3 Case f e = 48 f e = 36 f e = 6 a 649 498 78 b 683 535 88 c 76 559 99 a 6 7 69 b 7 34 7 c 79 40 75 a 8 64 35 b 85 67 36 c 89 70 37 able 3: Sigal latecy i ls for the cases cosidered i Sectio 5. 7. Coclusios
he framewor of polyphase filter bas opes up a variety of possibilities to realize efficiet FM modulatio systems. he objective of this cotributio has bee to itroduce ad illustrate some importat aspects of these systems. he selectio of the filterig elemets at the trasmitter as well as at the receiver represets a desig tradeoff betwee may parameters. Furter ivestigatio is eeded to determie optimum tradeoffs i coectio with the VDSL system requiremets. 8. Summary his cotributio should be preseted uder G.ge.bis ad is for iformatio oly. he described FM modulatio techique ca be used for DSL trasmissio i geeral. I this cotributio, it has bee aalyzed for VDSL trasmissio. Appedix: Polyphase represetatio ad computatioally efficiet realizatios he polyphase represetatio [] leads to computatioally efficiet implemetatios of multirate decimatio ad iterpolatio filters, as well as of filter bas. Cosider a filter H(z) give i z-trasform otatio as H(z) = = - h he for ay iteger M, we ca decompose H(z) as: z -. -M - -M H(z) = h z + z h z + = - his ca be expressed as: M = - M+ + z -(M-) = - h M+ M- z -M. where H(z) M = m= 0 z -m E (z M ), E - (z) = e z, with e = h + = M m, 0 m M -. he last equatio for H(z) is called the (type ) polyphase represetatio of H(z) with respect to M, ad E (z), 0 m M -, are the polyphase compoets of H(z). We ow apply the polyphase represetatio to show how the computatioally efficiet realizatio of Figure 3(b) ca be derived from Figure 3(a) for = M. he sigal at the chael iput i Figure 3(a) is give, for f i = i/, by x = M- i= 0 = which ca be rearraged as A (i) h -M e jπi/m, x = = h M- -M i= 0 A (i) e jπi/m.
A chage of otatio =l M + m allows us to itroduce the polyphase compoets of h. With the otatios x = x ad h = h, m = 0,,, M-, we obtai: l M+ m l l M+ m l x l M- (i) jπim/m = hl - A e = hl = i= 0 = where a, 0 m M -, is the iverse discrete Fourier trasform (IDF) of A (i), 0 i M -. - a, Refereces [] M. Bellager, G. Boerot, ad M. Coudreuse, Digital filterig by polyphase etwor: applicatio to sample-rate alteratio ad filter bas, IEEE ras. Acoust., Speech, Sigal Processig, vol. ASSP-4, o., pp. 09-4, Apr. 976. [] P.P. Vaidyaatha, Multirate systems ad filter bas. Eglewood Cliffs, J: Pretice-Hall, 99. [3] S.D. Sadberg ad M.A. zaes, Overlapped discrete multitoe modulatio for high speed copper wire commuicatios, IEEE J. Select. Areas Commu., vol. 3, o. 9, pp. 57-585, Dec. 995. [4] M.V. Eyuboglu ad G.D. Forey, rellis precodig: Combied codig, precodig ad shapig for itersymbol iterferece chaels, IEEE ras. Iform. heory, vol. 38, pp. 30-34, Mar. 99. [5] M. Isasso, P. Deutge, F. Sjöberg, S.. Wilso, P. Ödlig, ad P.O. Börjesso, Zipper A flexible duplex method for VDSL, Proc. 998 IEEE It. Cof. o Commuicatios, paper S9-7, Atlata, GA, Jue 998. [6] G.H. Im ad J.J. Werer, Badwidth-efficiet digital trasmissio over ushielded twisted-pair wirig,, IEEE J. Select. Areas Commu., vol. 3, o. 9, pp. 643-655, Dec. 995. [7] Commercial Buildig elecommuicatios Cablig Stadard, EIA-IA Stadard P 840, EIA/IA 568-A, Mar. 994. [8] J.M. Cioffi, Asymmetrical digital subsriber lies, i he Commuicatios Hadboo, J.D. Gibso (Ed.), CRC Press Ic., pp. 450-479, 997.