Distributed Margin Optimization Using Spectrum Balancing in Multi-user DSL Systems

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1 Distributed Margi Optimizatio Usig Spectrum Balacig i Multi-user DSL Systems Domiique Würtz, Aja Klei Commuicatios Egieerig Lab Techische Uiversität Darmstadt Darmstadt, Germay Marti Kuipers Noia Siemes Networs GmbH & Co. KG Berli, Germay Abstract Optimizig the SNR margi i Dyamic Spectrum Maagemet (DSM) systems ca sigificatly improve stability of DSL etwors i the presece of slowly varyig oise. I this wor, we preset a ovel distributed approach to the margi optimizatio problem where the etwor operator specifies the desired ratios of user margis relative to each other ad, give this costrait, the Spectrum Balacig algorithm sees the solutio with the largest sum margi. By employig partial Lagragia relaxatio of a rate-adaptive problem each user optimizes its ow trasmit power spectrum while the Spectrum Maagemet Ceter (SMC) iteratively updates the target margis of each user so that the system evetually coverges to a optimal power allocatio with desired margi ratios. Numerical simulatios demostrate fast covergece ad show that sigificatly higher margis ca be achieved with our approach compared to purely selfish optimizatio. I. INTRODUCTION With costatly growig badwidth demads of curret Digital Subscriber Lies (DSLs) systems, the problem of fared crosstal (FEXT) betwee copper wires i telephoe biders has become icreasigly importat [1]. Dyamic Spectrum Maagemet (DSM) Level 1 ad 2 attempt to miimize iterferece betwee loops (i the followig also iterchageably referred to as users) by optimizig the trasmit power spectra of modems employig Discrete Multitoe (DMT) trasmissio [2]. Not oly does this allow a sigificat improvemet of the overall system performace, but it eables also accurate cotrol of the quality of service (QoS) parameters per loop based o the actual measured lie coditios rather tha relyig o overly pessimistic worst-case assumptios as doe i traditioal static spectrum maagemet. Cetralized DSM Level 2 algorithms lie Optimal Spectrum Balacig (OSB) [3] achieve optimal performace, but are impractical due the high complexity. Recetly, low-complexity schemes such as Distributed Spectrum Balacig (DSB) [] have bee developed which typically achieve a performace close to OSB ad allow a distributed optimizatio with low complexity similar to Iterative Water-Fillig (IWF) [5] i DSM Level 1. Stability i commuicatio systems is characterized by the probability of outage which is the probability that a system caot operate with a specified QoS aymore due to degradig chael coditios. I DSL etwors, stability is affected by slowly time-varyig oises origiatig from RFI igress or other loops i the bider beig switched o or off. These so-called quasi-statioary oises are typically dealt with by icludig a sigal-to-oise ratio (SNR) safety margi i the bit-loadig of DMT modems at iitializatio of a DSL sessio [1]. As such, the SNR margi is the ey parameter to cotrol log-term stability of DSLs. Margi-adaptive DSM assigs a fixed data rate to each user, usually chose accordig to some service level agreemet, ad uses the available power to maximize the SNR margis of each user. I practice, due to the vastly differet oise coditios across lies, it is ot desirable to assig equal margis to all users. How to exactly distribute margis across users i order to optimize overall system stability is a importat topic ad various wors cosider differet approaches to margi optimizatio with differet optimizatio goals. I [6], a multi-user margi optimizatio algorithm based o IWF is proposed which aims at maximizig the miimum SNR margi amog users i the system. The approaches preseted i [7], [8] directly optimize the outage probabilities i DSM Level 1 ad 2 systems, respectively, assumig statistics of quasi-statioary oise dyamics are available at the Spectrum Maagemet Ceter (SMC). I [9], a multi-user margi maximizatio (MMM) algorithm is proposed which fids a power allocatio with either DSM Level 1 or 2 coordiatio where the per-user margi values obey a ratio accordig to priorities specified by the provider. Ufortuately, this meta-algorithm is ot well suited for distributed spectrum optimizatio as it requires repeated executio of a base DSM algorithm at the SMC. For the same reaso, as admitted by the authors, it geerally suffers from a high computatioal complexity. I this wor, we preset a ovel low-complexity, distributed approach to solvig the margi optimizatio problem from [9] with DSM Level 2 coordiatio. Here, the etwor operator specifies the desired ratios of user margis relative to each other ad, give this costrait, the Spectrum Balacig algorithm sees the solutio with the largest sum margi. By employig partial Lagragia relaxatio of a rate-adaptive problem each user optimizes its ow trasmit power spectrum ad eforces its target rate while the SMC iteratively updates the target margis of each user so that the system evetually coverges to a optimal power allocatio with desired margi ratios. The remaider of the paper is structured as follows: Sec-

2 tio II defies the system model for the DSL bider. The spectrum maagemet problem for margi maximizatio is itroduced ad aalyzed i Sectio III. The proposed distributed Spectrum Balacig algorithm is preseted i Sectio IV. Fially, Sectio V discusses performace results for the ew scheme obtaied from umerical simulatios. γ 2 γ (γ t ) γ(γ t ) γ II. SYSTEM MODEL A N-user DSL bider is cosidered where users N employ DMT trasmissio over K subcarriers (toes). Assumig perfect sychroizatio of the trasmitters ad a sufficietly log cyclic prefix, the toes K ca be modeled as K parallel iterferece chaels where crosstal from other loops is treated as oise. Let g, = h, 2 deote the chael power gai of user ad g,m = h,m 2 ( m) the FEXT power couplig gai from disturber m to victim loop o toe. Let s deote the allocated trasmit power of user o toe ad σ the sum power of received alie oise ad iterferece ot maaged by the DSM system. The the sigal-to-iterferece-plus-oise ratio (SINR) at receiver o toe is obtaied by SINR = g, s m g,m s m + σ. (1) Protectio agaist oise variatio is achieved by icorporatig a SNR margi γ 1 (i liear scale) i the bit-loadig at iitializatio of a DSL sessio. Usig the Shao-gap approximatio [10], the umber of bits per symbol that ca be loaded o toe of user is give by ( ) b (γ) = log SINR (2) γγ where Γ > 1 deotes the SNR gap to capacity [10] which is a fuctio of the code ad target bit error rate (BER). The aggregate power P ad data rate R of user are give by P = s ad R (γ) = f s b (γ), (3) K K respectively, where f s is the symbol rate of the DMT system. III. PROBLEM STATEMENT Let µ = (µ 1,..., µ N ) (µ > 0 N ) deote the vector of positive per-user priorities specified by the etwor operator ad let s = {s K, N } deote the set of trasmit power levels to be optimized, respectively. A target data rate Rtarget is assiged to user by the etwor operator. Furthermore, the power allocatio of each user may ot exceed a spectral mas s,mas ad a maximum aggregate power Pmax i order to comply with DSL stadards. Give these costraits, this wor cosiders the multi-user margi optimizatio problem max γ s, γ () subject to R (µ γ) Rtarget P Pmax 0 s s,mas N, K µ γ t Fig. 1: Two-user margi regio with optimal margi tuple γ for priority vector µ. Projectio γ (γ t ) of operatig poit γ(γ t ) o µ. where γ is the overall positive margi scalig factor to be maximized. The oliear depedecy of R (µ γ) i the rate costraits o the variable γ to be maximized greatly complicates the problem compared to rate-adaptive spectrum maagemet problems as it itroduces a additioal couplig both across toes ad users. For give SINR ad target rate Rtarget, the effective margi γ of user is foud by solvig γ 1 R (γ ) R target = 0 (5) for γ. From (2) ad assumig b to be cotiuous, it follows that (5) establishes a strictly mootoe oe-to-oe mappig betwee rate R (γ ) ad margi γ of user. This relatio for a sigle user ca directly be exteded to the multi-user case: give a tuple (Rtarget N ) of target rates, a tuple of achieved rates (R (γ ) N ) is uiquely mapped to a tuple γ of achieved margis. From this follows, as explaied i [6], that a rate regio which represets the set of all feasible operatig poits, i.e. rate tuples, achieved by some DSM scheme ca be trasformed via (5) ito a equivalet margi regio. Furthermore, due to mootoicity of (5), the Pareto optimal poits o the boudary of the rate regio are mapped to poits o the boudary of the margi regio. The margi tuple γ correspodig to the uique optimal solutio of Problem () is the itersectio poit of the boudary of the margi regio ad the straight lie from the origi alog the vector µ [9], as depicted i Figure 1. Note that the special case µ 1 = µ 2 = = µ N correspods to the max-mi problem from [6] where the miimum margi amog users is to be maximized. To further characterize γ, cosider a stadard sum rate maximizatio problem max R (γt ) (6) s N subject to R (γt ) Rtarget P Pmax 0 s s,mas N, K for fixed target margis γ t = (γ t N ). Let γ(γ t ) = (γ (γ t ) N ) deote the effective margis accordig to (5) usig spectra s obtaied by solvig Problem (6) with give

3 γ t. It ca be easily verified that γ t = γ(γ t ), i.e. γ t is a fixed poit of the mappig γ(γ t ), iff γ t lies o the boudary of the margi regio. Now defie a ew mappig γ (γ t ) = proj µ γ(γ t ) = γ(γ t) µ µ 2 µ (7) which is the orthogoal projectio of γ(γ t ) o µ, as illustrated i Figure 1. It is ot hard to see that γ (γ t ) has γ as its uique fixed poit. This observatio is the ey i our iterative algorithm to fid the optimal value γ of Problem (). IV. ALGORITHM I the followig, we describe our ovel Spectrum Balacig algorithm to solve Problem () i a distributed fashio. Based o the observatios from Sectio III, we propose a iterative approach where i each iteratio the rate-adaptive Problem (6) is solved for a fixed vector γ t of target margis ad γ t is updated so that it evetually coverges to the fixed poit γ of the mappig γ (γ t ). Although the rate-adaptive Problem (6) is o-covex, ow low-complexity Spectrum Balacig algorithms ca be employed to yield ear-optimal spectra. Sice we are aimig at a distributed solutio, schemes lie DSB seem attractive. Lie other sub-optimal spectrum optimizatio schemes, DSB tacles the o-covex problem by iterative covex approximatio i a way that each user updates its ow trasmit power spectrum based oly o locally available iformatio as well as limited iformatio ifrequetly exchaged betwee users. Ufortuately, DSB as proposed i [] caot be directly applied to Problem (6) because DSB oly attempts to maximize a weighted sum rate without accoutig for the target rate costraits of each user. Therefore, we ow develop a exteded versio of DSB based o partial Lagragia relaxatio of the rate costraits where each user eforces its ow target rate by locally tuig its weight factor i the Lagragia dual of Problem (6). Followig a similar approach to [11], [12] which focus o distributed power miimizatio subject to data rate costraits, we obtai a covex approximatio of Problem (6) which ca be solved locally by user to update its ow spectrum by carryig out the followig steps: 1) Partial Lagragia relaxatio of the rate costraits of other users p, resultig i a ew objective ω p[ R p (γt ) Rtarget] p. p N R p (γ t ) + p 2) Fixig variables s p (p, K) ad Lagrage multipliers ω p associated with the rate costraits of other users p. 3) Approximatio of the o-cocave parts of the objective resultig from the above steps by a affie fuctio m W ms + U i the variables s ( K). The approximatio poit is chose as the optimal values of s from a previous iteratio. ) Droppig ay additive terms of the objective that are costat i s ( K), such as U, as these do ot ifluece the optimal solutio. Applyig all steps results i the covex sub-problem for user give by where max R (γ {s K} t ) W s (8) K subject to R (γ t ) R target P P max 0 s s,mas K R (γ t ) = f s K log 2 ( γ t Γ g, it s ) (9) W = f s g m, log(2) V m (10) m / ( 1 V = (1 + ω ) it 1 ) rec (11) it = g,m s m + σ (12) m rec = Γ 1 g, s + it. (13) it ad rec deote the measured iterferece power ad total received power of user, respectively, ad are both available i stadard compliat DSL modems. The term W ca be iterpreted as the price of icreasig power o toe ad hece icreasig iterferece o other lies. Problem (8) ca be efficietly solved via dual optimizatio. Based o the Karush-Kuh-Tucer (KKT) statioarity coditio, it ca be show that for fixed dual variables (ω, λ ) which correspod to rate ad power costrait, respectively, the values of s that maximize the dual objective are foud aalytically accordig to [ fs s log(2) = (1 + ] s ω,mas ) λ + W it Γ 1 g, (1) where [x] a 0 meas mi(max(x, 0), a) []. Algorithm 1 provides a possible realizatio of the local update scheme where the optimal value for ω is foud usig projected subgradiet method with step rule [ ( ω ω + κ Rtarget R + (γt ))] (15) where κ is the search step size ad [ ] + is the projectio o R + 0. Note that depedig o the curret iterferece it ad γ t, a feasible solutio might ot exist due to the data rate costrait. This has to be cosidered i the choice of a covergece criterio for {s } i lie 17 by quittig the loop if icreasig ω does ot chage s. However, the user power costrait ca ad must always be satisfied which is guarateed by the bisectio search which fids the optimal value of λ for fixed ω. Here, λ max should be chose such that λ max f s (1 + ω ). 0

4 Algorithm 1 Local algorithm of user 1: repeat 2: Receive messages W ( K) ad γ t from SMC 3: repeat : Iitialize λ λ max /2 ad λ λ /2 5: repeat 6: Update {s } usig (1) 7: Calculate power P usig (3) 8: if P > Pmax the 9: λ λ + λ 10: else 11: λ λ λ 12: ed if 13: λ λ/2 1: util λ coverged for give ω 15: Calculate rate R usig (9) 16: Update ω usig (15) 17: util spectrum {s } coverged 18: Calculate messages V ( K) usig (11), effective margi γ usig (5) ad sed to SMC 19: util global covergece of spectra s I order to improve the local approximatio coefficiets W ad steer the system to a solutio that satisfies the margi ratios specified by µ, assistace of the SMC is required accordig to Algorithm 2. The SMC receives messages V from users N which cotai local iformatio ad are computed usig (11). The prices W are the updated usig (10) ad distributed to the respective user. For further details about the derivatio of update rules for W ad V, see []. Havig obtaied a vector of effective margis γ(γ t ) resultig from rate-adaptive spectrum optimizatio as described above, ew target margis for the ext iteratios are foud by settig γ t γ (γ t ) with γ (γ t ) give by (7). A crucial advatage of our proposed scheme is that update of all quatities i the system such as target margis γ t, spectra s ad prices W ca be carried out totally asychroously without affectig covergece. This i particular meas that the distributed rate maximizatio accordig to Problem (6) does ot eed to have fully coverged before updatig γ t. While a geeral proof of covergece is beyod the scope of this paper, extesive umerical simulatios asserted reliable ad ofte very fast covergece i all cosidered scearios. Uiqueess of the fixed poit γ of γ (γ t ) assures that, oce coverged, we actually achieve the optimal margi tuple γ with desired priorities µ. I the above cosideratios, the margis γ t ad γ were assumed i liear scale. I case the provider wishes to defie the ratios of margis i logarithmic scale, i.e. db, our scheme ca be trivially modified by applyig the above update rule for γ t to the margi values i db, aturally leadig to a differet solutio tha optimizatio of liear margis. Algorithm 2 Cetral steerig loop of SMC 1: γt 1 N 2: repeat 3: Receive V ( K, N ) alog with γ(γ t) : Compute W ( K, N ) usig (10) 5: Update γ t proj µ γ(γ t ) 6: Trasmit W ( K) alog with γ t to lie N 7: util global covergece of spectra s Fig. 2: 8-user VDSL2 upstream sceario V. SIMULATION RESULTS I this sectio, the performace of the proposed distributed algorithm which we refer to here as proportioal margi (PM) DSB is studied ad compared to other state-of-the-art schemes. For this, MATLAB simulatios i a 8-user VDSL2 upstream sceario as depicted i Figure 2 have bee carried out. Due to the ear-far problem, this sceario offers sigificat potetial gais from Spectrum Balacig while IWF is ow to perform poorly due to the selfish optimizatio of each user. Table I summarizes the relevat system parameters used for the simulatios. Figure 3 shows the achieved operatig poits for differet algorithms ad priority vectors µ = (µ short, µ log ) i the 2- dimesioal margi space. Note that the ratios are specified for margis i db rather tha liear margis. Target data rates 10 Mbps ad 1 Mbps are assiged to short (600 m) ad log (1200 m) loops, respectively. MMM+IWF ad MMM+ISB are the schemes from [9] usig DSM Level 1 ad 2 coordiatio, respectively, whereas MHM is the DSM Level 1 algorithm from [6] which provides a max-mi solutio to the multi-user margi optimizatio problem. It shows that both MMM+ISB ad our PM-DSB scheme achieve the same operatig poits o the Pareto boudary ad as such clearly outperform MMM+IWF ad MHM. Ulie MMM+ISB, PM-DSB allows a low-complexity distributed optimizatio which does ot rely o repeated executio of computatioally expesive feasibility checs. TABLE I: Simulatio parameters cable type 26-AWG [13] FEXT model ETSI 1% worst-case without FSAN sum [13] bacgroud oise level 10 dbm/hz alie oise model ETSI MD EX [13] VDSL2 bad profile 998-ADE17M2x-B without US0 [1] symbol rate f s Hz umber of toes K 096 toe spacig f.3125 Hz SNR gap (for BER= 10 7 ) 9.8 db codig gai 3 db

5 SNR margi of log loops i db µ = (0., 0.6) feasible regio for IWF feasible regio for DSB/ISB MMM+IWF PM-DSB / MMM+ISB MHM µ = (0.6, 0.) µ = (0.5, 0.5) SNR margi of log loops i db i = 3 feasible regio for DSB effective margis γ(i) target margis γ t (i) 5 6 µ = (0.5, 0.5) SNR margi of short loops i db Fig. 3: Achieved operatig poits usig various margi optimizatio techiques. Figure illustrates the covergece behavior of our proposed iterative PM-DSB whe approachig the equal margi solutio. The algorithm starts at iteratio i = 0 with all users utilizig the highest power levels allowed by the spectral mass. At each iteratio step i = 1, 2,..., all users sequetially update their trasmit spectrum usig the local algorithm ad the SMC afterwards computes ew W values alog with ew target margis γ t for the ext iteratio. The resultig effective margis γ(i) ad target margis γ t (i) at each iteratio step i are mared by the squares ad diamod shapes, respectively. Margis for iteratios 0 2 are ot show sice the poits lie too far away from the optimal poit to allow appropriate visualizatio. For referece, usig the iitial spectra, the resultig effective margis of short ad log loops, respectively, are give by γ(0) = (11.5 db, 10.1 db). As ca be see i the figure, PM-DSB requires oly about five iteratios to fid a operatig o the Pareto boudary that satisfies the desired margi ratio give by µ. Fially, more simulatios have bee carried out i up- ad dowstream scearios for various choices of µ target rates R. Although with ratio vectors µ poitig ear the corers of the margi regio, the umber of required iteratios icreases i some cases, covergece has bee observed i all cases, assertig the robustess of our method. VI. CONCLUSION I this wor, we preseted a ovel low-complexity, distributed Spectrum Balacig scheme for multi-user margi optimizatio with specified ratios of user margis. As such, our approach overcomes some of the mai drawbacs of state-of-the-art algorithms lie MMM+ISB. Simulatio results attested optimal performace i problematic DSL deploymet scearios as well as fast covergece to the optimal solutio. Also, it showed that Spectrum Balacig clearly outperforms DSM Level 1 coordiatio i terms of achievable margis i 5 i = SNR margi of short loops i db Fig. : Effective margis ad target margis at iteratio steps i = 3,..., 6 of PM-DSB. such scearios. REFERENCES [1] P. Golde, H. Dedieu, ad K. Jacobse, Fudametals of DSL techology. Auerbach Publicatios, [2], Implemetatio ad applicatios of DSL techology. Auerbach Publicatios, [3] R. Cedrillo, W. Yu, M. Mooe, J. Verlide, ad T. Bostoe, Optimal Multiuser Spectrum Balacig for Digital Subscriber Lies, IEEE Trasactios o Commuicatios, vol. 5, o. 5, pp , May [] P. Tsiaflais, M. Diehl, ad M. Mooe, Distributed Spectrum Maagemet Algorithms for Multiuser DSL Networs, IEEE Trasactios o Sigal Processig, vol. 56, o. 10, pp , October [5] W. Yu, G. Giis, ad J. Cioffi, Distributed Multiuser Power Cotrol for Digital Subscriber Lies, IEEE Joural o Selected Areas i Commuicatios, vol. 20, o. 5, pp , ju [6] S. Paigrahi, Y. Xu, ad T. Le-Ngoc, Multiuser margi optimizatio i digital subscriber lie (DSL) chaels, IEEE Joural o Selected Areas i Commuicatios, vol. 2, o. 8, pp , August [7] S. Jagaatha, C. S. Hwag, ad J. Cioffi, Margi Optimizatio i Digital Subscriber Lies Employig Level-1 Dyamic Spectrum Maagemet, i Proc. IEEE Iteratioal Coferece o Commuicatios, May 2008, pp [8], Margi Optimizatio i Digital Subscriber Lies Employig Level-2 Dyamic Spectrum Maagemet, i Proc. IEEE Iteratioal Coferece o Commuicatios, May 2008, pp [9] M. Moteiro, A. Gomes, N. Lidqvist, B. Dortschy, ad A. Klautau, A Algorithm for Improved Stability of DSL Networs Usig Spectrum Balacig, i Proc. IEEE Global Telecommuicatios Coferece, December 2010, pp [10] T. Starr, J. M. Cioffi, ad P. J. Silverma, Uderstadig Digital Subscriber Lie Techology. Upper Saddle River, NJ, USA: Pretice Hall PTR, [11] M. Wolerstorfer, D. Statovci, ad T. Nordstrom, Dyamic spectrum maagemet for eergy-efficiet trasmissio i DSL, i Proc. 11th IEEE Sigapore Iteratioal Coferece o Commuicatio Systems, November 2008, pp [12] D. Würtz, A. Klei, ad M. Kuipers, Hybrid Rate ad Power Adaptive Distributed Spectrum Optimizatio for DSL Systems, i Proc. 16th Iteratioal OFDM Worshop, September [13] ETSI TS , V1.1.1, Access Termials Trasmissio ad Multiplexig (ATTM); Access trasmissio system o metallic pairs; Very High Speed digital subscriber lie system (VDSL2), Jauary [1] Very high speed digital subscriber lie trasceivers 2 (VDSL2), ITU-T Rec. G.993.2, February 2006.