IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010 3717

Semiblind Spectrum Balancing for DSLRodrigo B. Moraes, Student Member, IEEE, Boris Dortschy, Aldebaro Klautau, Senior Member, IEEE, and

Jaume Rius i Riu

Abstract—Digital subscriber lines (DSLs) technology is vastlyused for high-speed data transmission. Crosstalk is one of the mainproblems in DSL. The research community has done extensivework on how to optimize spectrum allocation across DSL frequen-cies and mitigate crosstalk, a subject that has been called dynamicspectrum management (DSM). This text presents a novel DSMsolution, referred to as semiblind spectrum balancing (2SB). 2SBperforms an optimization process with a virtual line, a fictitiousline that represents to each user the damage it causes for all otherusers. With the aid of message exchanges between modems and acentral agent, the method will adjust the virtual line’s parametersso that it represents the real crosstalk scenario in the binder. Inthis paper, we provide the conditions for how such a situation canbe achieved and show that it can do so with semicentralization,low complexity and limited crosstalk channel information—onlythe ratios between crosstalk channels are necessary. Performanceis very close to optimal.

Index Terms—Digital subscriber lines (DSLs), discrete multi-tone, dynamic spectrum management (DSM), optimization.

I. INTRODUCTION

D IGITAL subscriber lines (DSL) have reached the mark ofhundreds of millions users across the world, becoming

the most important technology for high-speed data transmission.The question of whether twisted-pair cable transmission will beable to maintain its lead will be, in a broad sense, answered byhow future service providers will handle the crosstalk problemin that century-old media. The research community has paid agreat deal of attention to this problem. The body of work thattries to optimize power allocation across the DSL spectrum so

Manuscript received March 12, 2009; accepted February 02, 2010. Date ofpublication March 15, 2010; date of current version June 16, 2010. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Amir Leshem. This work was partially supported by the Researchand Development Center, Ericsson Telecommunications S.A., Brazil. Some au-thors acknowledge financial support from the Swedish Agency for InnovationSystems, VINNOVA. This paper was presented in part at the IEEE InternationalCommunications Conference (ICC), Beijing, China, 2008.

R. B. Moraes was with Signal Processing Laboratory, Federal Universityof Pará, Belém, PA, Brazil. He is now with the SISTA/ESAT Laboratory,Katholieke Unviversiteit Leuven, 3001 Leuven-Heverlee, Leuven, Belgium(e-mail: rodrigo.moraes@esat.kuleuven.be).

B. Dortschy is with Ericsson Research, Broadband Technologies, EricssonAB, 16480 Stockholm, Sweden (e-mail: boris.dortschy@ericsson.com).

A. Klautau is with the Signal Processing Laboratory, Federal University ofPará, 66075.110 Belém, PA, Brazil (e-mail: aldebaro@ufpa.br).

J. R. i Riu is with Ericsson Research, Broadband Technologies, Ericsson AB,16480 Stockholm, Sweden, and with the Department of Electrical and Infor-mation Technology at Lund University, Lund, Sweden (e-mail: jaume.rius.i.riu@ericsson.com).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2045553

that the effect of crosstalk is minimized has been called dynamicspectrum management (DSM)1 [2], [3].

DSL in its most important types adopts discrete multi-tonetransmission (DMT), in which the available band is divided in anumber of narrow independent subchannels or tones. The DSMproblem that has received most attention from academia con-siders a synchronous situation, i.e., when all users’ transmissionis perfectly synchronized—cyclic prefixes and data blocks aretransmitted at the same time for all participating users in the net-work [4]—and it is this scenario we consider in this work. Allsolutions to the DSM problem will be qualified in four differentaspects: rate performance, centralization (if processing is donein a single point or distributed across the network), computa-tional cost and crosstalk channel information. This text presentsa scheme with good results on all four aspects: the semiblindspectrum balancing (2SB). The algorithm makes use of an op-timization routine with a virtual line (VL), a fictitious line thatrepresents for a given user the damage it causes to all other users.With the aid of limited message exchanges between modemsand a central coordinating agent, the VL can be adapted to meetas faithfully as possible the actual crosstalk characteristics inthe binder. In this text we provide the conditions for how suchscheme can yield a clever power utilization across the spectrumand show that it is capable of doing so with limited crosstalkchannel information—for scenarios with more than two users,only the ratios of the crosstalk channels are needed.

The remainder of this paper is organized as follows. Section IIintroduces notation and the problem of interest and discussessome of the previous solutions. Section III presents the proposedsolution. Section IV contains simulation results, and Section Vpresents final comments.

II. PROBLEM STATEMENT AND PREVIOUS SOLUTIONS

As mentioned, the most popular types of DSL standards, suchas ADSL and VDSL, adopt DMT modulation [5], which is thetechnique assumed in this work. The idea that constitutes thecore of this technique is the division of the available spectra ina number of -spaced independent subchannels or tones.

Given an -user DMT system with tones, letbe a matrix in which is the transmitter PSD of the th

user on tone . The th column representsthe power allocation of user across all tones and the th row

contains the PSD levels of all users in agiven tone. Let be the background noise’s PSD of the thuser on tone . Also, let be the crosstalk transfer functionbetween transmitter and receiver at tone . Both and

1Literature also refers to DSM as a signal-level coordination paradigm. See,e.g., [1] and references therein.

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3718 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

TABLE ICOMPARISON OF PREVIOUS SOLUTIONS ACCORDING TO THE FOUR PROPOSED FEATURES

are normalized by being divided by the squared mag-nitude of the direct channel of user . The bit loadingfor user on tone is a function of and is defined as

(1)

where is the SNR gap, which depends on the desired BER,coding gain and noise margin [5], and

(2)

is the total crosstalk noise at the receiver (uppercase), whichis a sum of the individual crosstalks (lowercase). The total bitrate of line is , where is the symbol rate.Throughout this work the practical issue of loading a tone withan integer number of bits is ignored.

The problem of interest is that of finding a matrix whichmaximizes the data rates of all users in the network given apower budget for each user. This problem has been referred to inliterature as the rate adaptive (RA) problem [2]. In mathematicalform it can be written as the maximization of the rate of one userin the network (in our notation, this will always be user 1), whilethe others achieve a minimum specified data rate

(3)

and

in which and are the total actually allocated powerand maximum power available to user , respectively; andand are the achieved and minimum desired rate for user

, respectively.The optimization problem in (3) can be rewritten as a

weighted rate sum maximization [6]

(4)

and

in which the values can be interpreted as priorities givento each user. They should be adjusted such that they are justenough for achieving minimum rates for all users. For

convenience, it is considered that , where is aconstant. If , the ’s are more straightforwardly inter-preted as proportions. The first user, which should have its ratemaximized, should get the “rest”, .

The first important solution to the DSM problem, the itera-tive waterfilling (IWF), was due to Yu et al. [7]. An importantbreakthrough came with the work of Cendrillon et al., the op-timal spectrum balancing (OSB) [6], [8]. OSB decoupled theproblem with the aid of Lagrange multipliers in a way that itcould be solved in each tone independently. OSB could provideprovably optimal results with reasonable complexity for small

and full centralization. The challenge was then to profit fromthe ideas OSB had put forward to come up with more practicalschemes.

Four solutions will be discussed here. The first is that byCendrillon and Moonen [9] and Lui and Yu [10], both pre-senting the same solution, the iterative spectrum balancing(ISB). Papandriopoulos and Evans proposed SCALE in [11].Cioffi et al. proposed the band-preference (BP) method [12]and, finally, Cendrillon et al. suggested the autonomous spec-trum balancing (ASB) [13]. We will later comment the kind ofstatic pricing adopted in ASB, which will be important to thiswork. Table I summarizes the classification of the algorithmsaccording to the aspects already mentioned.

Some other works have analyzed and optimized these solu-tions, specially OSB and IWF. For OSB, see, e.g., [14]–[18].For IWF, see [19]–[22].

DSM research has progressed faster than we can follow, andwe should mention that other interesting proposals have beenpublished in recent years, for example, [23] and [24].

III. PROPOSED SOLUTION: 2SB

2SB will adjust the network with a series of message ex-changes between modems and a central coordinating agent,which we will call spectrum management center (SMC).Modems make an optimization with the already introducedVL, which has its parameters adjusted so as to emulate the realcrosstalk situation in the binder.2

This part of the text will first introduce in Section III-A atwo-user near–far scenario, the typical testing ground for DSMsolutions. We then extend the algorithm for a -user case inSection III-B.

2Notice that the algorithm presented on this work differs quite a lot from theone published in [25]. What is presented here is an improved version of ourprevious proposal.

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MORAES et al.: SEMIBLIND SPECTRUM BALANCING FOR DSL 3719

Fig. 1. Downstream ADSL two-user near–far scenario. Crosstalk from the RTto the CO may be prohibitive.

A. 2SB in a Two-User Scenario

The ADSL downstream scenario depicted in Fig. 1 encom-passes most of the complexity of the problem. Two users, oneconnected to the central office (CO) and the other to a remotelydeployed terminal (RT), share the same cable binder. Due to theproximity of the transmitter (Tx) of the RT to the receiver (Rx)of the CO, crosstalk in this direction will be strong, to the pointwhere the transmission on the CO may be seriously compro-mised. A good solution is one in which the RT protects the COby reducing power in the frequencies the CO is active. In thisfashion, crosstalk is reduced and the binder overall capacity isbetter utilized.

The OSB algorithm solves this problem optimally by usingthe weighted rate sum of (4) and Lagrange multipliers for de-coupling the problem across frequency. Unfortunately, the op-timality comes at high cost, as seen in Table I.

A different solution is that of ASB, which introduced theconcept of the reference line. The reference line of the th useris a fictitious line that works as a type of pricing representingthe damage to be caused to all other users in the network. Agood analogy is with the Thèvenin-equivalent from circuittheory: The reference line should represent for user all otherusers in the network, just as the single Thèvenin impedancerepresents any combination of impedances in a circuit. In thisfashion, the optimization problem can be distributed acrossthe network. Each modem should solve a problem as in (4)in which the goal should be to maximize the data rate of itsreference line. However, for the near–far case, this raises thequestion of how to set the parameters of the RT’s referenceline. For the two-user near–far scenario, ASB will find agood spectrum design if the reference line of the RT closelyresembles the real crosstalk damage in the CO. Actually, ASBflawlessly emulates ISB if perfect estimates of the crosstalkchannel , the CO’s background noise, , and powerprofile, , are available. However, these are complicated toacquire. DSL crosstalk channels vary slowly with time and canbe eventually acquired with specialized channel measurementsor heuristic calculations (which are known to be highly inaccu-rate). There has been recent progress in standardized crosstalkchannel estimation methods (e.g., [26]), but it should be notedthat crosstalk information may be unavailable or inaccurate.Background noise and power profile usually change with time,which makes an accurate representation dependent on constantupdates.

2SB will use an optimization process with a VL, which isbasically the same concept as the reference line, but not static.For the two-user scenario, the VL of the RT will be adjusted sothat it accurately represents the impact on the CO. At this point,we need to define the VL mathematically. All users should have

an independent VL, with different channel characteristics andachieved rate. The VL is defined by three parameters: PSD ,channel gain and background noise . Bit loading for theVL of user will be calculated by

(5)

Power allocation for every tone and user should be foundwith an optimization of each user with its respective VL. Theoptimization should maximize the data rate of the VL, or, inother words, it should minimize the damage to be caused toit. In this fashion, a good equilibrium between network andindividual performance is achieved. The problem each modemshould solve is

and (6)

which, as (3) and (4), can be written as

(7)

in which is the weight or priority user has with respectto the VL. For the two-user scenario, the RT should find theminimum possible , which ranges from 0 to 1, so that its min-imum data rate is achieved. User 1, the CO, is to have its ratemaximized and thus (the VL does not play a role in thepower loading of the CO, and it basically does waterfilling). The

are Lagrangian variables which serve as control for power.They should be adjusted such that . We usebisection search for finding the appropriate values of both and

. Also, it is considered that power allocation is done in par-allel—all users allocate power at the same time.

We now focus on crosstalk. Since crosstalk from the Tx of theRT to the Rx of the CO should be kept as small as possible, onehas to find a way to quantify it. The simplest and most straight-forward way to do so is having access to both the PSD of the RTand the corresponding crosstalk channel, i.e.,

(8)

which is how previous solutions did it. We propose to evaluatecrosstalk in an indirect manner, through the crosstalk damageratio (CDR).

Definition 1: The CDR is a measure of the impact of crosstalkon bit loading, and is defined as

(9)

where is the bit loading of the CO on tone whenconsidering background noise plus crosstalk and is thebit loading when considering only background noise.

It is considered that every user has an estimate of its back-ground noise. The CDR ranges in a continuum from 0 to 1. Thecloser it is to zero, the less significant the crosstalk damage.The closer it is to 1, the higher the crosstalk damage. Con-sider a simple example: A given user on a specific tone loads

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3720 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

Fig. 2. Illustration of the crosstalk damage ratio (CDR). (a) If close to 1,crosstalk is big, if close to zero it is not. (b) 2SB will define an acceptableregion for CDR.

a certain amount of power which, when only considering back-ground noise only, amounts to 10 bits. If, when consideringcrosstalk, bit loading goes down to 8 bits, CDR would equal 0.2,which could mean that crosstalk in this tone is bearable. If bitloading drops to 2 bits, CDR would be 0.8, which indicates thattransmission on this tone is essentially destroyed by crosstalk.Fig. 2(a) illustrates this.

Definition 2: Two victim lines and that do not interferewith one another are CDR-equivalent when a common interferer

causes the same CDR in both and , i.e.,

(10)

The condition for CDR-equivalence is easily shown to be

(11a)

(11b)

The important fact here is that, in the condition of CDR-equiva-lence between victims and , the solution of (7) for interferer

optimizing with or is the same. That is easily demon-strated by showing that

and that, as a consequence of the definition of CDR-equivalence,. We argued before

that ASB emulates ISB if the parameters and ofthe reference line/VL of the RT perfectly match those of the CO.We now see that this condition can be loosened: the CO and thereference line/VL of the RT only need to be CDR-equivalent,i.e., the ratios in (11a) need to be equal. This simple fact is oneof the main building blocks of our proposal. We shall now seehow the VL of the RT can achieve (11a) without the knowledgeof or .

Before we proceed, we introduce the approximations used inthe following argument and in other points in this text.

Approximation 1: , where isthe set of tones where user is active, so that

(12)

Approximation 2: The product of crosstalks is zero, i.e.,.

Approximation 3: To estimate crosstalk on the CO, we shallconsider a high SNR and low CDR regime for the VL of the RT,i.e.

(13)

so that power loading on the RT is given by (see [13, eq. (12)])

(14)

Equation (14) can be interpreted as a frequency selective water-filling, and this can be seen better by rearranging as

(15)Notice that we will not make use of Approximation 3 for de-

termining RT’s power profile.Now to the main point: given and hence a CDR profile on

the CO, we ask, What would be an optimal level of fora globally good solution?

Proposition 1: For the two-user near–far scenario, if Approx-imations 1, 2, and 3 are considered, then at convergence

(16)

leads to a good spectrum design.Proof: With Approximation 1, CDR can be rewritten as

(17)

where, using (15), we get

(18)

The two last terms of the sum can be ignored, which leaves uswith

(19)

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MORAES et al.: SEMIBLIND SPECTRUM BALANCING FOR DSL 3721

For the best solution, the VL of the RT and the CO should beCDR-equivalent. Given this condition and using (11a), we get

(20)

We shall consider that for the purpose of estimating crosstalkfrom RT to CO, the water level can be ignored—this is a conse-quence of . That leaves us with

(21)

Substituting (21) into (17), we finally reach (16).There are two vital conditions for the demonstration above.

First, . This usually holds for typical near–farscenarios, specially for tones in which crosstalk is most dam-aging. Second, the CDR-equivalence between the VL of theRT and the CO. We have shown that a condition for (16) isCDR-equivalence between the two victims. The converse isalso true—i.e., we will have CDR-equivalence between thetwo victim lines if (16) is enforced (mathematically, we couldwrite (16) CDR-equivalence). The other condition forCDR-equivalence, (11b), should also hold true. Notice how(16) makes sense: first, it is proportional to the emphasis theRT needs in relation to its VL, , which is directly related tohow big is. Second, it is inversely proportional to ,which can be considered a measure of channel quality. Tonesin which this value is high should be considered more carefullyfrom excessive crosstalk, since they play a more important partin the final rate of the CO.

The proposition is in essence simple. Enforcing (16) defines aregion of acceptable crosstalk [see Fig. 2(b)]—it, of course, doesnot apply to tones in which the CO loads no power. With this rulein hands, it is possible to apply the scheme in a semicentralizedmanner. Modems will do an optimization with a VL. With theaid of message exchanges between both modems and a SMC,the method will adjust the RT’s VL parameters and sothat the RT’s VL and the CO achieve CDR-equivalence. Thereis no need for crosstalk channel information, i.e., —hencethe name semiblind. Section IV will provide strong evidencethat condition (16) will yield a good spectrum design.

The full scheme for the two-user scenario is detailed in Algo-rithm 1. There are basically three steps: optimization with VL(performed in the SMC); SMC sending the respective foreach modem; and modems applying and relaying crosstalkinformation to the SMC. Each of them will be commented inthe next paragraphs.

The PSD and channel gain of the VL are initialized with con-stant values in step 1 of Algorithm 1. These are the values thatwill determine the power allocation in the first iteration. Theydo not faithfully represent the actual crosstalk damage in thebinder and hence the resulting matrix of the first iteration willbe fairly poor in terms of (3). This situation should be adjustedso that the VL of the RT accurately represents the actual noisecharacteristics in the binder.

After solution of (7) in the SMC, the respective vectorsshould be reported to modems (step 5 in Algorithm 1). Modemsshould apply , measure and report their andvalues back to the SMC (step 15). So each iteration entails acommunication of 3 values back and forth between modemsand the SMC. 2SB will then adapt the parameters and(steps 7 to 12). Steps 7 to 9 adjust the former. The goal is CDR-equivalence between the VL of the RT and the CO (see (11)).

A right-sided Gaussian pricing curve given by

(22)

is applied for the adjustment of . For a similar functionis calculated—the only difference from is that the numeratorof the fraction is given by . The varianceof the Gaussian curve of acceptable crosstalk is a constant valuewhich should be available to the SMC from the start. The choiceof the Gaussian pricing curve has been observed in experimentsto produce good results.

B. 2SB: -User Scenario

For the -user case, some adjustments need to be made. Thescenario of interest in this case is that in Fig. 3. The scenariodepicted is the most general possible, but we will be specially

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Fig. 3. � -user scenario.

Fig. 4. Interferer–victim pair for RT 1. The victim is solely the CO.

Fig. 5. Interferer–victim pair for RT 2. Victims are both RT 1 and the CO, andthey will be represented by the greatest victim between RT 1 and CO for eachtone.

interested in situations with one CO and multiple RTs in dif-ferent locations.

Processing in modems is the same as that for the two-usercase but the task of the SMC has to be different. Its goal is thesame, to achieve CDR-equivalence between all interferer-victimpairs in the binder, but one complication arises. To illustrate theissue, consider the three-user network depicted in Figs. 4 and 5,with one CO and RTs 1 and 2. Both RTs interfere with the COand RT 2 interferes with RT 1. The problem is about interfererswith two potential victims—for instance, RT 2 interferes withthe two other users, but which one should it take under consider-ation for updating its VL? The problem can also be interpretedfrom the victim side: if the CO reports a high value for CDR,who should be punished?

What basically is going to be done is to break the interac-tion of the users sharing the binder into pairs of interferer-victim(s). For interferer RT 1, 2SB will select its potential vic-tims (in this case only the CO) and treat them like as in thetwo-user case. For RT 2, there are two potential victims, and2SB will form the pair of interferer (RT 2) and a victim whichwill be represented by the greatest victim between CO and RT 1for each tone. In this way, we simplify the network so that thetwo-user scenario again comes into play.

One fortunate fact is that, taking into account Approximations1 and 2, CDR is “summable”, i.e., the CDR of a victim when

considering crosstalk from interferers and equals the sumof the individual CDRs. Mathematically,

(23)

To demonstrate that, we write

(24)

Here, we use Approximations 1, 2, and 1 again.Equation (23) allows us to separate crosstalk damage and

assign the proper punishment for each interferer. If andinterfere with , then a fair measure for punishment for thecrosstalkers should be and , respectively.Each of these multipliers is a weight, and they are organized inthe crosstalk matrix . For a general three-user example (nottaking into account who interferes considerably with whom), agood way to initialize this matrix is

(25)

There should be one matrix for every tone. The th row containsthe weights for interferer , while the th column contains ’sfor a victim. Notice that each column should sum to 1. Fromnow on, we represent the th entry in (25) by .

The objective of is to locate crosstalkers and punish themaccordingly. Consider a simple example. Again for the three-user scenario in Figs. 4 and 5, consider that user 2 reports ahigh value of CDR on a given tone. Column 2 of needs to beadjusted so that it reflects how much of this damage was causedby each interferer. Initializing as in (25) assigns an equalshare of damage caused by users 1 and 3. This is probably nottru, since crosstalk from user 3 and user 2 is considerable, whileuser 1 to user 2 is negligible. A crucial part of the problem isnow to assign fair values for so that interferers are punishedproportionally.

Proposition 2: Consider a set of interferers with a commonvictim . If we consider Approximations 1 and 2, the share eachinterferer has in is given by , where

(26)

Proof: See the Appendix.

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MORAES et al.: SEMIBLIND SPECTRUM BALANCING FOR DSL 3723

Hence, in a general case, 2SB needs the ratios of the crosstalkchannels. The SMC should have access to these ratios and calcu-late the coefficients in (26) before adjustments on the VLs. Thissimplified channel information is a good advantage 2SB pos-sesses in comparison to previous optimal or near-optimal solu-tions. The requirement of information about all crosstalk chan-nels was mandatory for most DSM schemes cited in Section II.The point to be stressed here is that obtaining these ratios, nomatter what method is used, should be half as simple and half asfast than estimating each crosstalk channel individually. Also,there is a good way to do this estimation with the aid of CDRs,however we omit that due to space limitations.

SMC processing for the -user case is detailed inAlgorithm 2. Most of the steps have already been men-tioned and all except two were also present in the two-user casealgorithm. After power loading and message exchanges, thefirst difference from the two-user case is the calculation offor all tones. In the two-user case there was of course no needfor this, since there was only one interferer in the network. Instep 6, a given interferer should attain CDR-equivalence with itsgreatest victim. If the greatest victim is sufficiently protected,then all other victims are too. Next, the greatest victim for ischosen as the greatest value of andstored in maximum CDR, . It is which willbe compared to in steps 8 and 9 of the algorithm. We shouldalso identify the greatest victim for . These values are storedin in step 6. They are necessary for , which in turnwill be used to calculate and to adjust .

Complexity is , the same as that for IWF, ASB andSCALE. The amount of message exchanges is the same as forthe two-user case. We shall see, however, that convergencemight be slow.

IV. EXPERIMENTS

All simulations in this section consider downstream ADSL.It is used 0.5 mm cables (AWG 24 cable) and it is adopted aSNR gap of 12.8 dB. The values for and are set to 4.3125kHz and 4 kHz, respectively. Modems have at their disposal amaximum power of 20.4 dBm [27]. In each line, noise modelANSI A [28] is included.

Two different experiments were realized.

1) Random Two-User Scenario: The first experiment willevaluate how close the approximations made in 2SB for thetwo-user case are to the optimal solution. We simulate arandom scenario, in which the values of and in Fig. 1 areuniformly distributed random variables in [4, 5.5] and [2, 4],respectively. The distance from CO to RT, is also uniformlydistributed in . We simulated 200 network realizationsand calculated the solutions of IWF, ISB, SCALE, ASB and2SB for each one. We are interested in how close the fournonoptimal solutions are to ISB, which is considered theoptimal here. For each network realization the ratios

and were calculated, whereis the rate achieved in the CO for algorithm . The minimumrate of the RT was always set as 75% of the rate achieved whenwaterfilling with background noise.

For 2SB, the VL of the RT is initialized with 45dB for all tones, is set to a flat power level and is set tomatch . The variance of the Gaussian update function wasset to . The same initialization parameters were applied toASB, except , which was set to perfectly match .SCALE was implemented as suggested in [11], however thisreference does not take into account the search procedure of theappropriate values, hence we had to add one loop to SCALE.We programmed it with an outer loop which adapts ’s, an in-termediate loop which finishes when there is PSD convergence,and an inner loop which adapts power. This solution will have toiterate on the outer loop for finding appropriate weights (whichare initialized equal to 0.5 to both users) for the desired datarate. The intermediate loop comprises SMC-modems commu-nication, and we count iterations for SCALE as the number oftimes the algorithm goes through this loop. We will be interestedin how many times it is necessary to iterate on the intermediateloop once we know the appropriate values and for findingthe appropriate values. We consider PSD convergence as thestopping criteria for all solutions.

Fig. 6 plots the cumulative frequency distribution for the ra-tios of each of the four suboptimal solutions. The worst resultis that of IWF, which has the most spread distribution and neverachieves optimality. ASB has good results, though for a few sce-narios convergence was not attained—those are shown as zeroin Fig. 6. ASB never achieves optimality, and that is to be ex-pected since the parameters of the VL of the RT do not make itCDR-equivalent to the CO line. SCALE has very good results,mostly between 0.85 and 1. In about 50% of the channel realiza-tions, SCALE reaches equality in rate results with the optimal.

2SB has clearly the best results. In about 78% of the times,2SB attains the same rate performance as the optimal and in13.5% of the time 2SB reaches 99% of the rate performance ofthe optimal. This experiment is supposed to measure the qualityof approximations made for each algorithm (all suboptimal so-lutions considered in this experiment depend on some kind ofapproximation). These results attest the excellent quality of theapproximations made for our proposal. To further illustrate thispoint, Fig. 8 shows the final CDR on the CO for IWF, ISB and2SB for the “average” scenario, and equal to 4.75 and 3km, respectively, and equals to 3.35 km. The target rate forthe RT is set to 6.2 Mb/s. 2SB’s CDR follows ISB’s very closely.IWF’s CDR shoots up rapidly, which leads to poor performance.Both 2SB and ISB can provide 2.1 Mb/s to the CO, while IWF

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3724 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

Fig. 6. Relative frequency of the quality of suboptimal solutions when com-pared to the optimal ISB. The scenario is random and 200 realizations weresimulated. The variables � � � and � are uniformly distributed in [4, 5.5], [2,4] and [2, � ] km, respectively.

Fig. 7. Relative frequency of the quality of 2SB when compared to the optimalISB. Here we sample the performance of 2SB in 15, 30, and 45 iterations.

can only achieve about half of that. An interesting point to men-tion is that the only region in which 2SB’s CDR deviates a littlefrom that of ISB is around 0.2 MHz. In this particular part ofthe spectrum the noise model used is quite powerful, so that thecondition for Proposition 1 does not hold. Thathowever, does not compromise the final result.

On the negative side, PSD convergence is slow. On average2SB needed approximately 146 iterations for PSD convergence.That is very high and might be a problem, specially since 2SBrequires more message exchanges than other methods. SCALEneeded about 15 iterations once there were correct values ofthe ’s and 208.23 in total—also high, since it also dependson SMC-modems communication. Table II lists the averagenumber of iterations for convergence and average processingtime. We should remind that 2SB did not use any crosstalkinformation for this experiment, while SCALE did. We shouldalso mention that for 2SB the most significant adaptationsoccur on the first tens of iterations. The last iterations makefine adjustments that do not have great impact on the final rate.If the criteria for convergence was rate, 2SB would have beenmuch faster. This is illustrated in Fig. 7, where for the sameexperiment we sample the performance of 2SB at 15, 30 and

Fig. 8. CDR on the CO for IWF, ISB and 2SB for the “average” scenario forthe experiment in Section IV-1).

Fig. 9. Rate region for a three-user scenario. � � � km; � and � equal to 2and 4 km, respectively; and � and � equal to 4 and 3 km. It is desired � �

2 Mb/s.

45 iterations. On this figure we also show the results for thecomplete 2SB. By iteration 30, we can see that there is notmuch change in the final rate results and that 2SB is alreadybetter than SCALE for all points. On Table II we also showprocessing time and average number of iterations for 2SB whensampled at those instants.

2) Three-User Scenario: The proposed algorithm is com-pared with previous solutions in the scenario of Fig. 3 with

and km; and equal to 2 and 4 km, re-spectively; and and equal to 4 and 3 km. The number ofusers in each RT is one.

Fig. 9 depicts the rate regions for IWF, ISB, SCALE, and2SB. It is desired to have user 2 operating at 2 Mb/s for all pointsin the curves. 2SB has interference matrix initially set as in (25).It is again observed that 2SB performs very close to optimal.

Fig. 10 portrays the PSD design for ISB and 2SB for6 Mb/s. It can be seen that power allocation for 2SB is approxi-mately the same as that for ISB. This is also clear evidence of thegood quality of the approximations made. For this experiment,

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MORAES et al.: SEMIBLIND SPECTRUM BALANCING FOR DSL 3725

TABLE IIAVERAGE QUANTITIES FOR THE RANDOM EXPERIMENT

TABLE IIIPROPOSED SOLUTION

Fig. 10. PSD design for the three-user scenario. Top: ISB. Bottom: 2SB.

2SB requires the ratio , while SCALE and ISBneed these crosstalk channels individually.

V. FINAL REMARKS

We presented 2SB, a novel solution for the DSM problem.The method is based on the fact that two lines can reach CDR-equivalence given that a set of conditions apply. We have pro-vided these conditions and we have shown promising results.The algorithm performs very close to the optimal solution, butwith smaller computational cost and semicentralization. A goodadvantage of 2SB is that it only requires the ratios betweencrosstalk channels. Most previous solutions required full es-timation of the channel matrix, which in practice is compli-cated. Ratios should be half as fast and half as simple to acquire.A problem with the solution is the high number of iterationsneeded for PSD convergence. Table III classifies 2SB accordingto the features mentioned before.

APPENDIX

We shall consider the three-user scenario for simplicity. Ourgoal is to assign fair weights in for users 2 and 3 so that they

are punished accordingly. These weights will be represented byand . Following the notation on Section III-B, we obtain

(27)

(28)

The coefficients clearly sum to 1. They can be uniquely de-termined by solving

(29)

where

(30)

Here we use (27), (28), the definition of CDR, the factthat we can join a subtraction of log’s into a division andApproximation 1.

We now can solve (29) by elimination, which results in

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3726 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

(31)

Here we omit some steps due to space limitations. We useApproximation 1, the fact that we can join the sum of log’sinto a multiplication, Approximation 2, when

is small—in our case, , which is usuallysmall for a good DSM solution at convergence—and, finally,Approximation 2. The result is intuitive and it can be extendedfor -user networks easily, which leads to (26).

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[2] T. Starr, M. Sorbara, J. M. Cioffi, and P. Silverman, DSL Advances.Englewood Cliffs, NJ: Prentice-Hall, 1999.

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[5] T. Starr, J. M. Cioffi, and P. Silverman, Understanding DigitalSubscriber Lines Technology. Englewood Cliffs, NJ: Prentice-Hall,1999.

[6] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen, “Op-timal multiuser spectrum balancing for digital subscriber lines,” IEEETrans. Commun., vol. 54, no. 5, pp. 922–933, 2006.

[7] W. Yu, G. Ginis, and J. M. Cioffi, “Disributed multiuser power controlfor digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, no.5, pp. 1105–1115, 2002.

[8] R. Cendrillon, “Multi-user signal and spectral coordination for digitalsubscriber lines,” Ph.D. dissertation, Katholieke Universiteit Leuven,Leuven, Belgium, 2004.

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[11] J. Papandriopoulos and J. S. Evans, “Low-complexity distributed algo-rithms for spectrum balancing in multi-user DSL networks,” presentedat the IEEE Int. Conf. Commun., Istanbul, Turkey, 2006.

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[14] P. Tsiaflakis, J. Vangorp, M. Moonen, J. Verdilen, and K. V. Acker, “Anefficient search algorithm for the Lagrange multipliers of optimal spec-trum balancing in multi-user xDSL systems,” presented at the IEEE Int.Conf. Acoust., Speech, Signal Process., Toulouse, France, 2006.

[15] R. Moraes, B. Dortschy, A. Klautau, R. Zampolo, and J. R. I Riu, “Op-timal solution for the fixed margin problem in digital subscriber lines,”presented at the Int. Symp. Control, Commun., Signal Process., St. Ju-lians, Malta, 2008.

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[27] Asymmetrical Digital Subscriber Line Transceivers 2 (ADSL2), ITUstd. G.992.2, 2002.

[28] Noise Models for VDSL Performance Verification), ANSI-77E7.4/99.438R2, 1999.

Rodrigo B. Moraes (S’08) was born in Belém,Brazil, in 1982. He received the Bachelor’s degreeat the Federal University of Pará, Belém, Brazil,in 2005 and the M.Sc. degree at the PontificalCatholic University, Rio de Janeiro, Brazil, in 2009,both in electrical engineering. Since October 2009,he has been working towards the Ph.D. degree atthe Department of Electrical Engineering at theKatholieke Universiteit Leuven (K.U. Leuven),Leuven, Belgium.

In 2006, he was a visiting researcher at Ericsson’sBroadband Technologies laboratories. His research interests are in signal pro-cessing for communications.

Mr. Moraes has been awarded the FAPERJ Nota Dez Scholarship by state ofRio de Janeiro, Brazil, and the IEEE Travel Grants, both in 2008.

Boris Dortschy received the M.Sc. degree in electrical engineering from theRWTH Aachen University, Aachen, Germany.

From 1999 to 2004, he has been with the Institute of Communications andRadio-Frequency Engineering of the Vienna University of Technology as aTeaching and Research Assistant. Since 2004, he has been a Researcher at Eric-sson’s Broadband Technology Laboratories, Stockholm, Sweden. His researchinterests include modern coding and information theory, signal processing forwireline and wireless communication systems and management systems forDSL.

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MORAES et al.: SEMIBLIND SPECTRUM BALANCING FOR DSL 3727

Aldebaro Klautau (S’92–M’04–SM’08) receivedthe Ph.D. degree from the University of California,San Diego, in 2003.

In 1995, he was a Faculty Member of the FederalUniversity of Santa Catarina, Florianópolis, Brazil.Since 1996, he has been with the Federal Universityof Pará, Belém, Brazil, where he is currently the Vice-Director of the Department of Computer Engineeringand affiliated with the Computer Science (PPGCC)and Electrical Engineering (PPGEE) Graduate Pro-grams. He also directs the Signal Processing Labo-

ratory (LaPS) and the Embedded Systems Laboratory (LASSE). His researchinterests include machine learning for signal processing, with applications tospeech recognition, DSL, and software/cognitive radio.

Dr. Klautau has served as Technical Program Committee Member in severalconferences. He is a co-founder and the current Chair of the IEEE Joint Chapter,Northern Brazil.

Jaume Rius i Riu received the M.Sc. degreein physics from the Autonomic University ofBarcelona, Barcelona, Spain, in 1996, the M.S.degree in teaching and pedagogy from the Universityof Lleida, Lleida, Spain, in 1997, the ResearcherQualifying degree from the University of Barcelona,Barcelona, in 1998, and the Ph.D. degree in ex-perimental physics from the Royal Institute ofTechnology, Stockholm, Sweden, in 2002.

In 2003, he was a Postdoctoral Fellow with the De-partment of Physics, Oulu University, Oulu, Finland.

Since 2004, he has been with Ericsson A.B., Stockholm. From 2004 to 2006,he was the First Mile Technologies Work Package Leader with the EuropeanUnions 6FP MUSE Project, the Project Manager for a number of xDSL researchprojects, and a member of Ericssons European Commission Research SteeringBoard and of support teams for broadband business case analysis and prepara-tion. Since 2006, he has actively participated in Ericssons’ broadband networksstrategies and project planning. Since 2007, he has been Ericssons BroadbandForum Standardization Coordinator and the Project Manager for BroadbandForum-Related Activities.

Dr. Riu is one of a member of the Industrial Research Group, Swedish RoyalAcademy of Engineering and Sciences (IFG IVA), and the Broadband ForumBroadband Convergence Oversight Committee.

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