Multiuser Margin Optimization in Digital Subscriber Line (DSL) Channels Saswat Panigrahi, Yang Xu, and Tho Le-Ngoc, Fellow, IEEE. 1 Level 0 is SSM.

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST Multiuser Margin Optimization in Digital Subscriber Line (DSL) Channels Saswat Panigrahi, Yang Xu, and Tho Le-Ngoc, Fellow, IEEE Abstract This paper presents efficient multiuser margin optimization algorithms suitable for multicarrier digital subscriber line (DSL) systems using Dynamic Spectrum Management (DSM). The favorable monotonicity and fairness properties of multiuser margin are employed to formulate a box-constrained nonlinear least squares (NLSQ) problem for multiuser margin maximization, which is efficiently solved by using a scaled-gradient trust-region approach with Broyden Jacobian update. Based on this NLSQ formulation, a multiuser harmonized margin (MHM) optimization algorithm for resource allocation is developed. A Newton Raphson method is also developed for fast margin estimation and used within the MHM. The MHM algorithm converges efficiently to a solution for the best common equal margin to all users, while explicitly guaranteeing their target rate requirements. (This is the reason for the term harmonized.) Furthermore, its predominantly distributed structure can be implemented in DSL/DSM scenarios with only Level 1 coordination. Simulation results of various cases verify the convergence to the unique optimal solution within 5 10 iterations. Index Terms Broyden update, digital subscriber line (DSL), dynamic spectrum management (DSM), iterated water-filling, loading algorithms, margin adaptive, trust region methods. I. INTRODUCTION DISCRETE MULTITONE (DMT) modulation [4] in conjunction with efficient rate and power loading for given channel conditions, system constraints, and performance requirements has been widely used in xdsl applications such as asymmetric digital subscriber line (ADSL) [1], and more recently, very high bit rate DSL (VDSL) [3]. The existence of multiple twisted copper cables within the same bundle causes significant crosstalk into each other s channels. In the first decade of DSL, coexistence among multiple users and diverse services was ensured through specification of admissible spectral masks [2] for each user. Since these spectral masks are based on the worst case crosstalk scenario, they resulted in the aforementioned Static Spectrum Management (SSM) techniques to be unduly restrictive and thereby leading to conservative rates. This realization has recently fueled intense Manuscript received August 29, 2005; revised April 20, This work was supported in part by an NSERC CRD Grant with Laboratoires Universitaires Bell. This paper was presented in part at the IEEE GLOBECOM 2005, St. Louis, MO, November 28 December 2, S. Panigrahi is with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 2A7, Canada, and also with Ericsson Canada, Inc., Mount Royal, QC H4P 2N2, Canada ( saswat.panigrahi@ mail.mcgill.ca). Y. Xu and T. Le-Ngoc are with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 2A7, Canada ( yang. xu@mail.mcgill.ca; tho@ece.mcgill.ca). Digital Object Identifier /JSAC research activity in Dynamic Spectrum Management (DSM) [6], [8], [20], which seeks to jointly optimize transmit spectra in order to minimize crosstalk, and hence to improve the achievable rates. Based on the required amount of coordination and centralized control, DSM techniques can be classified from Level 1 to Level 3 1 [7]. In Level 1, only macro parameters such as data rates, total transmit power, and margin are reported and controlled centrally and other micro parameters such as actual subcarrier-specific power and rate allocation are autonomously managed by each individual user modem in a distributed manner [7]. Level-1 DSM schemes are desired for their lowest requirements of coordination and centralized control, especially when multiple competing service providers share the bundle in current DSL scenarios. Among Level-1 DSM techniques, iterated water-filling (IWF) [8], which is based on a game theoretic formulation of the DSL Gaussian interference channel, is possibly the most popular [6], [9], due to its predominantly distributed nature and significant rate enhancement as compared with SSM techniques. The noise (in the broad sense) in DSL environments is classified into two types capacity limiting and performance limiting 2 [5]. Thermal noise and crosstalk fall into the category of capacity limiting noise and is effectively dealt with by DSM techniques such as IWF. Performance limiting noise is constituted by impulse noise and radio frequency interference (RFI) pickup, which are nonstationary, geographically variable and unpredictable. Impulse noise consists of relatively high-energy bursts due to electromagnetic interference from physical phenomena, electrical switches, motors, and home appliances which are invariably present in the close vicinity of DSL modems. In addition to interleaving, which has the disadvantage of being the primary contributor of end-to-end delay (undesirable for interactive applications) [16], the primary defense against performance-limiting noise is the performance margin of the system. With VDSL system bandwidth going up to 12 MHz (and even higher for VDSL2) and increased antenna efficiency of network cables at these frequencies, RFI pickup becomes a major issue [5]. HAM radio interference can also appear outside reserved RF bands (due to circuit nonlinearities, imperfect filtering [16]). These two factors can make interference duration well above what interleaving and error correction can handle [16]; thus leaving margin as the main defense. In addition, for 1 Level 0 is SSM. 2 In the strict sense, all noise would be capacity limiting. However, in the discussion in this paper, as well as [5], the term capacity-limiting noise implies the predictable forms of noise and performance-limiting noise implies the unpredictable disturbances typically experienced in DSL scenarios, as explained in this paragraph /$ IEEE

2 1572 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 constant bit rate DSL applications, maximizing the margin and thereby minimizing the system probability of error (e.g., see [10, Lemma 4.1]) while satisfying a certain bit-rate demand is desirable [4], [10] [15]. Further margin allows error-free transitions when certain modems change their mode from passive to active. One might contend that since margin is so important, why margin should not be set to a high but constant value such as 12 db? The reason for not doing so is that, in a way, margin and rate maximization are conflicting requirements. Setting margin at a very high value can limit the maximum rates that can be reached by users. Setting it a too low a value will risk performance. Hence, there is the need for optimization. Most applications such as video conferencing, streaming, etc., have fixed requirements on rate known as target rate. But during this application, performance-limiting noise can harm the quality of the session. Hence, for such applications, maximizing the margin, while respecting the constraint on the target rate, is the ideal solution. This is exactly what the existing single-user margin maximization techniques strive to do. The importance of margin maximization (MM) due to the above reasons, has led to the development a number of margin adaptive loading algorithms [11] [15]. One of the first solutions [11] to the MM problem was based on reaching the solution by iterating over the margin and using the signal-to-noise ratio (SNR)-gap formulation. In [12], the authors approached the problem directly in terms of minimizing probability of error by maximizing the SNR profile leading to slightly improved results. In [13], the equivalence of the MM problem and the energy minimization (EM) problem was established. This equivalent reformulation being convex unlike its original counterpart was used to develop optimal MM algorithms in [14] and [15]. The energy saved by solving the EM problem, could be directly transferred into gain in margin, by multiplying all subcarrier energies with the ratio of the total energy constraint to the total utilized energy, i.e., optimal value of EM problem. This ratio itself was the final margin of the system. Irrespective of whether SSM or DSM is in use, all of the factors described in this section to motivate margin, still deem DSL performance equally vulnerable to unpredictable disturbances. However, as all of the single-user margin maximization algorithms (described in this section) were developed in the SSM era, they have an inherent fixed crosstalk and noise profile assumption, and hence are essentially single-user algorithms. No direct extensions to the new DSM scenario are possible, where each user s power allocation decides other users crosstalk profile. Furthermore, the duality between the MM and EM is more involved in the DSM scenario. The energy saved by solving the EM problem cannot be converted into gain in margin by any simple multiplication, because any such multiplication would destroy the stationarity or equilibrium that exists between the crosstalk profiles of different users due to the DSM algorithm, e.g., IWF. This in turn can result in other users loosing rate or margin or both. 3 Hence, the multiuser margin maximization problem deserves an independent study instead of extensions from the existing single-user counterparts. 3 Due to this reason, in DSM scenarios [9], there is some skepticism about modems working in margin adaptive (MA) mode, which has been popular in SSM [4], [10] [15]. In this paper, we attempt to solve the multiuser margin maximization problem in the min max sense and develop a DSM algorithm with only Level-1 coordination, which can provide the best common equal margin to all users, while guaranteeing the target rate requirement satisfaction for each user. We formulate the multiuser margin maximization problem in the min max sense in Section II. In Section III, we derive a Newton Raphson method for fast margin estimation, while maintaining the target rate requirements of all users, and examine the monotonicity and inherent fairness properties of the margin over convex multiuser rate-regions. In Section IV, these favorable properties are employed to reformulate the min max problem as a nonlinear least squares (NLSQ) problem with box constraints, which is efficiently solved by using a scaled-gradient trust-region approach. Based on the NLSQ formulation, we further develop the multiuser harmonized margin (MHM) optimization algorithm suitable for resource allocation in multicarrier DSL systems using Level-1 DSM. Section V presents the performance and convergence results in realistic DSL scenarios and concluding remarks are made in Section VI. II. PROBLEM FORMULATION Consider the DSL interference channel [8] of users (i.e., transmitters and receivers) using a DMT system with subcarriers. Throughout this paper, unless otherwise specified, superscripts refer to user number and subscripts to subcarrier number. The channel response from user to user on the th subcarrier is denoted by. For user on subcarrier, the controlled transmit power spectral density (PSD) used is denoted by and the background noise PSD encountered is. The set of all is denoted by. The intercarrier spacing is assumed to be small enough for,, and to be nearly flat over for each. The total utilized power by user is, where The total utilized power vector for the users is denoted by. The total power constraint on user is and together for the users,. It is important to note that is specified by standards, e.g., in [3], dbm for upstream. But, in a certain scenario, depends on the particular DSM algorithm in use, e.g., a user might choose out of compassion for more needy users. However, independent of the allocation method in use, the following must hold: where the above vector inequality is componentwise. The received SNR is (1) (2) (3)

3 PANIGRAHI et al.: MULTIUSER MARGIN OPTIMIZATION IN DSL CHANNELS 1573 where crosstalk is viewed as noise. For a given a SNR profile, the maximum achievable rate is (4) margin among ;, by varying, while respecting the constraints on total power (2) and target rate (5), in an as distributed as possible manner. The problem can be formally stated as follows. Objective function where is the SNR-gap [4], [11] and. The rate region is defined as the set of rates achievable by a particular algorithm, i.e.,. The boundary of the rate region is defined as Constraints where is used to denote component-wise inequality. At any stage, each of the users has a target (demanded) rate referred together as. The performance or SNR margin of the th user denoted by is defined 4 as the factor by which the noise can be increased before the system error-rate rises above the acceptable threshold (specified implicitly in ). Mathematically, is the solution to For given and, the left-hand side of (5) is monotonously increasing in (and obviously continuous). Hence, (5) uniquely (though implicitly) specifies. Noticing the similarity between (4) and (5), one might be tempted to think that can be eliminated and an expression for can be obtained only in terms of and. Such an expression would make the margin independent of the particular algorithm in use and would remain the same for single-user or multiuser. While no such exact expression is possible, an approximation is available Reliability of (6) will be discussed in Section III-B. By using the monotonicity in (5), the continuity of (4) and (5), and a few operations, the following three conclusions can be made about,, and, irrespective of the that generated them (i.e., independent of algorithm and scenario). (C1) if and only if. (C2) if and only if. (C3) if and only if. Margin by its nature is a safety margin [16] and simultaneously a performance enhancer (in terms of error rate) [12]. Thus, going by the philosophy of distributed DSM whereby our concern is each user s well-being and not some single overall system metric, it is natural to model the problem as a min max problem. Our objective will be to maximize the minimum 4 The subscript M here indicates Margin and is not a subcarrier number. Margin is a single positive scalar for a user, i.e., it is user-specific and not subcarrier-specific. (5) (6) (P1) By explicitly accounting for the target rate constraint in (P1) we guarantee that the luxury of margin maximization does not come to any user at the unacceptable cost of failure of meeting target rate for any other user. One might suspect from the min max formulation that the users closer to the central office will always dominate and get a better at the solution to (P1). However, we will observe in Section III that for a convex rate region, the solutions to (P1) have the property of, and hence they inherently guarantee the fairness among users. III. BEHAVIOR OF MARGIN OVER A RATE REGION As shown in the previous section, for the multiuser margin maximization problem considered in this paper, the multiuser scenario directly leads to a constrained min max optimization problem (P1). However, in general, algorithms and analysis available for constrained min max optimization are more complicated than those for constrained nonlinear minimization problems. This is because the objective function to be minimized in min max is already a max of many component functions, and consequently becomes nondifferentiable at some points even if all the individual component functions are differentiable. In this section, we study the behavior of margin over a rate region and derive the properties that help to bring the min max problem (P1) to a very special class of constrained nonlinear minimization problems, which is easier to solve in Section IV. A. Monotonicity and Fairness Assuming that the approximation (6) holds accurately, the following 2 claims can be made about the properties of margin over a rate region. (C3) is monotonously increasing in for any for fixed. (C4) For every, there exists a unique point, which is the solution to (P1) and at this point.

4 1574 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 Fig. 1. Relationship between rate and margin. (a) 2-user case; (b) 3-user case. (C4) directly follows functional form in (6). The remainder of this section is devoted to proving (C5). The strategy is to progressively narrow down the set, where the solution to (P1) can exist, from the entire rate region to progressively smaller subsets of the rate region, and finally to a point. The main steps of the proof are as follows. Step 1) Explain that for a given, the rate region is equivalent to a margin region Step 2) Use (C1), (C2), and (C3) to identify, which is a subset of, where all users have positive margin. Prove that solution to (P1) is in. Step 3) Prove that the solution to (P1) lies in, which is naturally a subset of. Step 4) Prove that among all the points on, there exists a unique point, where. Prove that this unique point is the solution to (P1). The next four paragraphs perform the above four steps of the proof in sequential order. Consider the two-user rate region in Fig. 1(a). Point A represents. For any, it is possible to find 2 points B,. B represents the maximum rate achievable for user 2 when user 1 has a minimum rate of T1. C represents the analogous situation for user 1. First of all, it is important to understand that once is specified, the entire rate region is transformed into a margin region. This is because, by definition of, each point in specifies a corresponding, which in turn translates into user margin by (3) and (5). By (C1), the straight lines AB and AC represent the only points where the margins [in decibels (db)] for user 1 and user 2 are zero, respectively. By (C2), the region demarked by A, B, C, and (excluding points on AB and BC) represents the subset of, where all the users are guaranteed to have positive margin. denotes this region and its closure, and can be represented as. Finally, by (C3), all other points represent negative (in db) margin for at least one of the users, and hence cannot contain the solution to (P1), e.g., X has negative margin for user 1, but positive margin for user 2, Y has negative margin for user 2 and positive margin for user 1, and Z has negative margin for both users. Thus, the solution must exist in. For any point in and not on, it is possible to find a point on such that ;. Hence, due to (C4), will have a greater margin, for both the users (and hence greater minimum margin), than. As a result cannot be the solution to (P1). Thus, the solution must exist in represented by BC in Fig. 1(a). At point B, the margin achieved by user 1 and user 2 is, where is the maximum margin user 2 can have, while user 1 having a non-negative margin. Similarly, at point C, the margin achieved is. If the is convex, as we move from B to C along, monotonously decreases from to 0, and monotonously increases from 0 to. Thus, by continuity arguments there must exist a point between B and C on [labeled as Optimum in Fig. 1(a)] with the margin, i.e., equal margin for both users and. For any point between Optimum and B, and hence is a better solution to the min max in (P1). Similarly, for any point between and C,. Thus, is the unique global solution to (P1) over. The argument is easily extended step by step for. As an example, see Fig. 1(b) for a three-user case. Here, the argument would begin by the identification of 3 points that demarcate, instead of the 2 points B and C in the two-user case, and in the general -user case there would be such points. Therefore, (C5) is proved. In summary, we have the following mapping from a rate point on the rate region to the margin for a given and target rate. The mapping from to can be made inherently distributed; see (3). For, a very suitable distributed choice is IWF, i.e., the inner loop in [8], as discussed in Section I. IWF further has the convenient property that the total power tuple has a one-to-one correspondence with the rate tuple, i.e.,. Thus, by adjusting alone, we can span entire and control. The IWF algorithm [8] is presented in Table I with minor modifications to suit our needs. Step 4 depends directly on the choice of and the water-filling can be implemented using [17, Table I]. Step 5 ensures that at the end of the IWF, each user has its own SNR profile available, i.e., which it would need anyway to calculate implicitly or explicitly for Step 6.

5 PANIGRAHI et al.: MULTIUSER MARGIN OPTIMIZATION IN DSL CHANNELS 1575 TABLE I SUBALGORITHM IWF TABLE II SUBALGORITHM NEWTON RAPHSON MARGIN ESTIMATION (NRME) B. Accuracy of Margin Approximation in (6) The approximation in (6) follows from the representation of the entire SNR profile by its geometric mean. This single-parameter representation is seen to be fairly accurate when we have a nearly continuous SNR profile, e.g., in ADSL. Furthermore, in a multiuser scenario, (6) suggests that depends only on and not on,. This is not strictly true. However, due to the diagonal dominance 5 of the DSL channels, for a major part of the rate region, changes very little 6 with,, as compared with its rate of change with respect to. Thus, in summary, the approximation in (6) is a reliable indicator of trend and behavior of margin, but not of its exact value. In such a situation, the alternative is to obtain the exact value of margin, using an iterative approach to solve (5). Since (5) has continuous first derivative and a unique solution, and a reasonably good starting point is available in (6), we propose the usage of Newton Raphson method, which is one of the fastest methods to solve nonlinear equations, as shown in Table II. Note that in Table II, all user related superscripts are dropped because once the SNR profile is available; the margin calculation does not need to distinguish between users. The in Step 5 denotes the differentiation of (5) with respect to. Step 7 is the Newton Raphson update. Steps 8 10 are required while measuring very small (close to zero), to prevent an ambitious Newton Raphson update from pushing the margin to an unacceptable value (i.e., negative in linear scale). The algorithm was tested in many varied scenarios and reached the solution in less than four iterations in most cases for bits. C. Practical Validation of (C4) and (C5) Consider the scenario of 8 VDSL 26 AWG lines (similar to [8]), four of which are at a distance of 1500 ft from the Central Office (collectively referred to as user 1) and the other four at 5 Diagonal dominance here implies that the direct channel s strength is much higher than that of the interference channel, i.e., H H. This is amply demonstrated for DSL channels in [8, Fig. 6]. 6 This claim is illustrated by the nearly horizontal and nearly vertical nature of the margin contours over a rate region in Fig. 3, which will be introduced shortly. Fig. 2. Rate region and target rate points. a distance 3000 ft. (collectively referred to as user 2). We conduct an extensive experiment by analyzing every power combination of User 1 and user 2 from 60 to 11.5 dbm at a step-size of 0.5 dbm, where dbm [3] in the upstream direction. Crosstalk Noise Model A is assumed and the FEXT transfer functions and upstream frequency band specifications are chosen as specified in [3]. With each power combination, we execute subalgorithm IWF (Table I) and each resulting unique rate pair is plotted as (small) dots in Fig. 2, and thus span the entire. The target (demanded) rate is 4 Mb/s for each of the 1500 ft lines and 1 Mb/s for each of the 3000 ft lines. This point is shown in Fig. 2 as point A and corresponds to point A in Fig. 1(a). The straight lines correspond to lines AB and AC in Fig. 1(a). For each of these points in, we evaluate the for each user by using subalgorithm NRME (Table II). The constant margin contours over for both users are plotted in Fig. 3. We plot only the positive margin (in db) contours. The nearly horizontal contours represent the contours for user 2 and the nearly vertical ones are

6 1576 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 i.e.. The plots of and clearly reflect (C4), i.e., and increase with (lower axis) and (upper axis), respectively. As shown in Fig. 4 and also in Fig. 3, the has the unique maximum at, where db. This verifies (C5). In other examples presented in Section V and many other tests, (C4) and (C5) have been seen to hold for DSL channels. 7 IV. ALGORITHM FOR MULTIUSER MARGIN OPTIMIZATION Given subalgorithm IWF (Table I) and subalgorithm NRME (Table II), we now have the following mapping from the design variable to the objective function (7) Fig. 3. Margin contours. Fig. 4. Behavior of margin on rate region. for user 1. The contours are labeled with the margin value in db that they represent. We can see that the 0 db margin contours are perfect straight lines following from (C1) and the zero margin boundaries predicted in Fig. 2. In Fig. 3, is represented by the region formed by the intersections of the horizontal and vertical contours. We see that the contour values of user 2 keep increasing with and the same effect is observed for user 1. Thus, (C4) holds. For verifying (C5), we highlight the contour representing db for user 1 and user 2. Both these contours intersect on at. A point above this intersection such as P corresponds to db. A point below this intersection such as L indicates db, and any point to the interior like K represent both and db. Thus, assumes its highest value at. The behavior of and is very important, particularly, on the intersection. In Fig. 4, the lower axis denotes the set for positive and (in db), and the upper axis denotes the corresponding on the rate-region boundary, and have one-to-one mapping in IWF. However, is naturally easier to control. In the last step of the mapping, the usage of subalgorithm NRME (Table II) causes a simplification of (P1). The termination condition in Step 11 of subalgorithm NRME (Table II) ensures that the target rate constraints are met. Therefore, we can eliminate this constraint from (P1) as along as we use NRME to evaluate. With the removal of the target rate constraint, we are left with the constraint. This form of constraints is much simpler than general nonlinear constraints and is referred to as bound constraints or box constraints. At least for nonlinear minimization with box constraints, significant research has been done [18], [19]. The min max problem (P1) while being the most natural modeling of the situation cannot directly apply these methods. Furthermore, the trust region methods in [18] and [19] require the Jacobian and the Hessian. In our case, since the margin is a very complicated multistage mapping [see (7)] from, the Jacobian or Hessian is impossible to obtain in a closed form. NLSQ problems present a convenient class of problems whereby dependence on Hessian of component functions is minimized under certain conditions. (C4) and (C5) help us reformulate our problem (P1) into such a problem. A. Bound-Constrained NLSQ Formulation (C5) indicates that the optimal point lies on with, i.e., the variance of the vector of margins will be zero at the optimal point. Therefore, we would want to keep the search always on. This can be ensured by maximizing the mean of the vector from (C4) because the mean has continuous growth towards. Thus, we have the following NLSQ formulation: (P2.a) (P2.b) (P2.c) 7 For general xdsl channels, the rate region generated by IWF is not guaranteed to be always convex. [21, Fig. 7] illustrates an example of nonconvex rate region generated by IWF for the scenario of two downstream ADSL users. However, our experimental and theoretical results indicate that DSL channels with the strong diagonal dominance (i.e., direct channel responses more powerful than crosstalk responses) tend to provide convex or near-convex rate regions.

7 PANIGRAHI et al.: MULTIUSER MARGIN OPTIMIZATION IN DSL CHANNELS 1577 In the above formulation, is the variance of the margin vector divided by the square of their mean. in (P2.c) are functions of through (7). Clearly, at the optimal point, the minimum value of, also called the residual of the NLSQ problem, becomes zero by (C5). The Jacobian is an matrix with the component. From, the gradient and the Hessian of are calculated as TABLE III SUBALGORITHM SG TRUST REGION METHOD FOR NLSQ (SGTRNLSQ) (8) where is the Hessian of the component function. Due to our zero-residual problem, the in the summation in (9) will be very small near, and hence the Hessian can be approximated as. The functional dependence on is dropped henceforth in notation whenever obvious from context. The Jacobian could be estimated by using finite-difference, but this would increase the number of objective function evaluations by at each iteration. Since in our case, each objective function evaluation in (P2.a) will require an operation due to (7), doing this at each iteration is unaffordable. Instead, we use a secant method (the Broyden s rank-one update) stated below (9) (10) The estimation of, i.e., the Jacobian at the very first iteration, is a nontrivial task. Finite-differencing could be used to estimate the initial Jacobian. However, from (C4) and (C5), a much simpler approach can be used to obtain a good initial guess, as will be discussed later in Section IV-C. The above NLSQ problem with bounds, i.e., (P2) can be considered as a special case of the algorithm in [19] as follows. B. Trust-Region (TR) Reflective Newton Methods Using trust-region (TR) method, solving an unconstrained minimization problem requires the definition of a simple quadratic model and the neighborhood where this model can be trusted. The gradient and Hessian are used to define the model, and the scaling matrix and TR size are used to determine the neighborhood. At each iteration, minimizing the above quadratic (model) with ellipsoidal constraints (neighborhood) yields a step update, and then both the model and the neighborhood are updated. For a general constrained minimization problem, in addition to the ellipsoidal constraints arising from the neighborhood, a constrained version of the above problem would have to be solved at each iteration. However, for the special case of bound-constrained problem using TR method, a special scaling matrix can be defined so that, at each iteration, we have to solve a quadratic problem with only ellipsoidal constraints (just like in unconstrained TR) without having to explicitly handle the bound constraints [18]. Applying this ap- proach to our problem, we define the following special scaling matrix at any iteration: where, using the bounds in (P2.a) and [18], we obtain (11) The quadratic subproblem to be solved for determining the step update at each iteration is (12) where,, is the Jacobian of, and. At each iteration, all the above quantities are updated. In [19], it is proposed to choose among the best of three possible solutions for : 1) original solution to (12); 2) scaled gradient (SG) solution (both truncated to remain strictly feasible); and 3) reflected solution. It is observed in [19] that solutions 1) and 3) predominantly lead to much faster convergence for large-scale problems, but are complicated. On the other hand, solution 2) is much simpler since the step direction is already determined by the scaled gradient, i.e.,. The only one-dimensional (1-D) operation to be performed is to determine the optimum step-size. The scaled gradient converges to zero, and thus first-order optimality ([19, p. 8]). The SG solution method is used in the subalgorithm SGTRNLSQ summarized in Table III. With a few operations, the problem in Step 1 of Table III can be seen as a 1-D problem of the form, where each of the quantities is a scalar. The well-known solution is if. Otherwise, it is either or. Thus, in Step 1 is obtained in less than three evaluations of a 1-D quadratic. Steps 2 and 3 are computationally trivial and are used to ensure that the updated results in a strictly feasible. As will be shown in Section V, using sub-algorithm SGTRNLSQ (Table III) based on the SG method needs only 5 10 iterations. In most cases, the much more complex best of three approach gives the same number of iterations. This can be due to the following reasons. The number of variables for which [19] showed the great improvements of 1) and 2) is in hundreds and thousands while, in DSL, is much smaller.

8 1578 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 TABLE IV ALGORITHM MHM The best of three approach can have faster convergence with exact Jacobian availability while, in our problem, we can only have approximate Jacobian estimate. The accuracy or tolerance for the stopping condition in [18] and [19] is 10. For DSL applications, the maximum accuracy in margin is 10 [11] or at best 10 [15]. Thus, the primary utility of the trust region method for our problem is the scaling matrix that ensures the appropriate behavior of SG around correct and incorrect bounds [19]. Furthermore, at each step to solve for, we deal with a simple 1-D quadratic rather than having to evaluate after using subalgorithm IWF (Table I) and subalgorithm NRME (Table II). C. Multiuser Harmonized Margin (MHM) Optimization Algorithm The above derivations are used to develop the complete MHM optimization algorithm shown in Table IV. The unshaded regions denote the operations taking place at the center, e.g., the central office (CO), or a spectrum management center (SMC) or DSLAM depending on a particular xdsl network structure [6]. The shaded regions denote the autonomous operation in each of the modems. The exchange between the center and modems involves only macroparameters such as and. The control of microparameters such as PSD and bit allocation over subcarriers is autonomously performed by the individual user modems. In other words, the functional structure of the MHM algorithm is suitable for implementation in xdsl systems using Level-1 DSM [7]. The initialization is performed by Step 0 in Table IV. All modems initially start at their maximum power constraint. This initial point is common in general, e.g., also used in [8]. Furthermore, as the optimal solution is on in our case, it is desired to start with that is guaranteed to be a point on. The estimation of the initial Jacobian is a nontrivial task. It could be performed by finite-differencing but this would involve iterations of MHM to estimate. For simplicity, we consider the initial guess as a diagonally dominant matrix, where is the identity matrix and is the matrix of all 1 s and is much larger than 1, i.e., the diagonal elements of are, while all off-diagonal elements are 1 s. This heuristic guess is based on the fact that a xdsl channel matrix is diagonally dominant, whereby has more effect on than, and margin of each user is an increase in its own power and a decrease in power of other users power (represented by 1in ). Though heuristic, this approach (motivated primarily by numerical experiments) saves the iterations that would be required for finite-differencing, and gives good results in all tested cases ( is used Section V). At each iteration in Table IV (Steps 2 13), each individual user modem, i.e., user, obtains from the center (in Step 2), performs IWF to produce and for all its subcarriers (in Step 3), and subsequently derives using NRME and report it to the center (in Step 5). Obviously, each user knows its own. Once the center receives the from all users, it computes the NLSQ component functions (P2.c), updates the Jacobian using the Broyden update (10), and calculates the scaled gradient step for to produce a new set of for the next iteration in Steps 7 13 in Table IV. The stopping condition (Step 14) is determined by the residual,, which must reach zero at by (C5) and Section IV-A. Note that the Euclidean norm of also approaches zero at. At the end of the MHM, each user s margin is expected to reach an equal. At this point, the PSD allocation and bit loading are decided, respectively, by the IWF and NRME procedures in the last iteration. The usage of NRME guarantees that of each user is met at every iteration as explained in Section IV. Thus, the margin maximization does not come at a cost of any of the users not meeting their. Furthermore, the common equal margin ensures no user looses margin due to another. V. ILLUSTRATIVE RESULTS We consider the upstream scenario with four DSL lines of 1500 ft (user 1) and four DSL lines of 3000 ft (user 2) described in Section III with dbm. We consider six target tuples fairly well dispersed over, as shown in Fig. 2. For each of these target rates, we run the MHM algorithm and the results are reported in Table V. The common equal maximum margin is reported in db. The required number of iterations (denoted by Iter in MHM algorithm) is reported next. denotes the value of the residual of the NLSQ function in (P2.a) at the end of MHM. This is the stopping condition in MHM and used to test if is truly the common equal margin among all users. The total utilized power vector that achieves the optimal point,, is reported in dbm. denotes the actual location of the optimal point on the rate region. As denoted by (7), the energy allocation used corresponds to this point. Also, verifies that the optimal point lies on. The example of the (4,1) megabits per second target rate pair was discussed in Section III, and we had noticed that a unique

9 PANIGRAHI et al.: MULTIUSER MARGIN OPTIMIZATION IN DSL CHANNELS 1579 TABLE V PERFORMANCE AND CONVERGENCE OF MHM Fig. 6. Margin contours and MHM trajectory for target pair (10 and 4 Mb/s). Fig. 5. Margin contours and MHM trajectory for target pair (8 and 6 Mb/s). optimum existed in Fig. 3, where the db margin contours of both users intersected on. Table V confirms that MHM actually reached this optimum point with in seven iterations. The of (21.07,4.87) corresponds to the actual location of the optimal point on as seen in Fig. 3 as well. The contour plots of both users margins for the next two target rate pairs (8,6) and (10,4), are shown in Figs. 5 and 6, respectively. A survey of the contour values shows that there is an intersection of 3.79 db margin contour lines for both users in Fig. 5 on and the same is observed for 6.38 db margin contours in Fig. 6. A quick survey of the contours in the neighborhood of these intersection points in both figures shows that, for each feasible point around this intersection at least one of the margins drops, and hence cannot be a solution to (P1), thus establishing the unique optimality of the intersection of equal margin on as expected from (C5). The trajectory of MHM in both figures shows how MHM approaches and eventually converges to this optimal solution, which is also confirmed by Table V with and 6.38 db. Both Figs. 5 and 6 indicate that the final solution is on and every single iteration in the trajectory of MHM also corresponds to a point on as expected in (C5) and the NLSQ reformulation (P2). For each of the target rates in Table V, we can see that MHM algorithm converges in 5 10 iterations. A study of the column reveals that for each target rate pair, MHM converges to with at least one of the two users using their maximum power constraint of 11.5 dbm which itself guarantees that the solution point is on [8]. The positive value of verifies that the optimal point lies in and the low value of 10 in each case ensures that this point indeed provides a common equal margin to all users. Thus, all features of (C5) are satisfied. We further observe that the value of optimum is inversely related to the distance between the target rate point and, in terms of the area of its. As shown in Fig. 1 and Table V, among the target points under consideration, the innermost (4,1) has the largest db, while the closest to, (20,5), has the minimum db. A similar ordering of for intermediate points can be observed. The MHM algorithm always converges to the unique common optimum margin to all users as expected from (C5), if the rate region is convex. In particular, for all target points under consideration, 8 the MHM algorithm is seen to behave well and reach the unique optimum, which ensures the best equal margin for all users efficiently, while guaranteeing each users target rate requirement due to NRME. VI. CONCLUSION In this paper, we studied the problem of multiuser margin maximization in the min max sense. We examined the monotonicity and fairness properties that margin exhibits over the rate region. Based on these properties, we remodeled the problem as a NLSQ problem and developed an efficient solution using a scaled-gradient TR method and Broyden Jacobian update. We also derived a Newton Raphson method to estimate the margin, 8 In the MHM algorithm, it is not required to determine, from the beginning, whether the target rate is achievable or not. If the target rate is not within the rate-region, then MHM will STILL converge to an equal margin; BUT in this case, this final equal margin will be less than 1 (negative in db). From this negative margin, the service provider can clearly conclude that the target rate is outside the rate region. Negative margin means that the bit-error rate (BER) will be greater than the target BER (10 ). What the service provider does when this happens (negative margin) depends on service provider policy or user subscription types.

10 1580 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 while maintaining the target rate requirements of all users at each iteration, and made use of the IWF technique for subcarrier power allocation. By integrating these techniques, we further developed the MHM optimization algorithm and functional structure to achieve the best common equal margin for all users. This common equality does not follow from a compromise by any user and reflects the inherent fairness property of the margin. Due to this property, it will also be acceptable to all service providers, who share the bundles, as a fair scheme. Simulation results of various cases verify the convergence to unique min max optimal point within 5 10 iterations. Furthermore, the MHM algorithm can be implemented in xdsl systems using DSM scenario with only Level 1 coordination. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their detailed and constructive comments, which helped significantly improve this paper. REFERENCES [1] Asymmetric Digital Subscriber Line (ADSL) Metallic Interface, ANSI Standard T [2] Spectrum Management for Loop Transmission Systems, Comm. T1 Standard T , Jan [3] Very-High Speed Digital Subscriber Lines (VDSL) Metallic Interface, ANSI Standard T1E1.4/ R5, [4] T. Starr, J. M. Cioffi, and P. J. Silverman, Understanding Digital Subscriber Line Technology. Englewood Cliffs, NJ: Prentice-Hall, [5] J. W. Cook, R. H. Kirkby, M. G. Booth, K. T. Foster, D. E. A. Clarke, and G. Young, The noise and crosstalk environment for ADSL and VDSL systems, IEEE Commun. Mag., vol. 37, pp , May [6] K. B. Song, S. T. Cheung, G. Ginis, and J. M. Cioffi, Dynamic spectrum management for next-generation DSL systems, IEEE Commun. Mag., vol. 40, pp , Oct [7] K. J. Kerpez, D. L. Waring, S. Galli, J. Dixon, and P. 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Yu, M. Moonen, J. Verliden, and T. Bostoen, Optimal multi-user spectrum management for digital subscriber lines, IEEE Trans. Commun., Apr. 2006, to appear in. [21] W. Yu and R. Lui, Dual methods for nonconvex spectrum optimization of multicarrier systems, IEEE Trans. Commun., 2006, accepted for publication. Saswat Panigrahi received the B.Tech. degree (with National Academic Excellence Award) in electrical engineering from the Indian Institute of Technology (IIT), Kanpur, in 2003, and the M.Eng. degree (Dean s Honour List) in communications from McGill University, Montréal, QC, Canada, in Since October 2005, he has been working on R&D at Ericsson Canada. His current research interests include multicarrier systems, coding theory, and crosslayer optimization. Yang Xu received the B.E. degree from the Department of Telecommunication Engineering, Chongqing University of Posts and Telecommunications, Chongqing, China, and the M.E. degree from Faculty of Information Engineering, Beijing University of Posts and Telecommunications, Beijing, China, in 1998 and 2001, respectively. He is currently working towards the Ph.D. degree at McGill University, Montréal, QC, Canada. His research interests include multicarrier systems, resource allocation, and MIMO interference channel. Tho Le-Ngoc (F 97) received the B.Eng. degree (with Distinction) in electrical engineering and the M.Eng. degree in microprocessor applications from McGill University, Montréal, QC, Canada, in 1976 and 1978, respectively, and the Ph.D. degree in digital communications from the University of Ottawa, ON, Canada, in From 1977 to 1982, he was with Spar Aerospace Limited as a Design Engineer, and then a Senior Design Engineer, involved in the development and design of the microprocessor-based controller of Canadarm (of the Space Shuttle), and SCPC/FM, SCPC/PSK, and TDMA satellite communications systems. From 1982 to 1985, he was an Engineering Manager of the Radio Group in the Department of Development Engineering, SRTelecom, Inc., and developed the new point-to-multipoint DA-TDMA/TDM Subscriber Radio System SR500. He was the System Architect of this first digital point-to-multipoint wireless TDMA system. From 1985 to 2000, he was a Professor in the Department of Electrical and Computer Engineering, Concordia University. Since 2000, he has been with the Department of Electrical and Computer Engineering, McGill University. His research interest is in the area of broadband digital communications with a special emphasis on modulation, coding, and multiple-access techniques. Dr. Le-Ngoc is a Senior Member of the Ordre des Ingénieur du Québec, a Fellow of the Engineering Institute of Canada (EIC), and a Fellow of the Canadian Academy of Engineering (CAE). He is the recipient of the 2004 Canadian Award in Telecommunications Research, and recipient of the IEEE Canada Fessenden Award in 2005.