IN modern digital subscriber line (DSL) systems, twisted

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1 686 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 Optimized Resource Allocation for Upstream Vectored DSL Systems With Zero-Forcing Generalized Decision Feedback Equalizer Chiang-Yu Chen, Student Member, IEEE, Kibeom Seong, Student Member, IEEE, Rui Zhang, Member, IEEE, and John M. Cioffi, Fellow, IEEE Abstract In upstream vectored DSL systems using zero-forcing generalized decision feedback equalizers (ZF-GDFE), different decoding orders cause performance tradeoffs among the users. In this paper, these tradeoffs are characterized by formulating optimization problems with practical constraints. Lagrange dual decomposition and a two-step algorithm are used to solve the dual problems optimally with the computational complexity linear in the number of DMT tones. However, solving the tonal subproblem, which is shared by all the proposed optimization problems, associates with a high-complexity exhaustive search of! orderings, where is the number of users. Thus, this paper proposes two low-complexity algorithms in order to find suboptimal orderings: successive ordering search (SOS) with complexity ( 4 ) and modified greedy search (MGA) with complexity ( 3 ). Numerical results show that MGA performs well enough in finding the achievable rate region. For problems related to feasibility check, SOS is suitable for closely approximating the optimal solution. Index Terms Digital subscriber line (DSL), dynamic spectrum management (DSM), multiple access channel (MAC), dual decomposition, vectored transmission, QR decomposition, generalized decision feedback equalizer (GDFE), transmission optimization, resource allocation. I. INTRODUCTION IN modern digital subscriber line (DSL) systems, twisted pairs are significantly shortened because of the deployment of optical network units (ONU) or remote terminals (RT). The channel condition in the high-frequency band is dramatically improved so that high data rates can be delivered. In addition, the system performance becomes more interference-constrained. To combat interference in DSL effectively, many techniques of Dynamic Spectrum Management (DSM) are proposed [1]. Among the technologies of DSM, vectored DSL (also called vectoring) is the most powerful interference Manuscript received January 29, 2007; revised September 4, This paper was presented in part at IEEE Global Telecommunications Conference (Globecom), San Francisco, CA, The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Alex Gershman. C.-Y. Chen and J. M. Cioffi are with the Department of Electrical Engineering, Stanford University, Stanford CA 94305, USA ( chiangyu.chen@stanford.edu; cioffi@stanford.edu). K. Seong is with Qualcomm, Inc., San Diego, CA USA ( kibeoms@qualcomm.com). R. Zhang is with Institute for Infocomm Research, Singapore ( rzhang@i2r.a-star.edu.sg). J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA ( ). Digital Object Identifier /JSTSP cancellation technique because it actively cancels the interference via signal level cooperations rather than just avoids the interference by polite spectrum use. In upstream vectored transmission, the receivers are collocated at a central office (CO) or an ONU; thus, joint decoding is possible. Similarly, joint encoding at CO/ONU is possible for downstream vectored transmission. Therefore, the upstream channel can be modelled as a multiple-access channel (MAC), while the downstream channel can be described as a broadcast channel (BC). The interference seen in a vectored DSL system can be generally divided into two categories in-domain and out-of-domain interference (the latter is also called alien noise or alien crosstalk). The in-domain crosstalk is generated by either near-end or far-end transmitters in the same binder. In very-high-speed DSL (VDSL) standard [2], near-end crosstalk (NEXT) can be completely avoided by using frequency-division duplexing together with cyclic extension and frame synchronization [3]. That leaves far-end crosstalk (FEXT) and alien crosstalk the major impairments. In [4], a FEXT cancellation scheme using QR decomposition is proposed. In upstream transmission, the receiver in CO/ONU successively decodes the received signals based on the QR decomposed channel and previous decisions. The receiver is essentially a special case of the zero-forcing generalized decision feedback equalizer (ZF-GDFE) [5] or the zero-forcing vertical Bell Labs layered space-time (V-BLAST) architecture [6]. The equivalence of GDFE and V-BLAST is shown in [7]. When the noise between users is white, the upstream channel matrix is column-wise diagonally dominant (CWDD) [4]. Consequently, the subchannel gains hardly depend on the decoding order; the achievable rate region is very close to a hyper-rectangle, which means that there is no tradeoff among the data rates of users in the same vectored system. Moreover, the performance of ZF-GDFE can be closely approximated by zero-forcing linear equalizers (ZF-LE) [8]. However, in practical systems, alien disturbers (e.g., T1 users, out-of-domain DSL, and radio-frequency ingress) inject highly correlated noise to the vectored lines. Data rates can be increased further by predicting the noise in one line based on the noise observed in other lines [9]. When considering the channel and the noise whitening filter together, the equivalent channel matrix is no longer CWDD. As a result, the achievable data rates of ZF-LE may be much less than those of ZF-GDFE. Also, the subchannel gains depend on the ordering of QR decomposition, which means adopting different orderings over the tones produces various rate tuples. For a /$ IEEE

2 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 687 DSL system with vectored lines and tones, there are total possible rate tuples, where each of them corresponds to a set of decoding orders. The large number of possibilities makes resource allocation problems challenging. On the other hand, in downstream transmission, the information for different users can be jointly encoded based on the QR decomposition and a Tomlinson Harashima precoder [4]. Because of the row-wise diagonally dominant (RWDD) property of the downstream channel, the achievable rate region is very close to a hyper-rectangle [4]. Hence, a linear precoder [10] can approach the performance of [4] very closely. However, since the transmitter cannot predict noise, no alien crosstalk cancellation can be achieved even when the noise among the users are highly correlated. This property is consistent with the traditional information theoretical viewpoint, where the capacity region of BC is irrelevant to the noise correlation [11]. Therefore, the tradeoff of data rates among the users does not exist in downstream, and finding the best QR orderings does not provide much upgrade on performance. Since the potential benefit of using non-linear receivers over linear receivers is greater in upstream, this paper discusses resource allocation problems in upstream vectored transmission. Other than afore-mentioned ZF-GDFE, a minimum-mean-square-error (MMSE) GDFE is also of great interest since it is a canonical receiver structure [5]. That is, the boundary of capacity region can be achieved by MMSE-GDFE with proper allocation of transmit power. Recently, an efficient transmission-optimization algorithm is proposed [12]. However, there remain two difficulties in applying this technique together with MMSE-GDFE to practical systems. First, since the noise power seen at one user depends on the signal power of the others, the optimization process with MMSE-GDFE requires using interior point methods. The optimal transmit power distribution is iteratively updated, causing very high computational complexity. Second, in DSL systems, an SNR gap is adopted in bit loading to characterize the effects of channel coding [13]. Also, an SNR margin is reserved in order to keep the system from being disturbed by a sudden change of channel or noise statistics [1]. Unfortunately, the algorithm in [12] cannot solve the best achievable rate region with non-zero gap and margin. To overcome the difficulties of using MMSE-GDFE, an alternative approach with both low computational complexity and easy integration of SNR gap is desired. This paper studies transmission optimization in upstream vectoring using ZF-GDFE. As long as the decoding ordering is determined, the transmit power of different users is decoupled so that the PSD distributions have nice and simple single-user waterfilling expressions. Moreover, the SNR gap and margin can be directly applied to the rate formulas. Owing to the use of margin and bit-loading, tones with bad channel conditions are turned off automatically such that DSL lines always operate at high SNR. Thus, the degradation of using ZF-GDFE compared to MMSE-GDFE should be negligible. In this paper, three major optimization problems weighted sum-rate maximization (WSRmax), weighted sum-power minimization (WSPmin), and admission problem (AP) are discussed. By applying Lagrange dual decomposition [12], [14], [15], all these problems can be decomposed into the same tonal subproblem and their dual problems can be solved with complexity linear in the number of tones. Moreover, the duality gap can be proved to be zero [16], which means the solution to the dual problem is also the primal optimum. However, solving each tonal subproblem requires finding the optimal ordering, which can be achieved by using a tonal exhaustive search (TES) algorithm with complexity of. Therefore, the overall complexity still remains very high for systems with a large number of users. To reduce the computational complexity, two suboptimal solutions with polynomial complexity a successive ordering search (SOS) with complexity and a modified greedy algorithm (MGA) with complexity are proposed. Simulation results show that SOS, with slightly higher complexity than MGA, approaches the optimal solution in all three problems as well as the extensions of AP, including the rate maximization problem given a rate profile and the margin-maximization problem. On the other hand, MGA is more suitable for solving WSRmax and providing a rate region very close to that obtained by TES. These algorithms can also be directly applied to resource allocation of wireless systems with minimal changes. The rest of this paper is organized as follows: Section II defines the channel models. Section III formulates three optimization problems WSRmax, WSPmin, and AP. Lagrange dual decomposition, the dual problems, and the algorithms of solving dual problems are also explained in this section. Section IV presents TES, SOS, and MGA to solve the tonal subproblem and compares their complexity. Section V shows two practical applications of AP and the extension to wireless communications. Section IV shows simulation results and comparisons of optimal and suboptimal solutions. Finally, Section VII summarizes the paper. Notations: This paper uses upper case letters to denote matrices. Lower boldface letters are used to indicate vectors. and represent Hermitian and transpose operations, respectively. denotes the identity matrix. indicates a square diagonal matrix with the diagonal elements equal to the elements of the input vector. Subindex is used to represent the user index while subindex is the tone index. II. CHANNEL MODEL AND ZF-GDFE An upstream vectored DSL system consists of distributed transmitters at the customer premises equipment (CPE) end and receivers at the CO/ONU. Assuming there is only one twisted pair for each user, the system is essentially a single-input-multiple-output (SIMO) MAC because each transmitted signal can be observed at all the receivers, either from direct channel or FEXT. An upstream vectored DSL channel using discrete multitone (DMT) modulation with users can be represented as independent matrices for different tones: where is the tone index and, and are, respectively, the received vector, transmitted vector, and noise vector; all of them are with dimensions -by-1. The noise (1)

3 688 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 Fig. 1. ZF-GDFE receiver structure of a 3-user system at tone n. is modeled by a zero-mean complex Gaussian random vector, with a non-singular covariance matrix. is a -bychannel matrix, of which diagonal elements represent the direct channel gain, and off-diagonal elements denote the FEXT components. Additionally, is CWDD (each off-diagonal element has an absolute value much less than that of the diagonal element in the same column) [4]; thus, non-singular in practice. In this model, different tones can be treated as independent subchannels as in single-user DMT systems. This tonal decomposition is made possible because of the synchronization of all users DMT symbols, a standardized option in VDSL2 [2]. At the receiver side, a noise whitening filter can be applied to generate an equivalent channel model with white noise: where is the equivalent channel matrix, and represents the whitened noise with covariance matrix. Since the noise whitening filter is invertible, no information rate is lost. The authors in [4] proposed a receiver based on QR decomposition of the channel matrix. Since the equivalent channel is square and non-singular, a full QR decomposition can be computed: where is a permutation matrix, and is the corresponding ordering vector. Both of them are defined later in this section. is a unitary matrix, is a diagonal matrix, and is a monic upper triangular matrix. For now, is first assumed as an identity matrix. When multiplying by, (2) becomes Since is upper triangular and the noise is still white, the decision-feedback strategy of decoding from (4) becomes: To decode user s signal, are required. Therefore, assuming the previous decisions are correct, (5) can be implemented by successively decoding from to (2) (3) (4) (5) together with feedback. This receiver is a special case of ZF-GDFE with feed-forward filter and feedback filter. Moreover, the subchannel gain for the th user at this tone can be represented by. In (4) and (5), the decoding order is from user to user 1. This is only one of the possible permutations of decoding order. Since the channel matrix is CWDD, the columns are close to orthogonal. If is white, applying the noise whitening filter keeps the CWDD property of. Therefore, modifying the decoding order does not change the subchannel gains much [4]. However, the presence of alien crosstalk in DSL environments, such as RF ingress and crosstalk from other types of transmission, is very common [9]. Thus, is no longer CWDD due to the colored. Moreover, these options of the decoding order correspond to distinct sets of subchannel gains. Define as a -dimensional vector denoting the permutation on, where indicates the input dimension to be decoded th in the order at tone. Clearly,, and for all. Each corresponds to a permutation matrix, where For example, is an identity matrix if and only if. With different ordering, the decoding process remains the same. A feed-forward filter, including noise whitening, is first applied to the received signal. Then, successive decoding from user to user is implemented. The representation of feed-forward matrix is still while the feedback matrix should be changed to, based on the ordering. Fig. 1 illustrates an example of the receiver structure of a 3-user system with ordering matrix at tone. Based on the above definitions and (3), the subchannel gain given any can be expressed as Generally speaking, the later a user is decoded in ZF-GDFE, the higher their subchannel gain. This is because the FEXT from (6) (7)

4 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 689 previously decoded users is cancelled by the feedback filter, while the FEXT from later decoded users is avoided by the rotation of the feed-forward filter. For the user decoded first, the signals are received from the direction orthogonal to all the other users channel vector. The subchannel gain could be very small, if the remaining orthogonal component of this user s channel vector is small. On the other hand, the signals for the user decoded last are received from the direction parallel to the corresponding channel vector, which means this user s subchannel gain reaches the maximum possible value. Owing to the zero-forcing nature, once the orderings for all tones are determined, the SNRs for all users are entirely decoupled. Therefore, a user s data rate is unaffected by any other users bit and energy distribution. To be more precise, let and be the PSD and rate allocated for user at tone. The data rate for user with an SNR gap can be expressed as Therefore, a very intuitive algorithm to solve transmission problems is first to fix the set and then to calculate the bit and energy distributions for single-user bit-loading, which could be a rate adaptive (RA), margin adaptive (MA), or fixed margin (FM) bit loading [13]. Then, an exhaustive search over all possible orderings is done for all tones to get the optimal objective value. The following sections propose significantly more efficient optimization algorithms to solve a variety of resource allocation problems for the systems using ZF-GDFE. III. PROBLEM FORMULATION AND LAGRANGE DUAL DECOMPOSITION In a single-user multicarrier system using GDFE or V-BLAST receivers, the resource allocation problem focuses on either rate maximization or power minimization [17], [18]. In multiuser systems, rate and power cannot necessarily be shared by users so that the rate maximization or the power minimization could allocate the resources unfairly. Therefore, different from single-user systems, the tradeoff among users must be fully understood in multiuser systems. The tradeoff of transmission rates among users is characterized by the achievable rate region. Tofind the achievable rate region for ZF-GDFE in upstream vectoring, an exhaustive search of all possible orderings together with time-sharing is prohibitively time-consuming. Instead, since the rate region is convex, it can be characterized by solving weighted sum-rate maximization problems (WSRmax) given individual power constraint. Problem 1: Weighted sum-rate maximization (8) (9) The search of all possible non-negative rate-reward vectors generates the boundary of the whole achievable rate region. There are three constraints in this problem: The first constraint is the individual power constraint, where denotes the maximum power for user. The second constraint is the PSD mask, where represents the maximum PSD allowed at tone. This constraint prevents DSL from allocating too much power in certain frequency band. The last constraint in (9) is the bit cap for bit-loading, where is the maximum possible number of bits loaded at one tone. It should be noted that the feasible set of PSD in the last constraint depends on the ordering. Since the objective function of Problem 1 is related to a hidden combinatorial maximization of concave functions with various sets of, it is not a convex optimization problem [19]. In [16], the authors showed that many non-convex optimization problems in a multicarrier system can be solved with zero duality gap if the time-sharing condition is satisfied. To examine if Problem 1 satisfies the time-sharing condition, assume two WSRmax problems with different individual power constraints and and their solutions and, respectively. Then, the time-sharing condition is satisfied if for any WSRmax with the constraint, there exists a feasible point with the objective value greater than or equal to the same affine combination of the optimal objective values of the WSRmax problems with power constraints and. That is,, satisfying the constraints in Problem 1 with, such that (10) The rate in the right-hand side of (10) can be achieved by timesharing of and proportional to their weights and. Although time-sharing is not feasible in practice, it can be approximated by frequency sharing when the number of tones goes to infinity. This is because as, the subchannel conditions for adjacent tones are similar. By assigning as the interleaved PSD and ordering of and in frequency domain, the equality in (10) holds. It should also be noted that the resulting feasible solution of frequency-sharing still satisfies the constraints of PSD mask and bit cap, meaning these two extra constraints do not affect the basic result in [16]. This time-sharing condition suggests the problem be solved in the dual domain, if a low-complexity algorithm is available. The Lagrange dual decomposition, which is used prevalently in multitone systems [14] [16], can help develop algorithms on

5 690 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER The La- solving the dual problem with complexity linear in grangian of this problem can be formulated as (11) where and represent the sets of all and, respectively. denotes non-negative Lagrange multipliers. A feasible should be in the domain in order to satisfy the last two constraints in (9). The Lagrange dual function associated to this problem then becomes (12) And the dual problem is simply To apply dual decomposition, let (13) In the ellipsoid method, first a large ellipsoid containing the optimal Lagrange multiplier is initialized as a candidate region. At the calls of ordering step, the fixed is assigned as the center of the ellipsoid. Then, the candidate region is cut based on the subgradient in (16). Finally, the volume of the ellipsoid vanishes (less than a small value ) and the ellipsoid converges to a certain Lagrange multiplier. The updates of the Lagrange multipliers and the candidate region are discussed thoroughly in [16]. The overview of the two-step algorithm can be summarized as follows. Algorithm 1: Two-step Algorithm 1: Initialization: Select a large ellipsoid and assign as the center of the initial ellipsoid 2: While (the volume of the ellipsoid), do 3: Lagrange multiplier step- update by the ellipsoid method 4: Ordering step- 5: For to 6: Solve and given (algorithms in Section IV) The Lagrange dual function (12) becomes (14) (15) Besides the achievable rate region and WSRmax problem, the achievable power region given target rate constraints and its associated weighted sum-power minimization (WSPmin) problem are also of interest. The problem is a dual of WSRmax and can be expressed as: Problem 2: Weighted sum-power minimization To solve the dual problem, a typical two-step algorithm in [12], [14], [16] can be utilized. In this paper, these two steps are called the Lagrange multiplier step and the ordering step, respectively. They are performed alternatively until the Lagrange multipliers converge. In the ordering step, is fixed while and are solved. Hence, the second term of (15) is a constant and independent dual-decomposed tonal subproblems in (14) need to be solved. Since there are subproblems, the total complexity is proportional to the number of tones. When compared to the exponential complexity of exhaustively searching every ordering over all the tones in the primal problem, the dual decomposition method already shows a great reduction of computations. The next section presents the detailed solution of the ordering step and its complexity analysis. On the other hand, in the Lagrange multiplier step, and are fixed while is updated to obtain a smaller Lagrange dual function. Thus, all kinds of subgradient searches such as the ellipsoid method in [16], [12] can be used. A vector is a subgradient of at if for any. Following the same proof as in Proposition 1 in [16], the subgradient for WSRmax can be chosen as (17) Similar to WSRmax, varying the power price vector can characterize the achievable power region. The first constraint in (17) is the individual rate constraint, which requires user to transmit with data rates no less than. The last two constraints impose PSD mask and bit cap as in Problem 1. Same as in WSRmax, the dual problem of WSPmin can be solved efficiently via Lagrange dual decomposition. First, the Lagrangian can be written as (16) (18)

6 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 691 where is the vector denoting the nonnegative Lagrange multipliers. The Lagrange dual function then becomes: a subgradient method. Following the similar steps in (11), (12), (18), and (19), the dual function of AP becomes (22) (19) where and are Lagrange multipliers for rate and power constraints. Also where (23) (20) The dual problem is to solve for the maximum of given. The same two-step algorithm as in WS- Rmax applies here, where the Lagrange multiplier step uses a subgradient method to update the Lagrange multiplier until convergence. The subgradient can be chosen as. The ordering step fixes and solves the tonal subproblem in (20). Essentially, (20) has the same expression as (14) except for the different input variables. Since the Lagrange multipliers are fixed in the ordering step, the two subproblems generate the same solution set, once and are given. Moreover, let the optimal Lagrange multiplier and the amount of power used by user be. The solution to WSPmin then not only corresponds to a boundary point of the power region, but also achieves a boundary point of the rate region for the same channel under a weighted-sum power constraint [20]. In addition, that boundary point can be characterized by solving a WSRmax with weight under the weighted-sum power constraint. For the previous two optimization problems, either power or rate is the constraint while the other is optimized. In many practical situations, both of them could be constrained, meaning DSL lines should give best effort to attain their target rates with limited power. In a multiuser system, users should coordinate to satisfy all the desired transmission rates given power constraints. To understand if a given set of constraints is achievable or not, one can solve the following admission problem (AP). Problem 3: Admission problem It is obvious that when the Lagrange multipliers are fixed, (23) again becomes the same dual-decomposed subproblem as in (14) and (20). The dual problem is to maximize subject to ; the aforementioned two-step algorithm can be used as well. In the ordering step, (23) degenerates to the same tonal subproblem as in (14) and (20). and can be updated in the Lagrange multiplier step with a subgradient. The subgradient is now a -dimensional vector since the variables of the dual function can be cascaded as. Therefore, it can be expressed as (24) When an AP is feasible, the dual function should always be non-positive. Otherwise, given a set of PSD, ordering and Lagrange multipliers that generate a positive dual function, a positive scaling of the Lagrange multipliers makes the objective function of the dual problem go to infinity, meaning no maximum of the dual problem is available. The other way to view this infeasibility is that if the primal problem is feasible, the positive dual function yields a negative duality gap, which implies the definition of the dual function is flawed under the given constraints. Therefore, whenever a positive dual function is observed during the optimization routine, the problem can be claimed as an infeasible one immediately. On the other hand, when the primal problem is feasible, the dual objective function is zero by setting both and zero vectors, which further shows that the duality gap is zero. (21) The zero objective function indicates that AP is a feasibility problem. Therefore, the objective value is zero if and only if the feasible set of is non-empty. Otherwise, the objective value goes to infinity. AP again can be solved by using Lagrange dual decomposition, where the Lagrange multipliers are updated by IV. SOLVING THE TONAL SUBPROBLEM In the previous section, three optimization problems are formulated, and it is shown that all of them can be solved using the Lagrange dual decomposition and a two-step iterative algorithm, of which the ordering step contains dual-decomposed tonal subproblems. This section presents detailed algorithms of finding the solutions to the subproblems. First, the subproblem can be rewritten as (25)

7 692 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 All the notation is the same as defined in previous sections except that the tone indices are discarded. Since the maximization can be done by maximizing over first with given, the optimal solution of the subproblem can be found in a straightforward manner. First, given an ordering can be found by the first derivative of the concave objective function, which is the waterfilling expression: (26) where denotes the water-level for the th user and. It should be noted that the water-level for user is the same over all the tones; thus, the waterfilling is implemented automatically for each user. A. Tonal Exhaustive Search The subproblem can be solved optimally by using a tonal exhaustive search (TES), which means to check all possible orderings for each subproblem and find their PSD in (26). The algorithm can be summarized as follows. Algorithm 2: Tonal Exhaustive Search 1: While not all orderings are examined, do 2: Get a new ordering 3: Calculate the QR decomposition and subchannel gains for 4: Calculate the PSD by (26) 5: If the objective value is increased 6: 7: 8: Return, and the maximum of the objective value There are many methods regarding computing QR decomposition in the literature, including using a sequence of Householder transforms [21]. In each Householder transform, a symmetric and unitary Householder matrix is found. And the original channel matrix is upper-triangularized column-by-column, i.e., the th Householder matrix zeros the elements below the diagonal of the th column. The resulting is essentially the multiplication of all the Householder matrices. That is (27) The complexity of implementing a QR decomposition via Householder transforms is. Therefore, the complexity of solving the subproblem via TES is. The factorial growth of complexity makes the algorithm intractable when the number of users is large. To avoid repeating the same calculations whenever the ordering step is called, a pre-process step may be added to compute all possible subchannel gains for all orderings before the optimization starts. Therefore, by checking the summation of terms for each of the orderings, the computational complexity is reduced to. However, the subchannel gains need to be stored during the whole optimization process, causing a memory size problem. Based on the structure of Householder transform, the subchannel gain for one user to depend only on the later-decoded (or early-decoded) users, while the relative orderings for the users decoded later (or earlier) does not have any effect on it. Thus, for each user there are total possible values of subchannel gain, and a total memory storing subchannel gains is required. In vectored DSL systems, there could be 10 to 20 vectored lines. Because of the exponentially growing computational complexity and memory size, TES is impractical when the number of users are large. Since TES still provides the optimal solution to our optimization problems, it can serve as a performance benchmark to evaluate other low-complexity algorithms for small. B. Successive Ordering Search To break down the exponential computational complexity of TES, a successive ordering search (SOS) algorithm with polynomial complexity is proposed. First, given a QR decomposition of a matrix, of interest is the determination of the QR decomposition of another ordering, where for all except and for some. That is, the ordering of the th last decoded user and the th last decoded user are switched. The reason for this need is justified later in this section. Without loss of generality, is assumed to be, and thus. Therefore, the QR decomposition can be expressed as where of and, and (28) and are the -th columns By comparing the first columns of (28), (29) (30) It can be seen immediately that and for all and satisfies (30), meaning the first columns of, and matrices can remain the same

8 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 693 after the switch of ordering. Comparing the th and the columns, the following equations can be obtained: th (31) Since both and are unitary matrices, and for all. Also, because matrices are monic, (31) can be simplified as (32) Therefore, the unknowns in (32) can be solved sequentially as (33) Finally, for the remaining columns in (28), (30) still holds for. Because the columns of matrices are orthogonal and the span of and is the same as the span of and and for. Moreover, for all and. Equation (34) completes the characterization of the new matrix: From the above derivations, it can be observed that when a user switches its ordering with their adjacent user, only two columns of matrix, two elements of matrix, and two rows of matrix have to be recalculated. The computational complexity is, less than the complexity of a QR decomposition. When this property is applied to find the objective value in the tonal subproblem, orderings can be checked more efficiently. SOS initializes the problem with a given ordering and its corresponding objective value. Then, it successively updates the ordering of each user, with all the other users relative ordering kept the same. For example, in a 3-user system with an initial ordering, the algorithm starts from updating user 1 s position while keeping user 2 and user 3 s relative orderings the same. Therefore, two different orderings and are compared with the initial ordering. When checking these two orderings, the efficient method of calculating the QR decomposition and subchannal gains in (33) and (34) can be applied. Of course, the ordering with the maximum objective value replaces the original ordering. Assume after the update of user 1 s position, ordering gives the largest objective value. Then, the algorithm proceeds to check the user 2 s position while keeping user 1 and user 3 s relative ordering the same, i.e., and are compared with the current ordering and the best one is selected. Continuing this process for user 3 completes the first iteration. After that, another round of iterations starting from user 1 again is implemented. Since the objective value always increases and is upper-bounded, SOS will finally converge to a certain suboptimal ordering. This algorithm can be summarized as the following: Algorithm 3: Successive Ordering Search 1: Initialize, calculate the objective value 2: While does not converge 3: For to 4: Find such that 5: For to 6: Switch the ordering so that 7: Keep the relative ordering of other users the same 8: Calculate the new QR matrices from (33) and (34) and the objective material 9: If the objective value is improved 10: Assign current ordering, update 11: Return, and the objective value In each iteration, SOS requires all users to switch their orderings to different position. There are roughly QR updates, each with complexity. Therefore, the total complexity of SOS is, where is the number of iterations. According to the simulation experience, SOS usually converges in 2 or 3 iterations. In fact, the number of iterations can also be assigned with a very small degradation of performance. Since SOS solves the subproblem in polynomial time, it can be utilized in practical systems with large. In addition, SOS avoids the memory problem in TES. C. Modified Greedy Algorithm The complexity of SOS can be further reduced by a modified greedy algorithm (MGA). The combination of greedy algorithm and QR decomposition was first studied for BLAST receivers in single-user systems [22], [23], where the receive dimension with the best channel SNR is selected successively to determine the decoding order. Later, the authors in [24] present a greedy algorithm that solves the sum-rate maximization problem (34)

9 694 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 in MIMO broadcast channels using zero-forcing dirty paper coding (ZF-DPC) scheme [25]. ZF-DPC is also based on QR decomposition of the channel matrix and can be considered as a dual of ZF-GDFE for MAC. In order to determine the ordering, the algorithm also successively selects the users with the best channel SNR. The comparison between the algorithm in [24] and another suboptimal transmission scheme in BC, zero-forcing beamforming, is analyzed in [26], [27]. Instead of picking the highest SNR, MGA iteratively assigns a user from to, depending on which user helps maximize the objective value of the subproblem the most. Like [22] [24], the assignment of ordering follows the procedure of QR decomposition. Therefore, only one full QR decomposition is performed. To be more specific, when selecting for all the users are checked with equal to the squared norm of the th column of the channel matrix, which means putting all the users in the last position to be decoded and calculating how much they contribute to the objective value. The user with the largest is chosen to be. This selection can be expressed as (35) where is assigned by (26). The Householder matrix can thus be computed based on the selected user. Continuing the process from to, the orderings can be determined as the following: where is defined as (36) complexity of MGA is. Similar to SOS, no coefficients are needed to be stored in memory between the calls of MGA. However, the complexity reduction over SOS causes performance degradation. Section VI shows some examples where MGA does not approximate TES as close as SOS. This is because MGA only checks one term of the objective at a time. If the selected user at the current stage can contribute much more than the remaining users in a later stage, it would never be discovered. SOS, on the other hand, always checks the whole objective value of the tonal subproblem when comparing two different orderings. Every user is switched to all the possible positions in each iteration. Consequently, any large contribution to the objective value from any user at any position of the ordering is likely to be detected through the iterative search of orderings, resulting in better performance. V. EXTENSIONS OF PROPOSED ALGORITHMS A. Practical Applications of Admission Problem Among the proposed optimization problems, WSRmax and WSPmin are mainly used for characterizing the achievable rate region and power region, respectively. In a system operation viewpoint, only one rate tuple inside the achievable rate region is selected by the service provider. The rate tuple could be selected based on fairness or the price paid by subscribers. Unfortunately, there is no intuitive way to find the relationship between these factors and the weight vectors in WSRmax, unless the whole achievable rate region in a -dimensional space is obtained. In order to determine the operating bit and PSD distributions more effectively, two additional optimization problems are presented, and their solutions can be found by solving AP properly. First, the rate maximization problem given a rate profile and a minimum target rate tuple is formulated as follows. Problem 4: Rate maximization given a rate profile and a minimum target rate tuple MGA can be summarized as follows. (37) Algorithm 4: Modified Greedy Search 1: For to 2: For to 3: Find a new user not yet examined in this step 4: Compute and by (37) and (26) 5: If is increased 6: Assign 7: Calculate the Householder matrix if 8: Return, and the objective value Besides the computation of one QR decomposition, the additional computations on subchannel gains and taking the maximum do not increase the order of the complexity. Thus, the (38) In this problem, a rate profile is defined as a vector, where for all and. denotes the minimum guaranteed rates for every user. must be inside the achievable rate region so that the objective can be positive. The objective and the rate constraint are visualized in Fig. 2(a). The candidate rate tuples are characterized by an arrow, starting from with a direction of. The optimization objective is to maximize the length of the arrow, while keeping the whole arrow inside the achievable rate region. Since the optimal value corresponds to a rate tuple at the intersection of the arrow and the boundary of the achievable rate region, a bisection search of can solve this problem. For each search, AP is solved to

10 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 695 Fig. 2. Two applications of AP. (a) Rate constraint for rate maximization problem given rate profile and minimum desirable rate. (b)margin maximization given a desired rate tuple. check the feasibility with rate constraint. This bisection search is made possible because for any is feasible from the convexity of the achievable rate region. In practice, the service provider may simply select and to determine the operating rates by solving Problem 4. For example, can be the bandwidth for data streaming such as voice streaming or IPTV service, while can be assigned proportional to the price paid for other applications. Another possible application of Problem 4 lies in the cross layer resource allocation area. In [28], a queue-proportional scheduling (QPS) policy is presented to achieve good throughput, delay, and fairness properties. This scheduling policy assigns each user a data rate proportional to that user s queue backlog size. When QPS is applied to upstream DSL systems, the desired rate tuple can be obtained by solving Problem 4 with the rate profile in proportion to the queue lengths. The other problem is that given a target rate tuple, it is desirable to allocate the resources properly so that the SNR margin is maximized. In (8) and our previous derivations, the SNR margin is given and combined with the SNR gap. In fact, SNR margin can be a design parameter. To maximize margin while keeping the given rate tuple achievable, a margin-maximization problem can be formulated as follows. Problem 5: Margin maximization (39) The margins for all the users are the same so that the BER performance is not dominated by the user with the lowest margin. The achievable rate region is a function of margin, which can be seen in Fig. 2(b). Since given the same PSD distribution, a user s data rate increases as the margin decreases, the achievable rate region for a lower margin is strictly larger than the region for a higher margin. Thus, the target rate tuple must be on the boundary of the achievable rate region with the optimal margin. To solve the margin-maximization problem, a bisection search on together with the AP can be used. In each AP, is replaced by and the feasibility of is tested. Eventually, it converges to the optimal margin. B. Extension to Wireless Communications Thus far, all the algorithms are presented in DSL context. In fact, they are also directly applicable to uplink wireless channels with multiple receive antennas at the base station. In a SIMO MAC experiencing frequency flat fading, the following changes to the current derivations are required. a) The tone index in DSL environment represents the index of fading states. b) The data rate (power) is no longer a sum of individual rates (power) of all the tones; it should be changed to a weighted sum over all fading states with the weight equal to the probability of each fading state. (c) Since the number of receive antennas is less than that of users in general, the -by- channel matrix is fat rather than square. Hence, there are at most positive subchannel gains if a full QR decomposition of the channel matrix is calculated. This implies the number of users transmitting at each fading state is no more than. Those active users at each fading state and their decoding ordering are automatically selected when solving the presented optimization problems, of which subproblems can be solved by use of TES, SOS, or MGA. This intelligent selection of active users also achieves multiuser diversity [29]. Moreover, the proposed algorithms can be applied to the dual downlink broadcast channels employing ZF-DPC [25]. VI. SIMULATION RESULTS This section tests the proposed algorithms in various veryhigh-bit-rate DSL (VDSL) systems complex DMT tones with subchannel width khz, upstream transmit power 14.5 dbm, a bit cap of 15 bits, carrier mask, and PSD mask are used, which all follow the VDSL standard [30]. Although (26) assumes that the number of bits per tone is continuous, DSL systems allow only an integer number of bits. This discrete bit-loading can be adopted with a slight modification. Let the

11 696 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 Fig. 3. Achievable rate regions for a 2-user system by solving WSRmax. Loop lengths are 1 K and 3 Kft. number of bits loaded for user be a non-negative integer. The PSD and bit can then be assigned as TABLE I SIMULATION RESULTS OF WSRMAX FOR A 6-USER DSL SYSTEM (40) This discrete bit-loading may affect the convergence behavior and optimality. Fortunately, in all of our simulations, the optimization process always converges and the resulting duality gap of using TES is negligible, implying that the granularity of this integer assignment is small enough. The achievable rate region of a two-user system is first shown in Fig. 3. The loop lengths of the first and the second lines are 1000 and 3000 ft, respectively. One VDSL disturber outside the vectoring domain with the loop length 1000 ft is introduced as the common alien noise source. The off-diagonal elements of noise covariance matrices are assigned based on two correlation coefficients: and. This high noise correlation is a reasonable assumption because a measurement in [9] shows the noise correlation coefficient is close to 1 when there is only one disturber. Even with multiple disturbers, usually there exists a dominant one with noise PSD much higher than the others. Besides the alien noise, a white noise floor with PSD dbm/hz is also added to all the users. The rate regions using TES, MGA, and the same ordering for every tone with ZF-GDFE are presented. With only two users, SOS is equivalent to TES since every possible ordering is checked. For comparison, the ZF-LE achievable rate region is also shown. The achievable rate region obtained by MGA is almost as large as the one achieved by TES. The loss in weighted sum rate is negligible when the weight for the long loop is large while it is more significant when the weight of the short loop is large. As the noise correlation decreases, the rate region shrinks. However, even in the conservative setting of, ZF-GDFE provides significantly larger date rates than ZF-LE receivers. Also, in Fig. 3, using the same ordering for every tone does not necessarily achieve the boundary point of the achievable rate region. For example, with, using the same ordering causes a 12% rate loss for user 1 when the rate of user 2 is maximized. This is because all the tones for user 2 reach their maximum possible PSD (confined either by PSD mask or by the bit cap) when user 2 is decoded later on every tone. The individual power constraint is thus not tight. On the other hand, when solving the optimization routine, the tones of user 2 that hit the bit cap when decoded later can yield the priority of decoding to the other user. The subchannel gain of user 2 at this tone is reduced while the subchannel gain of user 1 is increased. However, since the individual power constraint is not tight, more power can be allocated for user 2 to reach the bit cap as before. Therefore, by using more power but still obeying the power constraints, the data rate for user 2 remains the same. For user 1, the increase of subchannel gain can help loading more bits, so the overall data rate is increased. Solving the optimization problems indirectly determines the tones that should yield the decoding priority to user 1.

12 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 697 Fig. 4. Profile of the achievable rate regions for a 4-user system. Loop lengths are 3 K, 3 K, 1 K, and 1 Kft. In Table I, a result of WSRmax for a 6-user system is demonstrated. The loop lengths from user 1 to user 6 are 500, 1000, 1500, 2000, 2500, and 3000 ft, respectively. The weight vector is chosen randomly. Other simulation parameters are the same as the previous example, where. Five rate tuples using different receiver structures or optimization schemes are shown. Among them, the ZF linear receiver is simply to multiply the received signal by the inverse matrix of the channel matrix [8]. In addition, the fixed ordering is selected according to the well-known result in [31], where the user with the larger weight goes later in the decoding order for all tones. This argument is true for multiple access channel with MMSE-GDFE receivers and zero SNR gap. However, with ZF-GDFE and practical constraints such as non-zero gap and bit cap added, the same ordering for every tone is not necessarily optimal, which is already shown in Fig. 3 as well. Comparing the first three algorithms, it can be observed that the algorithm with higher complexity results in higher weighted sum-rate. SOS shows almost identical performance to TES, while MGA has about only 1.1% rate loss. Although it seems that using the fixed ordering for all the tones also gives a reasonably good weighted sum-rate (3.0% loss), this method can only produce at most rate tuples, which is insufficient to characterize the entire achievable rate region. The linear ZF receiver does not execute successive interference cancellation, and provides significant performance loss when the noise is correlated. In Fig. 4, simulation results obtained by solving rate maximization problems given rate profiles are shown. There are four users with loop lengths 3 K, 3 K, 1 K, and 1 Kft. The data rates for the users with the same loop length are equalized, i.e., and. A profile of the achievable rate region is then obtained by sweeping the ratio of to. Each of the points in Fig. 4 corresponds to a solution to Problem 4. Again, it can be seen that SOS performs very close to TES TABLE II SIMULATION RESULTS OF A MARGIN MAXIMIZATION PROBLEM and the loss is negligible. However, when the rate for user 3 is fixed, the loss of user 1 s rate by using MGA is as large as 21%, which implies MGA rejects too many rate tuples that are actually feasible. The result of another application of AP is shown in Table II. Margin-maximization problems are solved for a four-user system as in Fig. 4 with target rates 5 Mbps, 5 Mbps, 20 Mbps, and 20 Mbps. MGA, which causes 1.96 db margin loss with respect to TES, is not a suitable suboptimal solution. SOS, with only 0.10 db margin loss, is a better suboptimal scheme for problems associated with the check of feasibility (AP). VII. SUMMARY This paper investigates transmission optimization and resource allocation for upstream vectored DSL systems using zero-forcing-based successive interference canceller. Three main categories of problems WSRmax, WSPmin, and AP are formulated. Because of the zero-forcing nature of the receiver, the optimal resource allocation satisfies single-user waterfilling, while finding the decoding orders for all the tones is a complicated optimization problem. The Lagrange dual decomposition is applied to these problems so that they can be solved in the dual domain with the complexity linear in the number of tones. Thanks to the time-sharing property, the duality gap is virtually zero. However, solving the best ordering requires exhaustive searches, of which complexity grows with the factorial of the number of users. To reduce the complexity, two practical algorithms SOS and MGA are proposed to solve the tonal subproblem efficiently. Simulation results

13 698 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, DECEMBER 2007 show that both SOS and MGA can provide good suboptimal approximations to TES in WSRmax problems. As far as AP is concerned, with slightly higher complexity, SOS performs much better than MGA in approximating the optimal solution. In the problem formulations, practical constraints such as SNR gap, discrete bit-loading, PSD mask, and bit cap are all considered. Therefore, the proposed practical algorithms enable future DSL systems to assign data rates, bit distributions, and energy distributions effectively. ACKNOWLEDGMENT The authors would like to thank anonymous reviewers for their constructive comments on the initial manuscript. REFERENCES [1] T. Starr, M. Sorbara, J. M. Cioffi, and P. J. Silverman, DSL Advances. Upper Saddle River, NJ: Prentice-Hall, [2] Very-High-Speed Digital Subscriber Line Transceivers 2 (VDSL2) ITU-T Recommendaiton G [3] F. Sjoberg, M. Isaksson, R. Nilsson, P. Odling, S. 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Theory, vol. 44, no. 7, pp , July Chiang-Yu Chen (S 05) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 2002, and the M.S. degree from Stanford University, Stanford CA, in 2004, both in electrical engineering. He is currently pursuing the Ph.D. degree at Stanford University. His current research interests include multiple-input-multiple-output (MIMO) systems, multiuser communications, and Dynamic Spectrum Management (DSM) in DSL systems. Kibeom Seong (S 05) received the B.S. and M.S. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1997 and 1999, respectively. He is currently pursuing the Ph.D. degree at Stanford University, Stanford, CA. From 1999 to 2002, he served as a Faculty Member in the Department of Electrical Engineering, Korea Military Academy, Seoul. He is currently with Qualcomm, Inc., San Diego, CA. His research interests include communication theory, multiuser information theory, and dynamic resource management in wireless and wireline communication systems. Rui Zhang (S 00-M 07) received the B.S. and M.S. degrees in electrical and computer engineering from National University of Singapore in 2000 and 2001, respectively, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in Since 2007, he has been a research fellow with the Institute for Infocomm Research (I2R), Singapore. His recent research interests include cognitive radio networks, cooperative communication systems, and multiuser MIMO transmission systems.

14 CHEN et al.: OPTIMIZED RESOURCE ALLOCATION 699 John M. Cioffi (S 77-M 78-SM 90-F 96) received the B.S. degree from the University of Illinois, Urbana-Champaign, in 1978, and the Ph.D. degree from Stanford University, Stanford, CA, in 1984, both in electrical engineering. He was with Bell Laboratories from 1978 to 1984, and IBM Research from 1984 to He has been a Professor of Electrical Engineering with Stanford University since He founded Amati Communications Corporation in 1991 (purchased by Texas Instruments in 1997), and was Officer/Director from 1991 to He is currently Chairman and founder of ASSIA, Inc., a company responsible for the introduction and use of Dynamic Spectrum Management by several large telephone companies. He has served on the boards of directors of public companies Amati, Marvell, and Integrated Telecom Express. He currently is on the boards of directors of Teknovus, Teranetics, ClariPhy, and ASSIA. He is on the advisory boards of Wavion and Amicus. His specific interests are in the area of high-performance digital transmission. He has published over 280 papers and holds over 80 patents, most of which are widely licensed, including basic patents on DMT, VDSL, and Vectored transmission. Dr. Cioffi has been the recipient of various awards: International Marconi Prize (2006), Holder of Hitachi America Professorship in Electrical Engineering at Stanford (2002), Member of the National Academy of Enginereering (2001), IEEE Kobayashi Medal (2001), IEEE Third Millennium Medal (2000), IEE JJ Tomson Medal (2000), University of Illinois Outstanding Alumnus (1999), IEEE Fellow (1996 Committee T1 Outstanding Achievement Award of the ANSI (1995), Outstanding Achievement award from the American National Standards Institute for contributions to ADSL (10/95), NSF Presidential Investigator ( ), IEEE Communications Magazine Best Paper Award (1991), and IEEE ISSLS Best Paper Award (2004), Faculty Development Award from IBM Research ( ).