Spectrm Balancing for DSL with Restrictions on Maximm Transmit PSD Driton Statovci, Tomas Nordström, and Rickard Nilsson Telecommnications Research Center Vienna (ftw.), Dona-City-Straße 1, A-1220 Vienna, Astria Emails:{statovci, nordstrom, nilsson}@ftw.at Abstract The importance of a power spectral density (PSD) mask restriction is often overlooked when optimizing the spectrm sage for mltiser digital sbscriber lines (DSL) systems. However, by developing the optimization strategies based only on the PSD constraints (masks) we can tremendosly redce the comptation complexity compared to the methods only based on the total power restriction. In this paper we introdce a maskbased spectrm balancing () algorithm and demonstrate the near optimm performance of this optimization approach. Frthermore, we show that besides standards compliance, PSD restriction is also needed to ensre the convergence of iterative spectrm balancing methods, which se dal decomposition optimization. I. INTRODUCTION In recent years a nmber of spectrm balancing methods have been introdced for digital sbscriber line (DSL) systems in order to optimize the spectrm tilization for mltiser network scenarios. The problem is to design the transmit spectra for U DSL systems (sers) sharing a cable bndle and ths interfering with each other. This is typically described as an optimization problem where the aim is to the sm of weighted bitrates. Sch an optimization is sally constraint by some limits on the sed power. Two approaches exist: one is to restrict the total power and the other is to set a PSD mask where the power is restricted over freqency. These two approaches can also be combined. Since the introdction of DSL transmission techniqes, 20 years ago, standardization bodies have set both kinds of limits on the sed power. The total power limit is mostly aimed at redcing power consmption and lessening the demands on the analog front-end. The PSD limits are mostly set for radio freqency interference egress and to redce the distrbance of other (legacy) DSL systems. In all standards p till now, inclding the latest very high-speed DSL standard known as VDSL2, the total power nder the PSD mask aims to match the total power allowed. Therefore it seems reasonable to base the spectrm balancing optimization solely on the PSD mask instead on total power. Exploring this concept, we have developed a new spectrm balancing algorithms that we call mask-based spectrm balancing (). Frthermore, or motivation was driven by instability problems we have enconter when implementing the iterative spectrm balancing () [1], [2] which only ses a total power constraint. This work was partially financed by the Astrian Kpls program. When evalating spectrm balancing algorithms the network scenario (the nmber of sers and cable topology) might skew the reslts in varios directions. For example, if only two sers are selected (might be good for reasons like comptational convenience or being able to visalize the rate regions) methods that can emphasizes freqency division mltiplexing, like optimal spectrm balancing (OSB) [3] and, shines. If we instead have a scenario with many sers they will all fight for the sed freqencies and no one gets a freqency by itself, and the simple methods like iterative water-filling [4] or normalized-rate iterative algorithm [5] will do almost as well as the optimal ones. In this paper we will still se a two ser scenario. We do this mostly for rate region visalization reasons. This will be to the disadvantage of, as discssed above, and to or experience becomes even closer to the optimm when more sers are inclded into the simlation scenario. Another problem when doing these kind of evalations is that the largest benefit with OSB is for scenarios with long distance between nodes and when total power is tilized, that is, for long lines. Ths, few investigations has been made for realistic scenarios for cabinet deployed VDSL2 (max distance of 500-800 m). However, when sch scenarios are tested, OSB and behaves nexpectedly and shows convergence problems. To highlight these problems and becase we think it is more realistic scenario, we will also simlate a VDSL2 cabinet deployment scenario with relatively short loops. The paper is strctred as follows. In the following section we show how to calclate bitrates in mltiser DSL environments and give a short introdction to the dal decomposition based optimization for DSL. Then in Section III we describe the need for PSD restriction and conseqences for not inclding it into optimization. In Section IV we describe or maskbased spectrm balancing () algorithm and following that we present some simlation reslts where we compare with. In the end we give some conclding remarks. II. PRELIMINARIES In this paper we will only analyze DSL systems that se a freqency division dplex (FDD) transmission scheme. For sch systems, based on Shannon s capacity formla the nmber of total bits that can be transmitted in a discrete mltitone (DMT) symbol (at a certain bit-error rate) for a particlar
ser is determined as R = ( log 1 + Hn P n ) ΓN n, with (1) n I N n = HvP n v n + P,V n, (2) v=1 v where I denotes the set of sbcarriers sed in a particlar transmission direction and it comprises N sbcarrier; Γ is the signal-to-noise ratio gap; N n, P n, P,V n denote the PSD of ser in sbcarrier n of noise, transmit signal, and the sm of backgrond and alien noises, respectively; Hv n denotes the sqared magnitde of channel transfer fnction from ser v to ser, i.e., it represents either the direct channel (with v = ), or far end crosstalk (FEXT) copling. As shown in [3], the optimization goal for DSL can be formlated as an optimization problem where the objective is to the sm of weighted bitrates: P n ;,n sbject to: w R, n I (3a) P n T max,, (3b) P n 0,, n I (3c) where w denotes the weighting vale (or short: weight ) assigned to ser and T max denotes the total power constraint for ser. Withot loss of generality, the weights can be selected sch that U w = 1. We increase the bitrate of ser compared to the bitrates of the other sers by increasing its w. The total power constraint T max is sally selected to be the same for all sers. To solve this constrained optimization problem, the total power constraint from (3b) is incorporated into the cost fnction (3a) by defining the Lagrangian fnction [6]: ) L = w R + ( λ T max n I P n, (4) where λ denotes the Lagrangian mltiplier of ser. Based on the definition of Lagrangian, the optimization problem (3) can be written as L(w, λ, P n ), (5a) P n;,n sbject to: P n 0,, n I λ 0,. (5b) (5c) By collecting the terms that belong to the same sbcarrier (4) can be rewritten as L = n I L n + λ T max, (6) where L n is the Lagrangian on sbcarrier n and is given by L n = w R n λ P n. (7) Sch an approach in optimization is known as dal decomposition, where the optimization is divided into N per-sbcarrier optimization sbproblems that are only related throgh the weighs w and Lagrangian mltipliers λ. This in fact leads to an optimization which has a complexity that scales linearly with the nmber of sbcarriers. However, note that solving directly optimization problem (5), as has been done in [3], still has an complexity that increases exponential with the nmber of sers (lines). From the theory of the Lagrangian fnctions it is known that at the convergence point, Lagrangian mltipliers will be selected sch that either the total power of ser satisfies n I Pn = T max or λ = 0. We achieve different bitrate combination among the sers by changing weights assigned to them. By trying ot all weights combination among the sers, we gain the bondary of the bitrate region. The reason for this is that by dal decomposition optimization only bitrates that lies on the rate region bondary can be fond, since only these bitrates the sm of weighted bitrates. III. THE NEED FOR CONSTRAINING THE PSD To solve the optimization problem (5) a nmber of algorithms have been proposed: optimm spectrm balancing (OSB) [3], iterative spectrm balancing () [1], [2], atonomos spectrm balancing (ASB) [7], and sccessive convex approximation for low-complexity (SCALE) [8]. For all these algorithms it is claimed that there is no need for the maximm transmit PSD mask constraint, since total power, is thoght to be sfficient to ensre the spectral compatibility among the DSL systems inclded into optimization. However, in this section we will show that this assmption might not be valid nder certain circmstances. De to low comptational complexity and performance almost identical to the OSB, iterative-based algorithms have received a special attention for spectrm balancing. Therefore, we will constraint orself to which works as follows: it iterates many times over all sers and in each iteration it s (7) for one ser while keeping the power allocation of all other sers fixed. At the end of each iteration the weight and Lagrangian mltiplier of the ser being optimized are pdated, cf. Algorithm 1. In the following discssions, we constraint orself to twoser case for easy explanations, bt all the claims made here are also valid for scenarios with mltiple sers. Now looking at the i-th iteration, for a two-ser case, the optimization problem for the first ser on a particlar sbcarrier n is constraint, T max 2 w R n sbject to: 0 2 λ P n, (8a) (8b)
As shown in [1], [2], the cost fnction (8a) is neither concave nor convex with respect to power allocation of the first ser P1 n. Therefore, the exhastive search is sed to find P1 n, which s (8a). Let s analyze the case when P2 n = 0. This occrs whenever the SNR of the second ser on the sbcarrier n is low either de to high noise level or high channel attenation. For this case, the optimization problem (8) becomes ( ) w 1 log sbject to: 0. 1 + Hn 11 Γ,V λ 1 (9a) (9b) The second derivative of (9a) can be calclated in a closed form and it is eqal to ( w 1 (H11) n 2 ) 2. (10) ΓP1,V n + Hn 11 Pn 1 Since (10) is smaller than zero for 0 P1 n <, the cost fnction in (9a) is a concave fnction with respect to P1 n. For this case, there is still no need for a maximm transmit PSD mask constraint to ensre the convergence of. However, Proposition 1 below shows that in order to ensre convergence of in any network scenario we need to set-p a PSD mask constraint. Algorithm 1 Iterative Spectrm balancing as in [1] (Notation is adopted for this paper) Preset vales: R target, T max, λ, w, for = 1 to U do For each n: fix Pj n, j, then P n = arg max P n L n for n I as in (6) {Solve by 1-D exhastive search } Update: w = [ w + ɛ ( R target )] + n I Rn Update: λ = [ λ + ɛ ( n I P )] n T max + {[] + : constraint to non-negative nmbers} ntil convergence end for ntil the PSDs of all sers have reach a desired accracy Proposition 1: algorithm converges in any network scenario only if we set a maximm transmit PSD mask constraint. Proof: We will show that does not converge when both λ 1 = 0 and P2 n = 0 (after the proof we describe when sch a sitation arises). Note that for optimization problem (8) the same also holds when λ 2 = 0 and P1 n = 0. With the made assmptions, the optimization problem (9) becomes w 1 log ( sbject to: 0. 1 + Hn 11 Γ,V ) (11a) (11b) The cost fnction (11a) is monotonically increasing on. Therefore, the optimization problem (11) achieves a maximm at =, and the search for it will not converge. Ths, to ensre the convergence of in any network scenario, we have to set a PSD mask constraint, which also concldes the proof. We describe now for which cases both λ 1 and P2 n get zero, based on s oter iterations (oter loop) as given in the psedo-code of Algorithm 1. Assme that in the (i 3)-th iteration, second ser performs power allocation and it tilizes all sbcarriers in set I. Frthermore, in (i 2)-th iteration, first ser does not tilize the total power to achieve a particlar target bitrate, R target 1 ; ths, λ 1 = 0. In (i 1)-th iteration, the second ser search for a new power allocation, and it does not se at least one sbcarrier. In (i)-th iteration, we start to calclate the optimal power allocation for the first ser with λ 1 = 0 and there is at least an n for the second ser which has P2 n = 0. IV. OPTIMIZATION UNDER A MAXIMUM PSD MASK CONSTRAINT As we have shown in Section III, the convergence of is only ensred if we set a maximm transmit PSD mask constraint in addition to the total power constraint. In this section we show that the optimization problem for DSL systems is tremendosly simplified if we select the total power constraint to be eqal to the power within the transmit PSD mask constraint. Under this assmption, the spectrm balancing optimization problem can be formlated as P n ;,n w R, (12a) sbject to: 0 P n P n,max,, n I, (12b) where P n,max denotes the maximm transmit PSD mask constraint for ser in sbcarrier n. Following this approach by setting a niqe transmit PSD mask constraint for each ser (line), we can inclde modems s implementation constraints into the optimization process. Frthermore, setting a niqe transmit PSD mask constraint for each ser does not increase the comptational complexity. However, sing the same maximm transmit PSD constraint for all sers is only a special case of optimization problem (12). Maximizing the bitrate in each sbcarrier independently also s the sm of bitrates over all sbcarrier. Therefore, the optimization problem (12) can be split into N persbcarrier optimization sbproblems by expressing the cost
fnction as w R = n I w R. n (13) Different methods have been proposed to search for the appropriate weighting vales w. Under the assmption that the target bitrates of all sers are known in advance (before rnning the algorithm), a sb-gradient method is proposed in [1]. Alternatively the algorithm in [5] ses bitrate relations in order to search for the weights withot any a priori knowledge on target bitrates. As the search for appropriate weights is not the focs of this paper we will in the following not search for the weights bt instead we vary them between 0 and 1 to pictre the complete rate region. The psedo-code of or proposed iterative scheme, which we call mask-based spectrm balancing (), is listed as Algorithm 2. The works as follows: For given weights and PSD mask constraints, it iterates many times over all sers and in each iteration it searches for the transmit PSD of a particlar ser that s the sm of weighted rates. which is only sed to ensre the convergence of in the light of discssion in Section III. For we have set a PSD mask constraint to 59 dbm/hz for all simlations. To have a fair comparison between and, the total power constraint by was set to be eqal to power within the PSD mask constraint in. All simlation are performed for the network scenario shown in Fig. 1 with only two sers. For sch two-ser case the bitrate region is two dimensional and is easy to draw conclsions from. Frthermore, to cover a broad range of network environments, simlations are done for the distance x between the modems eqal to: 200 m, 400 m, and 600 m. CO/Cabinet 1 x 1 2 Fig. 1. Simlation scenario with two sers. CO and x denote the central office and the distance between the modems, respectively. x 2 Algorithm 2 Mask-based spectrm balancing () Preset vales: w, P n,max,, n I {mask constraints} for = 1 to U do Calclate Noise N n for n I as in (2) P n = arg max P n w R n for n I as in (6) {Solve by 1-D exhastive search nder constraint (12b)} end for ntil the PSDs of all sers have reach a desired accracy Bitrate of 2 (Mbit/s) 100 80 60 40 20 x=600 x=400 x=200 is not the first algorithm to state a PSD mask constraint. For example, the ser niqe power back-off (UUPBO) [9] solves a similar optimization problem as. However, it sets an constraint on maximm transmit PSDs in addition to the total power constraint. This ensres the convergence of UUPBO in any network scenario. V. SIMULATION RESULTS AND DISCUSSIONS In order to evalate the performance of or proposed algorithm, mask-based spectrm balancing () algorithm, simlations have been sed. Simlation parameters are taken according to ETSI VDSL standard [10]. Ths, we se Γ = 12.8 db as the SNR gap, and the band plan 997, which ses two pstream bands. Moreover, to take into accont the alien noise, in addition to the backgrond noise at 140 dbm/hz, we have also added the ETSI VDSL Noise A, which is also specified in [10]. We compare the performance of the with the iterative spectrm balancing () for the pstream transmission direction. For we have set a PSD mask constraint to 0 dbm/hz, 0 0 20 40 60 80 100 120 140 Bitrate of 1 (Mbit/s) Fig. 2. Comparison of the rate regions between the MBS and for pstream transmission direction. Simlations are performed for the network scenario with two sers as in Fig. 1 and the distance x between the modems eqal to: 200 m, 400 m, and 600 m. From the simlation reslts presented in Fig. 2 it is obvios that otperform for these network scenarios. The reslts are not srprising, since ses more degrees of freedom and it tilizes them when searching for optimal transmit PSDs. In Table I a few, from Fig. 2, selected pairs of bitrates are shown. For the network scenario with x = 600 m we see in Fig. 2 that visibly otperforms when the operation point is selected on the left side of rate region. For the pairs of bitrates shown in Table I, otperforms by 6.6%. For this set of bitrates the transmit PSDs are shown in Fig. 3. The transmit PSDs of are always above the transmit PSDs of, which is also the reason why otperforms.
TABLE I COMPARISON OF THE WITH THE OSB FOR SOME PARTICULAR PAIRS OF BITRATES. PSD (dbm/hz) PSD (dbm/hz) Scenario User 1 User 2 Loss Algorithm (x in m) (Mbit/s) (Mbit/s) (%) 600 62.2 14.6 600 59.2 12.8 6.6 400 82.0 17.0 400 80.8 16.0 2.2 200 107.5 15.0 200 107.0 14.3 3.0 50 60 70 80 90 100 110 120 130 Transmit PSD of 1 140 0 500 1000 1500 2000 2500 3000 Sbcarrier index Transmit PSD of 2 50 60 70 80 90 100 110 120 130 140 0 500 1000 1500 2000 2500 3000 Sbcarrier index Fig. 3. Upstream transmit PSDs of and for sers bitrates given in Table I for x = 600 m. On the other hand, for the network scenario with x = 400 m and x = 200 m, and show similar performance. For these pairs of bitrates shown in Table I, only otperforms by 2 to 3 percent. Frthermore, for x = 400 m the bondary of the rate regions for and as plotted are not convex. The reason for this effect is that the doble precision arithmetic, which is also sed dring simlations is not always sfficient to find every operation point on the bondary of the rate region. It is worth mentioning that this holds also for other spectrm balancing algorithms based on the dal decomposition optimization approach. Comparing the psedo-code of and, it becomes clear that reqires mch higher comptational complexity. To indicate the difference in complexity, the simlation time to get a pair of bitrates is 3 seconds for while (for fixed weights) reqires 114 seconds. VI. CONCLUSIONS This paper introdced a new spectrm balancing algorithm for mlti-ser DSL systems: the mask-based spectrm balancing () algorithm. The optimization by is only based on PSD masks constraint instead on the total power constraint as sed by other spectrm balancing algorithms. We have shown that an optimization approach based on PSD masks instead on the total power makes sense both from a comptational complexity and a stability point of view. The stability isse of commonly sed spectrm balancing methods is frther analyzed and we have proven that stability isses exist. Throgh simlations we have additionally shown that the inevitable loss by compared to the optimal schemes (de to less degrees of freedom) is kept very low and is only a few percent for scenarios with short loops (sitable for cabinet deployed VDSL2). This small loss shold be compared to the complexity redction of two magnitdes for compared to. REFERENCES [1] R. Cendrillon and M. Moonen, Iterative spectrm balancing for digital sbscriber lines, in 2005 IEEE International Conference on Commnications (ICC 2005), vol. 3, 2005, pp. 1937 1941 Vol. 3. [2] R. Li and W. Y, Low-complexity near-optimal spectrm balancing for digital sbscriber lines, in 2005 IEEE International Conference on Commnications (ICC 2005), vol. 3, 2005, pp. 1947 1951 Vol. 3. [3] R. Cendrillon, W. Y, M. Moonen, J. Verlinden, and T. Bostoen, Optimal mltiser spectrm balancing for digital sbscriber lines, Commnications, IEEE Transactions on, vol. 54, no. 5, pp. 922 933, 2006. [4] W. Y, G. Ginis, and J. Cioffi, Distribted mltiser power control for digital sbscriber lines, Selected Areas in Commnications, IEEE Jornal on, vol. 20, no. 5, pp. 1105 1115, 2002. [5] D. Statovci, T. Nordström, and R. Nilsson, The normalized-rate iterative algorithm: A practical dynamic spectrm management method for dsl, EURASIP Jornal on Applied Signal Processing, vol. 2006, pp. Article ID 95 175, 17 pages, 2006. [6] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004, N 0-521-83378-7. [7] J. Hang, R. Cendrillon, M. Chiang, and M. Moonen, Atonomos spectrm balancing (asb) for freqency selective interference channels, in Information Theory, 2006 IEEE International Symposim on, 2006, pp. 610 614. [8] J. Papandriopolos and J. Evans, Low-complexity distribted algorithms for spectrm balancing in mlti-ser dsl networks, in Commnications, 2006. ICC 06. IEEE International Conference on, vol. 7, 2006, pp. 3270 3275. [9] D. Statovci, T. Nordström, and Nilsson, Dynamic spectrm management for standardized VDSL, in IEEE International Conference on Acostics, Speech, and Signal Processing (ICASSP 2007), vol. 3, Honoll, Hawaii, USA, Apr. 2007, pp. 73 76. [10] ETSI, Transmission and Mltiplexing (TM); Access transmission systems on metallic access cables; Very high speed Digital Sbscriber Line (VDSL); Part 1: Fnctional reqirements, ETSI, Tech. Rep. TM6 TS 101 270-1, Version 1.3.1, Jl 2003.