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IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 1, JANUARY 2010
A Power Allocation Method for DMT-Based DSL Systems Using Geometric Programming Roozbeh Mohammadian, Mehrzad Biguesh, Senior Member, IEEE, and Saeed Gazor, Senior Member, IEEE
Abstract—In this letter, we consider a power allocation problem for digital subscriber line (DSL) systems. The goal of this problem is to minimize the total transmit power under some constraints on minimum data-rate and maximum transmit power for each modem where we take into account various sources of interference. We convert our problem into an auxiliary geometric programming (GP) which gives the optimum solution for transmit powers in a neighborhood of a given feasible point. Then, we use an iterative scheme for obtaining the solution to our original problem by exploiting this auxiliary GP problem. Numerical examples show that the proposed method outperforms the existing schemes. Index Terms—DSL, geometric programming, power allocation.
I. INTRODUCTION IGITAL subscriber line (DSL) systems allow high speed data transmission over the existing copper twisted-pair networks. Cross-talk is the major obstacle toward reaching the promised high data-rates in these systems. Power Allocation (PA) is one of the methods used for mitigating the effect of cross-talk. Generally, existing PA algorithms use two main approaches, namely Rate Adaptive and Power Adaptive [1]. The rate adaptive approach attempts to maximize some form of the data-rate, subject to constraints on the maximum transmit power of individual users. In the power adaptive approach, the transmit powers are adjusted such that users are guaranteed to achieve at least some minimum data-rate while the total transmit power is minimized. Most researchers have put their efforts on the Rate Adaptive approach without concerning the power minimizing design objective [2]. Minimizing the transmit power results in lower power consumption by DSL systems which in return reduces the radiated crosstalk into other DSL systems. Thus, the power adaptive approach can significantly improve the overall performance of DSL systems [2], [3].
D
Manuscript received June 30, 2009; revised July 29, 2009. First published August 21, 2009; current version published October 07, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Behrouz Farhang-Boroujeny. R. Mohammadian is with the Wireless Communications Laboratory, Department of Communications and Electronics, Shiraz University, Shiraz, Iran. M. Biguesh is with the Department of Communications and Electronics, Engineering School, Shiraz University, Shiraz, Iran (e-mail:
[email protected]). S. Gazor is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6 Canada (e-mail: s.gazor@queensu. ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2009.2030863
In this manuscript, we consider a general PA problem in DSL systems and minimize the total transmit power under the constraint that modems achieve some minimum data-rate and have some limitation on their transmit powers. We formulate our problem for the case where the same frequency band is used for both downstream and upstream data transmission; however, the proposed solution can be also used for other cases. In the sequel, we encounter a non-convex optimization problem. The main contribution of this manuscript is to convert this problem into an auxiliary geometric programming (GP) problem which can be solved using fast methods in order to obtain the optimum transmit powers in a neighborhood of a given feasible point. The optimal solution to the original problem is obtained iteratively by employing this auxiliary GP problem. Our approach is similar to the iterative convex approximation [4], [5]. However, based on the concept of GP we choose a different convex approximation where our computer simulations show that it outperforms the methods studied in the field of DSL systems [5]. A new general DSL problem is proposed in [2] which is referred to as green DSL. In this problem, the goal is to maximize the difference between the sum of weighted data-rates and the sum of weighted transmit powers. It can be shown that our problem is a special case of the green DSL problem by properly redefining the weight vectors there. It is mentioned in [2] that the green DSL problem can be solved using various algorithms such as IW, ASB (2), SCALE, MIW, DSB, MS-DSB, and OSB. Each of these methods has its own properties, complexity, and performance. In [5] it is noted that none of these algorithms, except the OSB, are globally optimal method, and the MS-DSB algorithm converges to the global optimum solution with the highest probability. Based on these facts and to avoid the exponential complexity of OSB, we compare our method with MS-DSB algorithm. II. SYSTEM MODEL Utilizing the discrete multi-tone (DMT) modulation technique in DSL systems, the communication channel is divided into a number of frequency non-selective narrow-band subchannels, each of them carries a separate data stream. In a system with users and subchannels, the signal to interference and noise ratio (SINR) experienced by th customer premise (CP) modem at th subchannel is [1],
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MOHAMMADIAN et al.: POWER ALLOCATION METHOD FOR DMT-BASED DSL SYSTEMS
Here, is the channel transfer matrix at th subchannel shows the desired channel from the th central where office (CO) transmitter to the corresponding CP receiver at th subchannel and for shows the undesired far-end crosstalk (FEXT) caused by the th CO modem represents the near-end to the th CP modem. Also, crosstalk (NEXT) effect and leakage across CP modems on for represents the th subchannel. That is, the NEXT effect from th CP modem to th CP modem on th subchannel and represents the leakage from th and show the CP transmitter to its own receiver. transmit powers at th subchannel of th CO and CP modems, represents the noise power respectively. Also, is the tone of th CP receiver at th subchannel, where is the additive noise power spectral density at spacing and th subchannel for th CP receiver. In this model, the achievable downstream bit-rate of th CP modem at th subchannel can be calculated as [1],
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is the optimum solution for which at least the th user data-rate is greater than its threshold amount (i.e., ). In this case, we can properly decrease such that meets . With such a decrease in transmit power, all other data-rates will increase due to lower interference and in such a case with a lower total transmit power we have satisfied all the optimization constraints for all users in the system. Thus, the opoccur only if timal transmit powers and for all . Due to the nonlinearity of and in constraints and , the problem (3) is non-convex and it is complex to solve. To change this problem into a standard optimization problem, we express the constraints in (3) using (2) as (4) where
,
and
(1) where is a constant greater than one which denotes the SNR-gap to capacity and it depends on the coding gain, desired bit error rate, and noise margin [1]. Using (1) and assuming that the DMT symbol rate is , the total downstream data-rate for th CP modem can be expressed as,
Similar expressions can be replaced for the constraints . Using (4), we rewrite (3) as
(2) (5) Similar relation can be written for the total upstream data-rate of th CO modem. where III. THE POWER ALLOCATION PROBLEM For power allocation, we consider the minimization of the total transmit power by all CO and CP modems under the constraints on minimum data-rate and maximum transmit power for CP and CO modems. Mathematically, we express our PA problem as,
(3) and are the maximum allowed transmit Here, and are powers by th CP and CO modems, and the minimum required data-rates at the downstream and upstream directions of the th CP and CO modems, respectively. Since the Hessian matrix [6] of the objective function in the null matrix and noting that problem (3) is a -dimensional vector with all one entries, its gradient is a the probable optimum solution for this problem occurs at the border of the feasible region [7]. Additionally, it can be shown that at the optimal solution for this problem, all the data-rate constraints have to be active. This is due to the fact that is monotone increasing in while it is a decreasing func(for ) and (for all ). Assume that tion of
, and
and . We should stress that are functions of,
,
,
Interestingly, the above problem has some similarities with a geometric programming (GP) problem for which there are fast and efficient numerical algorithms [8]. In fact, the GP problem is easily transformed to a convex problem and solved globally using recently developed interior-point methods [9]. Taking the similarities in mind, we propose to approximate the nonlinear constraints in (5) with posynomial function of in order to obtain a standard GP optimization problem. Fortuand are positive and differentiable, nately, thus they can be approximated as monomial functions [9], [10] as, in a neighborhood of
(6) where
and,
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IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 1, JANUARY 2010
, respectively, can be written as and . The left side of these new inequalities are posynomial; thus, we use these inequalities to write the optimization problem (8) which is a GP problem in standard form and can be solved easily.
TABLE I PROPOSED PA ALGORITHM
(8) for for
Note that
.
Here, is some conjecture of the optimum transmit powers at the th subchannel. The approximation in (6) is obtained using the Taylor in terms of logarithms of series expansion of elements of . Also, using similar expression, we replace with its monomial approximation in a . neighborhood of Applying these approximations, we approximate (5) with the following problem:
(7)
. Two additional constraints and are added in (7), since the approximation in (6) for and the similar expansion of for are reliable only in a neighborhood around . These added constraints guarantee that the such that the solution of (7) stays in the neighborhood of monomial approximate expansions and are sufficiently accurate. Here, and are sufficiently small positive numbers. Noting that and are posynomials, it is obvious that (7) is a standard GP problem. Now, to find the solution of (5), we propose the iterative algorithm shown in Table I. This algorithm solves the GP problem (7) and uses the obtained solution for computing approximaand and getting ready for next round of iterations tion. In this way, it is expected to converge to the solution for our original PA problem (5). Similar approach for solving some optimization problems is already proposed in the optimization literature (see also [9]). In order to converge to the global optimal solution this proposed algorithm must be initialized in the feasible region around the optimal solution. In the sequel, we propose a problem where its solution can be used as an initial point for the above algorithm. It is easy to see that the inequaliand ties where
concludes and hence . Thus, the solution of the above problem also lays in the feasible region of the nonlinear problem (5). Since, we intuitively expect that they have close optimal solutions, we propose to use the solution to (8) as the initial feasible point for our described iterative algorithm. We must note that in some rare cases has no solution and in turn the problem (8) has no feasible solution. Thus, for those cases we propose to use a random initial point. IV. NUMERICAL EXAMPLE This section presents the simulation results of the proposed power allocation algorithm. For our numerical examples the following parameters are assumed. The twisted pair lines have a diameter of 0.5 mm (24 AWG). Each modem has a coding gain of 3 dB, a noise margin of 6 dB and a target error probability which results in . The tone spacing is of 4.3125 kHz and the DMT symbol rate is . The maximum transmit power of each modem is 20.4 dBm [11]. The and , is 140 dBm/Hz noise power spectral density, . The empirfor all users and subchannels, and ical models in [12] is used for computing the channel transfer functions. In the following simulations, except for the first one, we use (8) to obtain an initial feasible point. In all our extensive simulations we observed that our algorithm resulted in a feasible answer. To study the effect of initial point on the solution of our method when the number of users are small, we considered a users and subchannels, where scenario with for ,2. We ran the program 400 times with random initial powers and uniformly distributed over (0,20.4 dBm]. Using such assumptions for initialization, 325 cases out of 400 runs gave feasible initial point for our proposed PA algorithm. Fig. 1 shows the mean and the variance of the total transmit power of 4 modems versus iteration index. As seen, after about 6 iterations the algorithm converges independent of the starting feasible point and variance of transmit powers reduces which shows the uniqueness of the optimum solution regardless of the initial point. For our next numerical example, a DSL system with users and subchannels is assumed where
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MOHAMMADIAN et al.: POWER ALLOCATION METHOD FOR DMT-BASED DSL SYSTEMS
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TABLE II COMPARISON OF THE PROPOSED METHOD AND MS-DSB ALGORITHM [5]. HERE THE PROPOSED AND MS-DSB PA ALGORITHMS ARE COMPARED FOR . IN EACH BOX OF VARIOUS MINIMUM REQUIRED DATA-RATES R THE SECOND AND THIRD COLUMN OF THIS TABLE THE ACHIEVED TOTAL TRANSMIT POWER AND THE SIMULATION TIME ARE SHOWN
Fig. 1. Mean and variance of total transmit power versus iteration index for the proposed iterative method (vertical axis is in log scale).
V. CONCLUSION
Fig. 2. Achieved CP and CO data-rates versus iteration index for the assumed scenario (vertical axis is in log scale).
In this paper, we considered a power allocation problem for DMT-based DSL systems when the same frequency band is used for both downstream and upstream data transmission. Since the problem was not in the form of a GP, by means of monomial approximations of nonlinear data rate constraints, we changed our nonlinear PA problem into a geometric programming. The new problem gives the optimum solution in a neighborhood of any feasible point. To find the solution of our original problem, we proposed an iterative method to numerically compute the transmit powers. Numerical examples show that the proposed method outperforms the existing schemes. ACKNOWLEDGMENT
and are the required minimum data rates for all users. Using (8) for initial point, the achieved data-rates of all ten users are plotted versus iteration index in Fig. 2. It is observed that after approximately 14 iterations the minimum required data-rates and are achieved for all users in this scenario. The first and second simulations show that when the number of users are small the proposed method will result in the optimum solution. Now, we compare the performance of the proposed method and the MS-DSB algorithm [5] in the downstream direction. The simulations are performed for a five-user case up to a 25 user case. The users are distributed in five sets, which contains five users of equal length-loop, whose lengths varies from 500 to 2500 ft. The minimum required data-rates are set to be equal for all users. It should be mentioned that for each simulation we choose different value for the minimum required data-rates; this is due to the fact that when the number of users increases the volume of interference increases as well, which results in a lower achievable data-rate. The simulation results are summarized in Table II. From Table II, it can be seen that, while the MS-DSB algorithm is faster than our method, for lower number of users both schemes perform the same; however, when the number of modems increases our proposed method performs better than the MS-DSB algorithm.
The authors would like to express their cordial thanks to P. Tsiaflakis for his help on simulations. REFERENCES [1] T. Starr, M. Sorbara, J. Cioffi, and P. J. Silverman, DSL Advances. Upper Saddle River, NJ: Prentice Hall, 2003. [2] P. Tsiaflakis, Y. Yi, M. Chiang, and M. Moonen, “Green DSL: Energyefficient DSM,” in Proc. IEEE Int. Conf. Communications (ICC’09), Dresden, Germany, Jun. 14–18, 2009. [3] J. M. Cioffi, S. Jagannathan, W. Lee, H. Zou, A. Chowdhery, W. Rhee, G. Ginis, and P. Silverman, “Greener copper with dynamic spectrum management,” in AccessNets, Las Vegas, NV, Oct. 2008. [4] J. Papandriopoulos and J. S. Evans, “Low-complexity distributed algorithms for spectrum balancing in multi-user DSL networks,” in Proc. IEEE Int. Con. Communications (ICC’06), Jun 2006, pp. 3270–3275. [5] P. Tsiaflakis, M. Diehl, and M. Moonen, “Distributed spectrum management algorithms for multiuser DSL networks,” IEEE Trans. Sig. Proc., vol. 56, no. 10, pp. 4825–4843, Oct. 2008. [6] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [7] A. Antoniou and W. S. Lu, Practical Optimization: Algorithms and Engineering Applications. New York: Springer, 2007. [8] M. Chiang, C. W. Tan, D. P. Palomar, D. O’Neill, and D. Julian, “Power control by geometric programming,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2640–2651, Jul. 2007. [9] S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, “A toturial on geometric programming,” Optim. Eng., vol. 8, no. 1, pp. 67–127, Mar. 2007. [10] R. J. Duffin, E. L. Peterson, and C. Zener, Geometric Programming—Theory and Application. New York: Wiley, 1967. [11] Asymmetrical Digital Subscriber Line (ADSL) Transceivers, ITU Std. G.992.1, 1999. [12] P. Golden, H. Dedieu, and K. S. Jacobsen, Fundamentals of DSL Technology. Boca Raton, FL: Auerbach, 2006.
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