Analysis of Iterative Waterfilling Algorithm for Multiuser Power Control in Digital Subscriber Lines
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Analysis of Iterative Waterfilling Algorithm for Multiuser Power Control in Digital Subscriber Lines Zhi-Quan Luo1 and Jong-Shi Pang2 1 Department
of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, USA 2 Department of Mathematical Sciences and Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA Received 3 December 2004; Revised 19 July 2005; Accepted 22 July 2005 We present an equivalent linear complementarity problem (LCP) formulation of the noncooperative Nash game resulting from the DSL power control problem. Based on this LCP reformulation, we establish the linear convergence of the popular distributed iterative waterfilling algorithm (IWFA) for arbitrary symmetric interference environment and for certain asymmetric channel conditions with any number of users. In the case of symmetric interference crosstalk coefficients, we show that the users of IWFA in fact, unknowingly but willingly, cooperate to minimize a common quadratic cost function whose gradient measures the received signal power from all users. This is surprising since the DSL users in the IWFA have no intention to cooperate as each maximizes its own rate to reach a Nash equilibrium. In the case of asymmetric coefficients, the convergence of the IWFA is due to a contraction property of the iterates. In addition, the LCP reformulation enables us to solve the DSL power control problem under no restrictions on the interference coefficients using existing LCP algorithms, for example, Lemke’s method. Indeed, we use the latter method to benchmark the empirical performance of IWFA in the presence of strong crosstalk interference. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
In modern DSL systems, all users share the same frequency band and crosstalk is known to be the dominant source of interference. Since the conventional interference cancellation schemes require access to all users’ signals from different vendors in a bundled cable, they are difficult to implement in an unbundled service environment. An alternative strategy for reducing crosstalk interference and increasing system throughput is power control whereby interference is controlled (rather than cancelled) through the judicious choice of power allocations across frequency. This strategy does not require vendor collaboration and can be easily implemented to mitigate the effect of crosstalk interference and maximize total throughput. A typical measure of system throughput is the sum of all users’ rates. Unfortunately the problem of maximizing the sum rate subject to individual power constraints turns out to be nonconvex with many local maxima [1]. To obtain a global optimal power allocation solution, a simulated annealing method was proposed in [2]; however, this method suffers from slow convergence and lacks a rigorous analysis. More recently, a dual decomposition approach [3] was developed to solve the nonconvex rate maximization problem,
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