Existence of Fixed-Point and Convergence Analysis of Piece-Wise Wide-Sense Standard Power Update Functions | IEEE Journals & Magazine | IEEE Xplore

Existence of Fixed-Point and Convergence Analysis of Piece-Wise Wide-Sense Standard Power Update Functions


Abstract:

Many power update functions proposed in the literature are functions of the effective interference (the ratio of the interference to the path gain). It is well known that...Show More

Abstract:

Many power update functions proposed in the literature are functions of the effective interference (the ratio of the interference to the path gain). It is well known that for an upper-bounded wide-sense standard power update function (which is either standard or type-II standard function for each user), there exist a unique fixed-point and its convergence to its unique fixed-point is guaranteed. However, depending on the corresponding objective function, there exist power update functions which do not have the properties of monotonicity and scalability for all values of the effective interference (i.e., they are not wide-sense standard). For such a non-wide-sense standard power update function, the value-region of the effective interference for each user may be divided into several sub-regions, for each of which, a different wide-sense standard power update strategy may be employed, which we call it as a piece-wise wide-sense standard function. In this letter, we show that a bounded piece-wise wide-sense standard function is two-sided scalable and thus it poses a unique fixed-point and its convergence to its corresponding fixed-point is guaranteed.
Published in: IEEE Communications Letters ( Volume: 19, Issue: 6, June 2015)
Page(s): 1025 - 1028
Date of Publication: 27 March 2015

ISSN Information:


I. Introduction

SINCE the early work of [1], there have been many research studies on power control of wireless networks. Due to the complexity of most wireless systems, and also the lack of full knowledge of the network parameters, instantaneous calculation of the optimal power vector is generally so hard and sometimes even impossible. Therefore, many of the power update algorithms use distributed iterative methods for reaching the optimal or suboptimal equilibrium power vector. Existence of fixed-point and convergence to the corresponding fixed-point are the first essential properties that must be verified for any iterative power update function. While in iterative power update functions, the transmit-power vector at any time step is a function of the transmit-power vector of the previous time step, in most existing distributed algorithms, the power update function of each user reduces to be only a function of the effective interference (the ratio of the interference to the path gain between that user and its corresponding receiver) experienced by that user (e.g., [2]– [7]).

References

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