SIR-Based Power Control Algorithms for Wireless CDMA Networks : An Overview ∗

This paper summarizes and explains the main results on signal to interference (SIR) based power control algorithms, which are used to increase capacity and improve quality of service in cellular wireless radio systems. The classic works of Aein, Meyerhoff, Nettleton, and Alavi attracted considerable attention in the nineties. The modern approach to the power balancing control problem in wireless networks, formulated by Zander in 1992, matured in the papers of Foschini and Yates and their coworkers in the latter part of the nineties. However the field is still wide open for research as was indicated in the overview paper by Hanly and Tse (1999). The most recent approaches to solving the mobile power distribution problem in wireless networks use Kalman filters, dynamic estimators, and noncooperative Nash game theory. I. Background and Conventions Communication networks can be fixed or mobile; however, the same basic power control problem is common to both types. We do not concern ourselves here with motion of mobile stations. Accordingly, we use the terms “mobile” and “user” interchangeably. Information travels in two directions in a network: uplink (mobile-to-base) and downlink (base-to-mobile). The mathematical formulations of these problems are similar in satellite communications; they differ in cellular wireless systems. In cellular communications networks, an uplinkspecific concern is conservation of mobile battery power. Also, downlink codes are synchronous and can be made orthogonal; but uplink codes arrive at the base station asynchronously, resulting in cross correlation, and hence high in-cell interference potential unless power is adequately controlled. For this reason we concentrate on the uplink. Three types of multiple access techniques are commonly used, namely, time-, frequencyand code-division multiple access: TDMA, FDMA, and CDMA, respectively. TDMA and FDMA protocols assign a specific time, respectively, frequency slot to each user. These protocols are relatively wasteful in that when a user does not transmit in its assigned slot, no other user can make use of the resource. This work was supported by National Science Foundation grant ANIR-0106857. In contrast, in a CDMA system, all users share the same time-frequency space. In CDMA, the individual signals are distinguished by encoding and decoding using distinct code sequences assigned to each user. The code bandwidth is chosen to be much larger than the signal bandwidth, generating a spread spectrum signal. Most current cellular wireless networks use CDMA or TDMA techniques. Regardless of the access method, a common physical model is appropriate for use in power control. We characterize the system by a gain matrix G, the meaning of whose entries depends on access method and link direction. In TDMA and FDMA, interference arises from transmissions of users assigned the same slot in nearby cells; thus a matrix entry represents effective path gain between a pair of users. In CDMA, the effective path gain gij depends on distance and code cross correlation of same-cell users. We address our analysis specifically to CDMA networks as advantages of spread spectrum techniques include Capacity Combinations of powerful coding techniques and reuse of frequencies in every cell allows CDMA to provide higher capacity than TDMA. Privacy The code must be known in order to despread the received signal to recover the information. Interference rejection and anti-jamming Spread spectrum techniques are effective against both narrow band and wide band interference and jamming. Two significant interference mechanisms are termed “near-far” and “corner” effects. The corner effect is observed in the downlink with the mobile approximately equidistant from three base stations (i.e. at a corner of a hexagonal cell). The near-far effect dominates in the uplink. When mobiles transmit with equal power, the signals of mobiles close to the base station interfere strongly with those of mobiles far from the base station. Path loss, modeled using an inverse power law relationship, is the main phenomenon that determines the values of the gain matrix entries. Spatially averaged received power, prec, at a point located a distance r from a transmitter is prec ∝ ptrans rα (1) where ptrans is the transmitted power and the path loss exponent α is two in free space, somewhat higher in indoor systems, and generally in the range of three to five for outdoor networks [1]. Two additional phenomena that affect transmissions are shadowing and fast fading. The presence of obstructions such as buildings, hills and trees causes slow multiplicative gain fluctuations called shadowing, which can be thought of as changing the user’s effective position. Fast multipath fading results from signal reflections whose relative phases change with user motion. Assuming many multipaths and ideal Rake processing, these fluctuations can be ignored. As the effects of unmodeled shadow fading and fast fading do not change the general nature of the power control problem, we use the inverse power law model (1). II. History and Background Cochannel interference resulting from frequency reuse is a major factor limiting network capacity in cellular radio systems, so a power control algorithm that reduces cochannel interference has the potential to increase network capacity. Because each user contributes to the interference affecting other users, effective and efficient power control strategies are essential for achieving both quality of service (QoS) and system capacity objectives. The need for dynamic control of transmitted power in spread spectrum mobile communication systems was first encountered in the area of satellite communications. To fill this need, SIR balancing (also called power balancing) algorithms were proposed by Aein [2] and Meyerhoff [3] in the early 1970’s. A decade later, their results were adapted by Nettleton and Alavi [4], [5], and [6] for spread spectrum mobile cellular systems. The power balancing algorithms equalize, where possible, the SIR’s of all users. Although communication systems are stochastic, the power control problem leads to a purely deterministic eigenvalue problem or a linear equation, as we shall see below. Open-loop power control in wireless networks has been employed to combat path loss and shadow fading [7]. The average power control techniques of Gilhousen et al. [8] and Viterbi et al. [9] maintain received local mean constant, mitigating the effect of shadowing and near-far effects. Closed loop power control is used in wireless communication networks to compensate for fast fading and timevarying channel characteristics, and to reduce mobile battery power consumption. The closed loop control structure in IS-95 (a currently implemented CDMA standards used in wireless networks) consists of an outer loop algorithm that updates the SIR threshold every 10 ms and an inner loop which, based on SIR measurements, updates required powers at 800 Hz [10]. The algorithms that we present would replace the inner loop control algorithm. They would require additional power control bits if centrally implemented, but have the advantage that they could be implemented in a distributed manner, with the inner loop control algorithm implemented at the mobile rather than the base station. In this case, the mobile would need SIR and target SIR signals transmitted from the base station. In indoor systems, the path loss exponent depends on the degree of clutter and presence or absence of a line of sight (LoS) path. III. Power Control as an Eigenvalue/Eigenvector Problem This section will show that if the effect of noise power on the interfering signal experienced is sufficiently small to be neglected, formulating the power control problem mathematically leads to an eigenvalue problem involving positive matrices. Results on nonnegative matrices can be found in Gantmacher [11], Minc [12], or Varga [13]. If we let γd and γu denote the desired downlink and uplink SIR’s, respectively, for all users, Nettleton and Alavi [5] showed that the corresponding balanced power vectors, pd and pu must satisfy the eigenvalue problems Gpu = 1 + γu γu pu and G T pd = 1 + γd γd pd (2) where the matrix G is a nonnegative matrix of known parameters whose size depends on the number of mobiles of each cell and whose entries depend on the distances from each user to each base station. (Elements of G can be thought of as effective gains.) From (2) we can see that with λ(G) denoting the eigenvalues of G, a solution to the SIR balancing problem, if it exists, is 1 + γu γu = 1 + γd γd ∈ λ(G). (3) In the physical power control problem, the powers pd and pu and the SIR’s γd and γu must all be positive, so (3) shows that a solution exists only if a) the matrix G has a real positive eigenvalue greater than one and b) the corresponding left and right eigenvectors are nonnegative. We assume without loss of generality that G is irreducible, since if G is not irreducible, it should be decomposed and the subsystems analyzed separately. G then being nonnegative and irreducible, we may apply the PerronFrobenius theorem to conclude that G has a unique real eigenvalue equal to its spectral radius ρ(G), whose corresponding eigenvector has all components of the same sign. The components can then all be chosen positive. It then follows from (3) that so long as the spectral radius ρ(G) satisfies ρ(G) > 1, one solution to the double link power balancing problem [5] is given by γu = γd = 1 ρ(G) − 1 . (4) The corresponding right and left eigenvectors of the matrix G then give the corresponding balanced powers. Of course, the eigenvectors are unique only up to a multiplicative constant, and hence would be chosen, subject to physical constraints, to minimize the power used. The next task is to show that the spectral radius ρ(G) is indeed greater than one. Consider a cellular wireless system in which n users share a channel. If the effect of noise power