Optimal distributed power control in cellular wireless systems

In this paper, an optimal fast closed-loop SIR-based power control scheme is proposed for CDMA systems. The state-of-the-art of this power control algorithm is that the power evolution is directly driven by the error between the actual SIR and the desired SIR. The amount of power change is proportional to the error, and the optimal gain is chosen to achieve the minimum error at the next time instant. Assuming perfect estimation and prediction, the error between the desired SIR and the real value of SIR goes to zero in one step. In practical situations, where the estimator need several (3-4) steps to make accurate estimation and prediction, the error between the desired SIR and the real value of SIR goes to zero in several (4-5) steps. The proposed controller is an estimator based controller. The performance of the estimator will have direct impact on the performance of the power control scheme. Simulations of a CDMA system are performed to demonstrate effectiveness of the proposed power control scheme. The simulation results, in all cases, confirm theoretical analysis. These results indicate that the proposed optimal power control scheme is a very promissing solution to the distributed fast closed-loop power control problem in wireless systems, both theoretically and practically. I. Review of SIR based Power Control in Cellular Wireless Systems In a cellular wireless network, certain quality-of-service (QoS) should be maintained for all active users in the network. A quantity that measures the user’s provided QoS is the signal-to-interference ratio (SIR). The initial work on SIR-based power control schemes for narrowband systems has been done by Zander [1]-[3]. Power control schemes can be centralized [1]-[3] or distributed [2],[4],[6] according to the nature of computations. A centralized controller has information (e.g., all the link gains) for each user and it decides control actions for all users. On the other hand, a distributed controller only uses local information to make a control action for a user. For example, [2],[4],[6] only use information about user’s own link gain and/or SIR to make a decision about their transmitted power. The algorithm convergence has to be studied carefully when a distributed power control scheme is used. The distributed power control algorithm by Foschini [6] was shown to converge either synchronously [6] or asynchronously [9]. The basic model in [6] is relaxed in [9] to allow asynchronism. Propagation delay is also considered in [9]. A framework for convergence of the generalized uplink power control was proposed by Yates [10]. In recent years, there have been increasing interests in applying code-division-multipleaccess (CDMA) techniques to cellular wireless networks. Unlike the narrowband systems, users in a CDMA system share the same frequency all the time by using a specific spread spectrum pseudonoise code for each user. The system performance mainly depends on socalled multiple access interference (MAI), which is caused by the cross-correlations between the desired signal and other signals. Open-loop power control has been employed to combat the path loss and shadow fading. The average power control in [10],[22] keeps the local mean received power to a constant value, which mitigate the effect of shadowing and near-far problems. In IS-95B [20] and CDMA2000 [21], fast closed-loop power control is proposed to fight medium to fast fading. In [11]-[12], it is shown that when power control rate is higher, it can partially accomodate the effect of fast fading. Herdtner and Chong [9] considered a simple distributed power control scheme in which the convergence condition was given. Wu [13] extended Zander’s problem formulation to CDMA systems by reordering the number of users in different cells. Song et.al. [14]-[15] and Gunnarsson et.al. [16]-[17] considered up/down power control and closed-loop power control with other nonlinear elements within the framework of nonlinear control systems. [14]-[15] gave guidelines for choosing the appropriate power control step size in IS-95. In [16]-[17], PID controllers were designed to overcome the effects of time-delay in the feedback loop. Filtering techniques have been also used to design power control schemes. Leung [18] proposed a power control scheme for TDMA data service based on the Kalman filtering technique. In [31], a linear prediction of received power is made at the base station to predict the power control bit one step ahead. Both approaches assume that the interference is Gaussian. In this paper, an optimal fast closed-loop SIR-based uplink power control scheme is proposed for CDMA systems. In Section 2, a proportional controller is proposed and the optimal power control gain is derived by unconstrained optimization. In Section 3, the constraints are imposed on the maximum and minimum values of transmitter power. By applying the Kuhn-Tucker conditions, we derive the optimal gain for the proportional controller. Using Bellman’s optimality principle, the power control scheme achieves optimality in the whole time interval of interest. The proposed controller is an estimator based controller. The performance of the estimator will have direct impact on the performance of the power control scheme. Section 4 gives our choice of the estimator. In contrast to the well-known Kalman filter, which assumes that system and measurement disturbances are white Gaussian stochastic processes and requires the knowledge of their statistics, the H∞ filter, used in the proposed scheme, facilitates the system filtering regardless of the nature of the system and measurement disturbances. It meets our estimation requirement since the fluctuation of the interference is not Gaussian and its statistics is not known in general [18]. Simulations of a CDMA system are performed to demonstrate the effectiveness of the proposed power control scheme. The results are shown in Section 5. II. Optimal SIR-based Distributed Power Control by Unconstrained Optimization In this paper, only uplink (mobile-to-base) power control is considered. Suppose that there are N active users in the system. The Signal-to-Interference Ratio (SIR) in a CDMA system is defined as Definition 1: The SIR of an active link from mobile station i to base station n in a wireless system is defined as γni = Lhnipi ∑ j 6=i hnjpj + σ 2 (1) where hni is the link gain from mobile station i to base station n. We assume that L is the processing gain in spread spectrum wireless systems, e.g., in CDMA 2000, L=64 or 128 or 256. Let us denote the denominator by Ii, which represents the received interference, then the SIR can be rewritten as γni = Lhnipi Ii (2) Assuming that a mobile only transmits to one base station during the time of power control, then γni can be simplified as γi. Let δi = Lhni Ii denote the channel variation [11]. δi will be estimated and predicted in the proposed power control scheme. Given that a mobile station and its base station communicate directly to each other, we assume that hni can be estimated accurately. We also assume that the received interference Ii can be measured at each time step. The received interference is the difference between the total received power and the desired signal power [10]. The main idea of the proposed algorithm is to make the transmission power proportional to the error between the actual SIR and the desired SIR. Define the transmission power change from time step k to k + 1 as ∆pi(k + 1) = pi(k + 1)− pi(k) (3) where pi(k) and pi(k + 1) are the transmission powers from mobile i to the base station at the time instants k and k +1, respectively. Let the error between the actual SIR (γi(k)) and the desired SIR (γ i ) be denoted by ei(k) for mobile i at the time instant k, ei(k) = γ tar i − γi(k) (4) We propose the following power control algorithm ∆pi(k + 1) = αi(k)ei(k) =⇒ pi(k + 1) = pi(k) + αi(k)(γ tar i − γi(k)) (5) where αi(k) is the gain to be determined through the optimization procedure in which the square of the error is minimized (theoretically reduced to zero in each discrete-time instant k). The main result of this paper is presented in the following theorem. Theorem 1: Define the performance criterion at the time instant k as Ji(k) = min αi(k) (ei(k + 1)) 2 , i = 1, 2, . , N. (6) The optimal gains of the power control algorithm (5) that minimize Ji(k) at every time instant k are given by