Quality of Service Optimization in the Broadcast Channel with Imperfect Transmit Channel State Information

This work considers a Broadcast Channel (BC) system, where the transmitter is equipped with multiple antennas and each user at the receiver side could have one or more antennas. Depending on the number of antennas at the receiver side, such a system is known as Multiple-User Multiple-Input Single-Output (MU-MISO for single antenna users, or Multiple-UserMultiple-InputMultiple-Output (MU-MIMO), for several antenna users. This model is suitable for current wireless communication systems. Regarding the direction of the data flow, we differentiate between downlink channel or BC, and uplink channel or Multiple Access Channel (MAC). In the BC the signals are sent from the Base Station (BS) to the users, whereas the information from the users is sent to the BS in the MAC. In this work we focus on the BC where the BS applies linear precoding taking advantage of multiple antennas. The Channel State Information (CSI) is assumed to be perfectly known at each user. However, the users do not cooperate, and the BS only has partial CSI obtained via a feedback link in Frequency-Division Duplex (FDD) systems, which is bandwidth limited. This limitation forces the users to apply certain methods, such as quantization, to reduce the amount of data to be sent to the BS. The combination of the information provided by the users is interpreted as stochastic CSI at the BS, so that the partial CSI is critical for the design of the precoders. Several criteria have been considered to evaluate BC performance in the literature, namely Signal to Interference-plus-Noise Ratio (SINR), Minimum Mean Square Error (MMSE) and rate. While some works compute the corresponding metric for each of the users, others consider the sum of all of them as the value of interest. In our case, we concentrate on rate as figure of merit. In particular, we are interested in guaranteeing certain per-user rates. That way, we avoid unfair situations of the sum rate criterion arising when the channels for some of the users are poor with assigned low or even zero rates. Moreover, reducing the amount of power required to fulfill the mentioned Quality-of- Service (QoS) restrictions is a desirable feature for a wireless communication system. Thus, we address the optimization problem consisting of minimizing the total transmit power employed at the BS to fulfill a set of given QoS constraints, expressed as per-user rates. The power minimization problem has been widely studied during recent years for both perfect and imperfect CSI at the BS scenarios. The assumption of perfect CSI is rather unrealistic so, as we mentioned previously, we consider that the users send the channel information to the BS by means of the feedback channel, usually available in recent wireless communication standards. Although some authors have employed bounded uncertainty models for the CSI, e.g. rectangular, ellipsoidal or spherical, and have taken advantage of that assumption to solve the power minimization problem, we do not assume a particular shape for that uncertainty, instead modeling it as a stochastic error. In the considered MU-MIMO system model the number of antennas at the BS is greater than the number of antennas at each user, e.g. MU-MISO. Moreover, the users do not cooperate to separate the received signals. Due to that and to the lack of degrees of freedom at the users, the use of transmit filters, also denoted as precoders, becomes necessary to remove inter-user interference. Thus, in this work we jointly design the linear precoders and receive filters minimizing the total transmit power subject to per-user rate constraints. This problem formulation is non-convex. As a consequence, it is difficult to deal with. For such a reason, we apply Jensen?s inequality to the rate constraints to obtain a MMSE based restriction. Consequently, our aim is to find the precoders and the filters that minimize the MMSE for all the users. To that end, several types of dualities based on SINR, Mean Square Error (MSE), or rate have been employed for the design of the filters as conversion formulas that allow switching between the BC and the MAC for convenience. We employ the MSE BC/MAC duality for imperfect Channel State Information at the Transmitter (CSIT). Furthermore, for the power allocation design, we take advantage of the standard Interference Function (IF) framework, proposed to solve the power control algorithm. In such a way, an algorithm is proposed to solve the power minimization problem in the BC. To check the feasibility of the QoS constraints, we propose a test that allows the convergence of the algorithm to be determined. Additionally, the proposed algorithm can be employed to solve the dual problem, i.e., find the balanced targets for given total transmit power. Finally, some applications of the power minimization problem arising from different scenarios are studied and solved by means of the proposed algorithm. Simulation experiments are carried out using the technical programming language MATLAB in order to demonstrate the performance of the proposed methods.