The on-going wireless data explosion drives the mobile network operators to employ spectrum- and energy-efficient wireless communications technologies in current and future cellular networks [1–11]. As we have discussed in Chapter 1, multiuser downlink beamforming, in which multiple mobile stations (MSs) are jointly served on the same time and frequency resources, represents one of the promising technologies for achieving spectrum- and energy- efficient wireless communications.
Multiuser downlink beamforming has been intensively investigated in the literature (see, e.g., [12, 13, 18, 24–33, 49–53, 55–58]) and has already been adopted in the latest cellular standard, e.g., in 3GPP LTE-A [7, 8]. Particularly, the problem of optimizing the beam- formers to maximize the (weighted) sum-rate of the downlink system under the transmission power budget of the base station (BS) has been examined in, e.g., [13, 49–53, 55–58], where the problem has been proved to be NP-hard. In those existing works, the instantaneous data rates of the MSs are assumed to be continuous and strictly increasing functions of the re- ceived SINRs at the MSs, e.g., employing Shannon’s capacity formula [13, 49–53, 55–58]. As a result, the instantaneous data rates and the received SINRs of the MSs resulted from the sum-rate maximization problem may take any arbitrary continuous values.
In practical cellular networks, to accommodate the variations in wireless channels, dis- crete rate adaptation in the form of adaptive modulation and coding (AMC) is widely adopted for controlling the block error rate (BLER) or to meet the prescribed BLER requirement specified by the cellular standards [7, 8, 19, 20]. In this dissertation, discrete rate adapta-
tion refers to the procedure of assigning an achievable date rate, i.e., a specific modula- tion and coding scheme (MCS), to each MS according to their channel conditions [7, 8, 20, 85–90, 107, 108]. With AMC, the achievable physical-layer instantaneous data rates of the MSs are determined by the specific MCSs assigned to them and thus take discrete val- ues [7, 8, 20, 85–90, 107, 108]. As a result, under AMC, the data rate of a MS is not a continuous function of the received SINRs and it is different from the Shannon’s capacity formula. For instance, in LTE systems, discrete data rate adaptation in the form of AMC is employed. The instantaneous data rates of the MSs are determined by the assigned MCSs and attain discrete values, rather than arbitrary continuous values, as shown in Fig. 3.1. Fur- ther, assigning a specific data rate (i.e., a specific MCS) to a MS in practical cellular systems generally requires the received SINR at the MS to be above a predetermined threshold for guaranteeing the prescribed block error rate (BLER) requirement of the wireless link, see, e.g., Fig. 3.1. −100 −5 0 5 10 15 20 1 2 3 4 5 6 7
Shannon capacity formula: log2(1 + SINR) 64-QAM w/ different channel coding rates 16-QAM w/ different channel coding rates QPSK w/ different channel coding rates
D at a ra te [b p cu ] u n d er B L E R o f 10 %
Received SINR thresholds [dB]
Figure 3.1: The achievable data rates vs. the thresholds of the received SINRs in LTE sys- tems [7, 8, 20, 85–87].
To promote practical applications that employ discrete rate adaption in the form of AMC and multiuser downlink beamforming, e.g., in LTE and LTE-A systems [7, 8, 20, 85–87], we consider in this chapter the joint optimization of discrete rate adaptation and multiuser down- link beamforming (DRAB) to achieve the maximum sum-rate of the downlink system with minimum total transmitted BS power. In our discrete rate adaptation framework, it is allowed that a zero data rate is assigned to a MS in a given time-slot, practically meaning that the MS is not admitted in that particular time-slot. As a result, user admission control is embedded in
3.1. Introduction 47
the discrete rate assignment procedure. Furthermore, minimum received SINR requirements corresponding to the assigned data rates (i.e., the assigned MCSs) of the admitted MSs are included in the DRAB problem to meet the BLER targets prescribed by the wireless stan- dards, e.g., by LTE [7, 8, 20, 85–90]. The DRAB problem that we consider in this chapter can be interpreted as a nontrivial extension of the conventional sum-rate maximization prob- lem (with continuous rate adaptation) [13, 49–53, 55–58], and it includes as special cases the joint downlink beamforming and admission control problem (with fixed rate and SINR re- quirements) [83, 109–112], as well as the problem of joint rate adaptation and power control (with fixed transmit beamformers) [85–90, 107, 108]. Note that inspired by our preliminary work in [113], joint discrete rate adaptation and multiuser downlink beamforming has also been studied in [114]. However, the practical minimum SINR requirements for ensuring the prescribed BLER targets were not considered in [114].
Since the admitted MSs are coupled through the co-channel interference in the downlink SINR constraints and the MCS assignment (i.e., the discrete data rate allocation) proce- dure involves binary decision variables, the DRAB problem naturally leads to a non-convex mixed integer nonlinear program (MINLP) [67–69]. That is, the DRAB remains a non- convex program even after relaxing the integer constraints and thus cannot be efficiently solved [67–69]. Similar to Chapter 2, instead of treating it as a general non-convex MINLP, we address the DRAB problem using the mixed-integer second-order cone program (MIS- OCP) framework [82]. We reformulate the SINR constraints in our discrete data rate adap- tation framework using the big-M approach [67–69] and develop a standard big-M MISOCP formulation of the DRAB problem, which supports the convex continuous relaxation based BnC method [67–69, 81, 82]. Based on the big-M formulation, we introduce auxiliary op- timization variables and develop an improved extended MISOCP formulation [68, 82] of the DRAB problem. The extended formulation exhibits several interesting structural prop- erties that are exploited in the algorithmic solutions, e.g., in the BnC method. We provide in-depth theoretical analysis to show that the extended formulation generally admits strictly tighter continuous relaxations than that of the standard big-M formulation (and thus yields significantly reduced computational complexity when applying the BnC method). Based on the analysis, several efficient strategies, e.g., customized node selection and branching rules, are proposed to customize the standard BnC method implemented in the MISOCP solver IBM ILOG CPLEX for further computational complexity reduction. We also de- velop low-complexity second-order cone program (SOCP) based inflation and deflation pro- cedures [83,84] to compute the close-to-optimal solutions of the DRAB problem for practical application in large-scale systems.
the performance of the heuristic algorithms. The parallel BnC method implemented in the MISOCP solver CPLEX [81], which are customized according to the DRAB problem and our proposed customizing strategies, is applied to both DRAB problem formulations for reference. Our simulation results show that the average sum-rates of the downlink system achieved by the proposed fast inflation and deflation procedures are very close to that of the optimal solutions computed by the BnC method in CPLEX, while the computational com- plexity of the heuristic algorithms is much less than that of the customized BnC method. Furthermore, the simulation results confirm that the extended MISOCP formulation yields a significant reduction in computational complexity, as compared to the standard big-M MIS- OCP formulation, when applying the customized BnC method on the two formulations. Par- ticularly, the numerical results show that the percentages of the certified optimal solutions (cf. Section 5.5.1) when applying the BnC method on the extended MISOCP formulation are significantly larger than that of the standard big-M MISOCP formulation for the consid- ered settings. This confirms the improvement of the extended formulation over the standard big-M formulation in terms of computational complexity.
This chapter is based on my original work that has been submitted in [115] and my original work that has been published in [113].