CROSS-LAYER OPTIMIZATION WITH PERFECT CSIT

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Lemma 6.3 (Full Spatial Multiplexing at High SNR) The optimal solution in Problem 6.2 with Gcap = Skrk and capacity region COTBF satis es |A| = nT when the SNR (P0/s 2) is high. Here, Adenotes the admitted user set of user indices z with positive power allocation. Proof Please refer to Appendix 6C. 6.4.2 Coverage-Optimized Scheduler Design

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For the purpose of illustration, we consider the following user-centric coverage utility: Gcov = min rk

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k A

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(6.23)

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where A = {k : pk > 0} is the set of users with nonzero power allocation. Hence, coverage is de ned with respect to the worst-case instantaneous data rate of all the selected users. The coverage-optimized scheduler design is equivalent to the following optimization problem. Problem 6.3 (Coverage-Optimized Scheduling) Given any channel matrix realization of all mobile users, { L1h1, . . . , LKhK}, select the power allocation (p1, . . . , pK) to minimize the worst-case outage probability P* (de ned in out Lemma 6.2) such that Skpk P0 and (r1, . . . , rK) C. We shall illustrate in Lemma 6.4 that the optimal admitted user set always has one user being selected. In other words, the available degrees of freedom should all be utilized to exploit spatial diversity in order to minimize the outage probability. Lemma 6.4 (Full Spatial Diversity) The optimal solution (p1, . . . , pK) in Problem 6.3 with Gcov(r1, . . . , rK) = mink Ark and the capacity region COTBF satis es |A| = 1, where A = {k : pk > 0}. Proof Please refer to Appendix 6D. 6.4.3 Common Framework for Both Capacity-Optimized and Coverage-Optimized Scheduling While both the network capacity and the network coverage are important performance measures, they are different optimization objectives, and therefore tradeoff is generally required in the scheduling optimization with respect to these two objectives. As a result of Lemma 6.3, we see that at high SNR, the capacity-optimized scheduler would fully utilize the degrees of freedom for spatial multiplexing (|A| = nT). On the other hand, from Lemma 6.4, the coverage-optimized scheduler would fully utilize the available degrees of freedom for spatial diversity (|A| = 1). On the basis of these observations, we unify the

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CROSS-LAYER SCHEDULING FOR MULTIUSER SYSTEMS WITH MULTIPLE ANTENNAS

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capacity-optimized and coverage-optimized scheduling designs into a common optimization problem, P(Q, nT), in the following theorem. Theorem 6.1 (Common Scheduling Framework) Consider the optimization problem P(Q, nT) in Problem 6.4. Problem 6.4 Select ( p1 , . . . , pK ) to maximize Gcap (r1 , . . . , rK ) =

(6.24)

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with constraints Skpk P0, (r1, . . . , rK) C and |A| Q where Q [1, nT] is a control parameter on the level of spatial multiplexing and C is the capacity region of the corresponding physical layer. When Q = nT and C corresponds to the OTBF capacity region, the solution of P(Q, nT) optimizes the capacity-optimized scheduling problem with respect to Gcap = Skrk in Problem 6.2.When Q = 1 and C corresponds to the OTBF capacity region, the solution of P(Q, nT) optimizes the coverage-optimized scheduling problem with respect to P* in Problem 6.3 and Gcov in (6.23). out Proof Please refer to Appendix 6E. Hence, from Theorem 6.1, we can deduce that if the physical layer processing is OTBF at the base station, the optimization problem in P(Q, nT) can result in a capacity-optimized scheduler design (when Q = nT) as well as a coverage-optimized scheduler design (when Q = 1) by adjusting the cardinality parameter Q. Hence, for both design objectives, we can focus on solving a single optimization problem P(Q, nT). 6.4.4 Optimal Solution Single-Antenna Systems For a single-antenna base station nT = 1, the capacity region CnT=1 is given by the capacity region of degraded broadcast channels as illustrated in Equation (6.12). Without loss of generality, assume |h1| > |h2| > . . . > |hK|. For Gcap(r1, . . . , rK) = Skmkrk, where mk is a constant weight for user k, the cross-layer optimization problem [131] can be rewritten as follows. Problem 6.5 (Cross-Layer Optimization Single Antenna)

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max L( p1 , . . . , pK ; l )

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(6.25)

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2 hk pk where L = k log 2 1 + - l k pk is the Lagrangian func2 2 hk j <k p j + s z tion and l is the Lagrange multiplier for the constraint Skpk = P0.

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