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For simplicity, we ignore the path loss terms and assume the channel fadings of the K users are i.i.d. The received signal at the kth mobile station can be expressed as Yk = 14h k w kU k + p j h k w jU j + Zk pk 244 4 3 j k Information 144 44 2 3
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= pk h k w kU k + p j h b w jU j - p j Dh b w jU j + Zk k k
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= pk h k w kU k - p j Dh w jU j + Zk j 1 k 4 444 44 2 3
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where the term Sj k pj hb wjUj is nulled out because of the ZF operation and k Vk denotes the residual multiuser interference due to imperfect knowledge of CSIT. The maximum achievable data rate7 of the kth user is given by the maximum mutual information between Yk and Uk conditional on CSIR hk and is expressed as follows: Ck = max I (U k ; Yk h k )
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P (U k ) P (U k )
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= max{H (Yk h k ) - H (Yk U k , h k )}
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2 pk h k w k = log 2 1 + s2 + z j k pj Dh k w j
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From (6.37), the instantaneous channel capacity is a function of hk and Dhk which are unknown to the base station. Hence, given any estimated CSIT Hb, there is still uncertainty on the actual capacity Ck and packet transmission outage is possible when the scheduled data rate rk exceeds the actual capacity Ck. To take into consideration of the packet outage, we de ne the effective goodput of the kth user as: r k = rk 1[C k rk ] (6.38)
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where 1(E) is the indicator function, which is equal to 1 if the event E is true and 0 otherwise. 6.7.2 System Utility Function
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For illustration purposes, we consider the average total goodput as the measure of system performance and the optimization objective. The average total goodput of the multiuser multiantenna system is given by
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The maximum data rate here refers to the Shannon capacity based on random codebook and Gaussian constellation.
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K U thp (r1 , . . . , r K ) = e r k k =1 K = e h b rk e h [1(rk < Ck ) h b ] k =1 K = e h b rk Pr[rk < Ck h b ] k =1 (6.39)
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where rk = e[rk] denotes the average effective throughput and e hb[X] denotes the expectation of the random variable X w.r.t. hb. De ne an instantaneous total goodput G(r1, . . ., rK) as G(r1 , . . . , rK ) = rk Pr[rk < Ck h b ]
k =1 K K
= rk [1 - Pout (rk , h b )]
k =1
where Pout(rk, hb) is the conditional outage probability (conditioned on the CSIT hb) and the average total goodput is given by U thp (r1 , . . . , rK ) = e hb [G(r1 , . . . , rK )]
The Scheduling Problem Formulation
Since the scheduler determines the optimal {A}, {pk}, and {rk} once per CSIT realization, the cross-layer scheduling problem with imperfect CSIT can be summarized as the following optimization problem Problem 6.7 (Cross-Layer Scheduling with Imperfect CSIT) Given any realb ization of the estimated CSIT for all mobile users {hb, . . . , hK}, determine the 1 optimal admitted user set A, the optimal power allocation {pk} and the optimal rate allocation {rk}, such that the instantaneously total goodput, G(r1, . . ., rK), is maximized subject to the total transmit power constraint Sk Apk P0 and the cardinality constraint |A| nT.
The Optimal Scheduling Solution
The optimization variables in Problem 17 include {A}, {pk} as well as {rk}. Hence, the optimization problem involves a mixed convex optimization (w.r.t. {pk} and {rk}) and combinatorial search (w.r.t. {A}). Unlike the case with perfect CSIT where the optimal scheduled rates {rk} are simply given by the instantaneous channel capacities {Ck}, the selection of scheduled data rate