  Block diagonalization (BD) is a practical linear precoding technique that
eliminates the inter-user interference in downlink multiuser multiple-input
multiple-output (MIMO) systems. In this paper, we apply BD to the downlink
transmission in a cooperative multi-cell MIMO system, where the signals from
different base stations (BSs) to all the mobile stations (MSs) are jointly
designed with the perfect knowledge of the downlink channels and transmit
messages. Specifically, we study the optimal BD precoder design to maximize the
weighted sum-rate of all the MSs subject to a set of per-BS power constraints.
This design problem is formulated in an auxiliary MIMO broadcast channel (BC)
with a set of transmit power constraints corresponding to those for individual
BSs in the multi-cell system. By applying convex optimization techniques, this
paper develops an efficient algorithm to solve this problem, and derives the
closed-form expression for the optimal BD precoding matrix. It is revealed that
the optimal BD precoding vectors for each MS in the per-BS power constraint
case are in general non-orthogonal, which differs from the conventional
orthogonal BD precoder design for the MIMO-BC under one single sum-power
constraint. Moreover, for the special case of single-antenna BSs and MSs, the
proposed solution reduces to the optimal zero-forcing beamforming (ZF-BF)
precoder design for the weighted sum-rate maximization in the multiple-input
single-output (MISO) BC with per-antenna power constraints. Suboptimal and
low-complexity BD/ZF-BF precoding schemes are also presented, and their
achievable rates are compared against those with the optimal schemes.
