  This paper considers minimum sum mean-squared error (sum-MSE) linear
transceiver designs in multiuser downlink systems with imperfect channel state
information. Specifically, we derive the optimal energy allocations for
training and data phases for such a system. Under MMSE estimation of
uncorrelated Rayleigh block fading channels with equal average powers, we prove
the separability of the energy allocation and transceiver design optimization
problems. A closed-form optimum energy allocation is derived and applied to
existing transceiver designs. Analysis and simulation results demonstrate the
improvements that can be realized with the proposed design.
