  This paper considers linear minimum meansquare- error (MMSE) transceiver
design problems for downlink multiuser multiple-input multiple-output (MIMO)
systems where imperfect channel state information is available at the base
station (BS) and mobile stations (MSs). We examine robust sum mean-square-error
(MSE) minimization problems. The problems are examined for the generalized
scenario where the power constraint is per BS, per BS antenna, per user or per
symbol, and the noise vector of each MS is a zero-mean circularly symmetric
complex Gaussian random variable with arbitrary covariance matrix. For each of
these problems, we propose a novel duality based iterative solution. Each of
these problems is solved as follows. First, we establish a novel sum average
meansquare- error (AMSE) duality. Second, we formulate the power allocation
part of the problem in the downlink channel as a Geometric Program (GP). Third,
using the duality result and the solution of GP, we utilize alternating
optimization technique to solve the original downlink problem. To solve robust
sum MSE minimization constrained with per BS antenna and per BS power problems,
we have established novel downlink-uplink duality. On the other hand, to solve
robust sum MSE minimization constrained with per user and per symbol power
problems, we have established novel downlink-interference duality. For the
total BS power constrained robust sum MSE minimization problem, the current
duality is established by modifying the constraint function of the dual uplink
channel problem. And, for the robust sum MSE minimization with per BS antenna
and per user (symbol) power constraint problems, our duality are established by
formulating the noise covariance matrices of the uplink and interference
channels as fixed point functions, respectively.
