Worst-case MSE precoder design for imperfectly known MIMO communications channels

This paper considers a worst-case precoder design problem for multiple-input multiple-output (MIMO) wireless communication systems with imperfect channel knowledge at the transmitter. When the MIMO channel is a full-row rank matrix, which arises generically when the number n_T of transmit antennas is greater or equal to the number n_R of receive antennas, and when channel state information is known perfectly at the transmitter, the channel can be equalized exactly by employing a precoder equal to the channel pseudo-inverse. However, in actual systems, it is necessary to take into account the channel estimation error. We consider here a worst-case precoder design problem where the goal is to find the precoder minimizing the equalization mean-square error for the least favorable channel located in a ball centered about the estimated channel. Lagrangian optimization is used to convert this min-max problem into a min-min convex minimization problem over a convex domain which can be solved in closed form. The robust precoder and associated least favorable channel have an intuitive interpretation since the least-favorable channel zeroes out the weakest subchannels, and the robust precoder implements a pseudo-inverse only for the remaining subchannels.