Optimal signal design for estimation of correlated MIMO channels

We address signal design for optimal estimation of correlated multi-input multi-output (MIMO) channels using pilot signals, assuming knowledge of the second-order channel statistics at the transmitter. Assuming a block fading channel model and minimum mean square error (MMSE) estimation at the receiver, we design the transmitted signal to optimize two criteria: MMSE and the conditional mutual information between the MIMO channel and the received signal. Our analysis is based on the proposed virtual channel representation, however, it is generalized to other known channel models like the one in D.Shiu et al. [Mar. 2000]. We show that optimal signaling is in a block form, where the block length depends on the signal to noise ratio (SNR) as well as the channel correlation matrix. The block signal corresponds to transmitting beams in successive time intervals along a fixed virtual transmit angles, whose powers are determined by (non-identical) water filling arguments based on the optimization criteria. Our analysis effectively provides characterization of the dominant subspaces of the channel as a function of the SNR and the scattering environment, which is critical to achieving capacity and space-time code design. In particular, at low SNR the block length reduces to one and all the power is transmitted on the beam corresponding to the strongest transmit angle, while at high SNR the block length has a maximum length equal to the number of active virtual transmit angles and the power is assigned equally to all active transmit angles. Consequently, from a channel estimation viewpoint, a faster fading rate can be tolerated at low SNRs relative to higher SNRs.