Capacity pre-log achieving schemes for stationary frequency-flat multi-antenna channels

We analyze the Mutual Information of stationary Frequency-Flat MIMO Channels, that are hence characterized by a Doppler spectrum. Absence of Channel State Information at Transmitter or Receiver (no CSIT/CSIR) is assumed. For peak-power limited SISO frequency-flat channels with stationary Gaussian fading, it has been shown by Lapidoth that at high SNR, the capacity is determined by a pre-log factor that is equal to the bandwidth of frequencies where the channel Doppler spectrum is zero. In this paper, we give simple upper and lower bounds for the capacity of MIMO channels. These bounds are very revealing about the multiplexing gain (pre-log factor) of the system. Then we extend Lapidoth's result to MIMO channels with the help of these bounds. In a general (block) stationary setting, the absence of CSIR decreases the pre-log with a factor equal to 1 minus the average number of parameters per symbol period that parameterize the channel. This reduction term is proportional to the Doppler bandwidth and the number of transmit antennas. We introduce channel parameterizations that induce a split in the transmitted symbols between "learning" symbols (that carry loglog(SNR) information) and "data" symbols (that carry log(SNR) information). The pre-log factor is the proportion of "data" symbols. This decomposition shows the optimality of certain training schemes for practical SNR values. The optimal pre-log requires optimization over tx antennas as a function of Doppler BW.