The capacity loss of uncorrelated equi-power Gaussian input over MIMO channel

We investigate the capacity loss of an uncorrelated Gaussian input with equal power (i.i.d. Gaussian input) over a multi-input multi-output linear additive noise (not necessarily Gaussian nor memoryless) channel. Previous work showed that this input is the best input in the case of Gaussian noise, assuming the channel matrix is known at the receiver but unknown at the transmitter. We show that i.i.d. Gaussian is a robust input also when the noise is not Gaussian. Specifically, we show that for n_t transmit antennas and n_r receive antennas, the capacity loss of an i.i.d. Gaussian input is smaller than min{n_t/2, (n_r/2)log₂(1 + n_t/n_r)} bits, for any noise and channel matrix. This bound is apparently not tight. Nevertheless, for the case of Gaussian noise we derive a stronger bound which is tight for a "critical" channel matrix: (n_r/2)log₂(n_t/n_r) bits for 1≤ n_r≤(n_t/e) and (n_t/2)(log₂(e))/e bits for n_r≥(n_t/e).