Optimizing multiple-input single-output (MISO) communication systems with general Gaussian channels: nontrivial covariance and nonzero mean

We consider a narrow-band point-to-point communication system with many (input) transmitters and a single (output) receiver (i.e., a multiple-input single output (MISO) system). We assume the receiver has perfect knowledge of the channel but the transmitter only knows the channel distribution. We focus on two canonical classes of Gaussian channel models: (a) the channel has zero mean with a fixed covariance matrix and (b) the channel has nonzero mean with covariance matrix proportional to the identity. In both cases, we are able to derive simple analytic expressions for the ergodic average and the cumulative distribution function (c.d.f.) of the mutual information for arbitrary input (transmission) signal covariance. With minimal numerical effort, we then determine the ergodic and outage capacities and the corresponding capacity-achieving input signal covariances. Interestingly, we find that the optimal signal covariances for the ergodic and outage cases have very different behavior. In particular, under certain conditions, the outage capacity optimal covariance is a discontinuous function of the parameters describing the channel (such as strength of the correlations or the nonzero mean of the channel).