Mean-variance optimal linear precoders for random MISO broadcast channels

We consider the problem of designing linear precoders for Gaussian multiple input, single output (MISO) broadcast (BC) channels with a random channel matrix, where the randomness models imperfect channel knowledge due to feedback delay, channel time variations, limited training, or quantized feedback. First, we introduce a design framework where the goal is to minimize outage probability (or risk) or maximize rate. It is shown that full-power precoders, i.e. those that meet the transmit power constraint with equality, satisfy the design criterion. Next, we motivate and solve a related mean-variance optimization problem by supposing the channel density is a Gaussian mixture model (GMM) and deriving efficient formulae to compute the objective and constraint functions.