Optimal Generalized Inverses for Zero Forcing Precoding

We consider the problem of linear zero forcing precoding design, and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudo-inverse. We begin with the standard design under the assumption of a total power constraint and prove that precoders based on the pseudo-inverse are optimal in this setting. Then, we proceed to examine individual per-antenna power constraints. In this case, the pseudo-inverse is not necessarily the optimal generalized inverse. In fact, finding the optimal inverse is non-trivial and depends on the specific performance measure. We address two common criteria, fairness and throughput, and show that the optimal matrices may be found using standard convex optimization methods. We demonstrate the improved performance offered by our approach using computer simulations.