Convex formulation of the stochastic MV-PURE estimator and its relation to the reduced rank Wiener filter

We investigate relation between the well-known reduced rank Wiener filter (RR-MMSE) and the recently proposed stochastic MV-PURE estimator. We show that, for a white input random vector, the forms of stochastic MV-PURE estimator and RR-MMSE filter are very similar. We also provide conditions, upon which the stochastic MV-PURE and RR-MMSE coincide. Moreover, we demonstrate that the MV-PURE estimator, analogously as the full-rank MMSE estimator, can be cast as a solution of a convex optimization problem, which suppresses the difficulty of optimization under the non-convex rank constraint.