A General Framework for Covariance Matrix Optimization in MIMO Systems

For multi-input multi-output (MIMO) systems, many transceiver design problems involve the optimization of the covariance matrices of the transmitted signals. Karush-Kuhn-Tucker (KKT) conditions based derivations are the most popular method, and many derivations and results have been reported for different scenarios of MIMO systems. We propose a unified framework in formulating the KKT conditions for general MIMO systems. Based on this framework, the optimal water-filling structure of the transmission covariance matrices are derived rigorously, which is applicable to a wide range of MIMO systems. Our results show that for MIMO systems with various power constraint formulations and objective functions, both the derivation logics and water-filling structures for the optimal covariance matrix solutions are fundamentally the same. Thus, our unified framework and solution reveal the underlying relationships among the different water-filling structures of the covariance matrices. Furthermore, our results provide new solutions to the covariance matrix optimization of many complicated MIMO systems with multiple users and imperfect channel state information (CSI) which were unknown before.