Unified Framework of KKT Conditions Based Matrix Optimizations for MIMO Communications

For multi-input multi-output (MIMO) communication systems, many transceiver design problems involve the optimization of the covariance matrices of the transmitted signals. The derivation of the optimal solutions based on Karush-Kuhn-Tucker (KKT) conditions is a most popular method, and many results have been reported for different scenarios of MIMO systems. In this overview paper, we propose a unified framework in formulating the KKT conditions for general MIMO systems. Based on this framework, the optimal water-filling structures of the transmission covariance matrices are derived rigorously, which are applicable to a wide range of MIMO systems. Our results show that for seemingly different MIMO systems with various power constraints and objective functions, the derivations 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 which were unknown before.