On MMSE, mutual information and Popov function in continuous-time multivariable linear systems

By analyzing information transmission in stable and unstable, open- and closed-loop linear continuous-time Gaussian systems, connections within the MMSE of causal estimation, mutual information rate, and the integral of a Popov function are discussed. Yovits´ equality for scalar stable systems and Anderson´s equality for scalar unstable systems concerning MMSE are extended to multivariable systems. As a consequent result, it is concluded that the mutual information rate between system state and output is changed by feedback, and the variation of time-average information transmission is defined by a Bode- like integral function which is related to the causal filtering MMSE and the Popov integral function under a special feedback gain.