Fundamental limits in Gaussian channels with feedback: confluence of communication, estimation, and control

The emerging study of integrating information theory and control systems theory has at­ tracted considerable attention by researchers, mainly motivated by the problems of control under communication constraints, feedback communication, and networked systems. Since in most problems, estimation interacts with communication and control in various ways and can­ not be studied isolatedly, it is natural to investigate systems from the perspective of unifying communication, estimation, and control. This thesis is the first work to advocate such a perspective. To make matters concrete, we focus on communication systems over Gaussian channels with feedback. For some of these channels, their fundamental limits for communication have been studied using information theoretic methods and control-oriented methods but remain open after several decades of research. In this thesis, we address the problems of identifying and achieving the fundamental limits for these Gaussian channels with feedback by applying the unifying perspective. We establish a general equivalence among feedback communication, estimation, and feed­ back stabilization over the same Gaussian channels. As a consequence, we see that the infor­ mation transmission (communication), information processing (estimation), and information utilization (control), seemingly different and usually separately treated, are in fact three sides of the same entity. We then reveal that the fundamental limitations in feedback communica­ tion, estimation, and control coincide: The achievable communication rates in the feedback communication problems can be alternatively given by the decay rates of the Cramer-Rao bounds (CRB) in the associated estimation problems or by the Bode sensitivity integrals in the associated control problems. Utilizing the general equivalence, we design optimal feed­ back communication schemes based on the celebrated Kalman filtering algorithm; these are the first deterministic, optimal feedback communication schemes for these channels (except for the degenerated AWGN case). These schemes also extend the Schalkwijk-Kailath (SK) coding scheme and inherit its useful features, such as reduced coding complexity and improved perfor­ mance. Though for different types of channels, these generalizations are along different lines, they all admit a common interpretation in terms of Kalman filtering of appropriate forms. Thus, we consider that Kalman filtering, the estimation side, acts like the unifier for various problems. In addition, we show the optimality of the Kalman filtering in the sense of information transmission, a supplement to the optimality of Kalman filtering in the sense of information processing proposed by Mitter and Newton. We also obtain a new formula connecting the mutual information in the feedback communication system and the minimum mean-squared error (MMSE) in the associated estimation problem, a supplement to a fundamental relation between mutual information and MMSE proposed by Guo, Shamai, and Verdu. To summarize, this thesis demonstrates that the new perspective plays a significant role in gaining new insights and new results in studying Gaussian feedback communication problems. We anticipate that the perspective and the approaches developed in this thesis could be ex­ tended to more general scenarios and helpful in building a theoretically and practically sound paradigm that unifies information, estimation, and control.