Control-theoretic Approach to Communication with Feedback: Fundamental Limits and Code Design

TL;DR

This paper explores the fundamental limits of feedback communication through a control-theoretic lens, establishing bounds on capacity, and designing codes for Gaussian channels and multiple access channels using control principles.

Contribution

It introduces a control-theoretic framework for feedback communication, deriving capacity bounds, and generalizing code design to multi-user channels using control solutions.

Findings

MMSE capacity bounds by feedback capacity

Bode's integral formula relates feedback capacity to system instability

Kramer's code derived from LQG control for multi-user channels

Abstract

Feedback communication is studied from a control-theoretic perspective, mapping the communication problem to a control problem in which the control signal is received through the same noisy channel as in the communication problem, and the (nonlinear and time-varying) dynamics of the system determine a subclass of encoders available at the transmitter. The MMSE capacity is defined to be the supremum exponential decay rate of the mean square decoding error. This is upper bounded by the information-theoretic feedback capacity, which is the supremum of the achievable rates. A sufficient condition is provided under which the upper bound holds with equality. For the special class of stationary Gaussian channels, a simple application of Bode's integral formula shows that the feedback capacity, recently characterized by Kim, is equal to the maximum instability that can be tolerated by the controller under a given power constraint. Finally, the control mapping is generalized to the N-sender AWGN multiple access channel. It is shown that Kramer's code for this channel, which is known to be sum rate optimal in the class of generalized linear feedback codes, can be obtained by solving a linear quadratic Gaussian control problem.