Rate-Distortion Function for Gaussian Markov Processes

Suppose in the communication system of Fig. 1, the source emits a sequence of continuous-valued random variables. The exact specification of such variates requires an infinite number of binary digits. Hence exact transmission would require a channel of infinite capacity. Since no physical channels possess infinite capacity, we see that exact transmission is impossible through this system. However, if we are willing to accept some error in our specification of the source output, then finitely many binary digits are necessary. In the study of digital encoding systems, a useful quantity to know is the fewest number of binary digits necessary to represent an analog signal within a certain error. Such a quantity would give us a performance criterion with which to compare existing systems, and also tell us how much improvement is possible. The quantity we seek is given by Shannon's rate-distortion function. 1 ' 2 The rate-distortion function gives, for any bit rate, the minimum possible error achievable. In this paper we study the rate-distortion functions for the important * This research was partially supported by the Air Force Office of Scientific Research under Contract AF 49(638)-1600. This paper is part of a dissertation submitted in 1969 to the Faculty of the Polytechnic Institute of Brooklyn, in partial fulfillment of the requirements for the Ph.D. degree in systems science. 3059