Monitoring the Brownian Motion in Real Time: A Rate-Distortion Perspective

In this paper, we study the real-time remote monitoring of the Brownian motion from a rate-distortion perspective. Specifically, the distortion in the conversion of analog wiener processes into bit sequences is characterized by the mean square error (MSE) of the real-time reconstructed or estimated processes. We find that the real-time MSE is jointly determined by the age of the freshest sample and the quantization error. We then optimize the joint sampling and quantization policy for Brownian motions over the finite time horizon. For the infinite time horizon, we present a layered joint sampling and quantization method, which is capable of attaining both finite MSE and data rate simultaneously.