Kalman-Bucy Filtering with Randomized Sensing: Fundamental Limits and Sensor Network Design for Field Estimation

TL;DR

This paper establishes fundamental performance limits for continuous-time Kalman filtering with randomly varying measurements, providing insights for sensor network design in field estimation tasks.

Contribution

It derives a closed-form upper bound on estimation covariance in a general continuous-time setting with random measurement processes, and introduces a grid-independent lower bound on spatially averaged clarity for sensor placement.

Findings

The bounds are tight for discrete-time Kalman filters at low measurement rates.

The derived limits guide sensor network design before deployment.

Simulations validate the theoretical bounds and their practical relevance.

Abstract

Stability analysis of the Kalman filter under randomly lost measurements has been widely studied. We revisit this problem in a general continuous-time framework, where both the measurement matrix and noise covariance evolve as random processes, capturing variability in sensing locations. Within this setting, we derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman filtering. We then apply this framework to spatiotemporal field estimation, where the field is modeled as a Gaussian process observed by randomly located, noisy sensors. Using clarity, introduced in our earlier work as a rescaled form of the differential entropy of a random variable, we establish a grid-independent lower bound on the spatially averaged expected clarity. This result exposes fundamental performance limits through a composite sensing parameter that jointly captures the effects of the number of sensors, noise level, and measurement frequency. Simulations confirm that the proposed bound is tight for the discrete-time Kalman filter, approaching it as the measurement rate decreases, while avoiding the recursive computations required in the discrete-time formulation. Most importantly, the derived limits provide principled and efficient guidelines for sensor network design problem prior to deployment.