Kalman Filtering over Fading Channels: Zero-One Laws and Almost Sure Stabilities

TL;DR

This paper studies the probabilistic stability of Kalman filters over fading channels with correlated packet dropouts, establishing zero-one laws and deriving stability conditions based on packet arrival rates.

Contribution

It introduces zero-one laws for almost sure stability of Kalman filtering over correlated fading channels and provides necessary and sufficient stability conditions.

Findings

Zero-one laws apply to upper and lower a.s. stabilities.

Stability conditions depend on packet arrival rates.

Equivalence between absolute and standard a.s. stability under one-step observability.

Abstract

In this paper, we investigate probabilistic stability of Kalman filtering over fading channels modeled by $\ast$-mixing random processes, where channel fading is allowed to generate non-stationary packet dropouts with temporal and/or spatial correlations. Upper/lower almost sure (a.s.) stabilities and absolutely upper/lower a.s. stabilities are defined for characterizing the sample-path behaviors of the Kalman filtering. We prove that both upper and lower a.s. stabilities follow a zero-one law, i.e., these stabilities must happen with a probability either zero or one, and when the filtering system is one-step observable, the absolutely upper and lower a.s. stabilities can also be interpreted using a zero-one law. We establish general stability conditions for (absolutely) upper and lower a.s. stabilities. In particular, with one-step observability, we show the equivalence between absolutely a.s. stabilities and a.s. ones, and necessary and sufficient conditions in terms of packet arrival rate are derived; for the so-called non-degenerate systems, we also manage to give a necessary and sufficient condition for upper a.s. stability.