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On Stability and Performance of the Optimal Linear Filter Over Gilbert–Elliott Channels With Unobservable Packet Losses

This article investigates the optimal linear filter (OLF) over Gilbert–Elliott channels where packet losses follow a two-state Markov chain. For a system with packet losses, according to whether the packet losses can be observed by the filter, the system is divided into two types: 1) a system with observable packet losses (an OPL system) and 2) a system with unobservable packet losses (a UPL system). We first obtain the OLF for UPL systems over Gilbert–Elliott channels, and then analyze how the loss of packet-loss-status observability affects the estimation stability and performance. Unlike the OLF for OPL systems, which may remain stable for an unstable OPL system when the packet loss rate is less than a threshold value, we show that the OLF for UPL systems is stable if and only if the system is stable. We prove that from a mean sense perspective, the unobservability of packet losses will degrade the estimation performance, and an upper bound of the performance degradation is obtained. Furthermore, we analytically characterize how the packet recovery/failure rate affects the estimation performance for the two special cases. Finally, numerical examples are provided to illustrate the obtained results.