State Estimation Over Delayed and Lossy Channels: An Encoder–Decoder Synthesis

State estimation over a communication channel, in which sensory information of a stochastic source is transmitted in real-time by an encoder to a decoder that estimates the state of the source, is one of the basic problems in networked control systems. In this article, we investigate the performance of state estimation under two primary network imperfections: packet loss and time delay. To that end, we make a causal frequency-distortion tradeoff that is defined between the packet rate and the mean square error, when the source and the channel are modeled by a partially observable Gauss–Markov process and a fixed-delay packet-erasure channel, under two distinct communication protocols: one with and one without packet-loss detection. We prove the existence of a globally optimal policy profile, and show that this policy profile is composed of a symmetric threshold scheduling policy and a non-Gaussian linear estimation policy, which are used by the encoder and the decoder, respectively. Our structural results assert that the scheduling policy is expressible in terms of
$\boldsymbol{\mathit {3d-1}}$
variables related to the source and the channel, where
$\boldsymbol{\mathit {d}}$
is the time delay, and that the estimation policy incorporates no residuals related to signaling. The key finding is that packet-loss detection does not increase the performance of the underlying networked system in the sense of the causal frequency-distortion tradeoff. We prove this by showing that the globally optimal policy profile remains exactly the same under both of the communication protocols.