Remote estimation subject to packet loss and quantization noise

In this paper we consider the problem of designing coding and decoding schemes to estimate the state of a scalar stable stochastic linear system in the presence of a wireless communication channel between the sensor and the estimator. In particular, we consider a communication channel which is prone to packet loss and includes quantization noise due to its limited capacity. We study two scenarios: the first with channel feedback and the second with no channel feedback. More specifically, in the first scenario the transmitter is aware of the quantization noise and the packet loss history of the channel, while in the second scenario the transmitter is aware of the quantization noise only. We show that in the first scenario, the optimal strategy among all possible linear encoders corresponds to the transmission of the Kalman filter innovation similarly to the differential pulse-code modulation (DPCM). In the second scenario, we show that there is a critical packet loss probability above which it is better to transmit the state rather than the innovation. We also propose a heuristic strategy based on the transmission of a convex combination of the state and the Kalman filter innovation which is shown to provide a performance close to the one obtained with channel feedback.