Performance Bounds for Sampling and Remote Estimation of Gauss-Markov Processes over a Noisy Channel with Random Delay

TL;DR

This paper develops optimal sampling policies for Gauss-Markov processes, including unstable cases, over noisy channels with delays, providing bounds and analyzing the impact of noise on estimation accuracy.

Contribution

It extends existing solutions to unstable Ornstein-Uhlenbeck processes and incorporates noisy samples, deriving new threshold policies and performance bounds.

Findings

Optimal threshold policies vary for stable, unstable, and Wiener processes.

Additive noise degrades estimation performance, increasing mean-square error.

Derived upper bounds for estimation error with noisy samples.

Abstract

In this study, we generalize a problem of sampling a scalar Gauss Markov Process, namely, the Ornstein-Uhlenbeck (OU) process, where the samples are sent to a remote estimator and the estimator makes a causal estimate of the observed realtime signal. In recent years, the problem is solved for stable OU processes. We present solutions for the optimal sampling policy that exhibits a smaller estimation error for both stable and unstable cases of the OU process along with a special case when the OU process turns to a Wiener process. The obtained optimal sampling policy is a threshold policy. However, the thresholds are different for all three cases. Later, we consider additional noise with the sample when the sampling decision is made beforehand. The estimator utilizes noisy samples to make an estimate of the current signal value. The mean-square error (mse) is changed from previous due to noise and the additional term in the mse is solved which provides performance upper bound and room for a pursuing further investigation on this problem to find an optimal sampling strategy that minimizes the estimation error when the observed samples are noisy. Numerical results show performance degradation caused by the additive noise.