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

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 real-time 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.