In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet transmissions are unreliable and could take random time durations. Our objective is to devise an optimal causal sampling policy that minimizes the long-term average mean square estimation error. This optimal sampling problem is a recursive optimal stopping problem, which is generally quite difficult to solve. However, we prove that the optimal sampling strategy is, in fact, a simple threshold policy where a new sample is taken whenever the instantaneous estimation error exceeds a threshold. This threshold remains a constant value that does not vary over time. By exploring the structure properties of the recursive optimal stopping problem, a low-complexity iterative algorithm is developed to compute the optimal threshold. This work generalizes previous research by incorporating both transmission errors and random transmission times into remote estimation. Numerical simulations are provided to compare our optimal policy with the zero-wait and age-optimal policies.

翻译：本文研究一个采样问题：信源从Wiener过程中提取样本，并通过无线信道将其传输至远程估计器。由于信道衰落、干扰和潜在冲突，数据包传输不可靠且可能经历随机时长。我们的目标是设计一种最优因果采样策略，以最小化长期平均均方估计误差。该最优采样问题本质上是递归最优停止问题，通常求解难度较大。然而，我们证明最优采样策略实际上是一种简单的阈值策略：当瞬时估计误差超过阈值时立即采集新样本。该阈值为不随时间变化的常数。通过探索递归最优停止问题的结构特性，我们开发了一种低复杂度迭代算法来计算最优阈值。本研究通过将传输错误和随机传输时间同时纳入远程估计框架，推广了先前的研究成果。数值仿真将我们提出的最优策略与零等待策略和年龄最优策略进行了对比。