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：信源对维纳过程进行采样，并通过无线信道将采样数据发送至远程估计器。由于信道衰落、干扰及潜在冲突，数据包传输不可靠且传输时长具有随机性。我们的目标是设计最优因果采样策略，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个简单的阈值策略：当瞬时估计误差超过阈值时即启动新采样。该阈值保持恒定，不随时间变化。通过分析递归最优停止问题的结构特性，我们开发了一种低复杂度的迭代算法来计算最优阈值。本研究将传输错误与随机传输时间纳入远程估计模型，拓展了既有研究成果。数值仿真实验将我们的最优策略与零等待策略及年龄最优策略进行了对比分析。