Real-time monitoring has received considerable attention recently due to its potential in automatic driving, tele-surgery, and factory automation in the 6G era. In remote estimation or reconstruction of stochastic processes, the statistical properties of stochastic processes to be monitored play a central role. Among the stochastic processes interested by real-time applications, the Brownian motion, also known as the Wiener processes is a typical one. In this paper, we are interested in how to monitor Brownian motions efficiently, timely, and reliably. To achieve this goal, we reveal that the real-time estimation error is jointly determined by the quantization error and freshness of data samples. Based on this observation, we present an optimal joint sampling and quantization scheme that efficiently balances the quantization distortion and the age-of-information (AoI). Furthermore, we find that the error accumulation will lead to infinite distortion as monitoring time increases. To overcome this, a multi-layer error correction method is presented for infinite-time monitoring, in which bounded distortion can be achieved with limited data rate. Finally, to conquer the accumulation of transmission errors in unreliable channels, we present an error correction mechanism based on periodic feedback. Diffusion approximation is then adopted to determine the optimal feedback rate and interval.

中文翻译：

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布朗运动的实时监测

实时监控因其在 6G 时代的自动驾驶、远程手术和工厂自动化方面的潜力而受到了相当大的关注。在随机过程的远程估计或重建中，要监测的随机过程的统计特性起着核心作用。在实时应用感兴趣的随机过程中，布朗运动，也称为维纳过程是一个典型的过程。在本文中，我们感兴趣的是如何有效、及时和可靠地监测布朗运动。为了实现这一目标，我们揭示了实时估计误差是由量化误差和数据样本的新鲜度共同决定的。基于这一观察，我们提出了一种最佳的联合采样和量化方案，可以有效地平衡量化失真和信息年龄（AoI）。此外，我们发现随着监测时间的增加，误差累积会导致无限失真。为了克服这个问题，提出了一种用于无限时间监控的多层纠错方法，其中可以在有限的数据速率下实现有界失真。最后，为了克服不可靠信道中传输错误的累积，我们提出了一种基于周期性反馈的纠错机制。然后采用扩散近似来确定最佳反馈率和间隔。提出了一种用于无限时间监控的多层纠错方法，其中可以在有限的数据速率下实现有界失真。最后，为了克服不可靠信道中传输错误的累积，我们提出了一种基于周期性反馈的纠错机制。然后采用扩散近似来确定最佳反馈率和间隔。提出了一种用于无限时间监控的多层纠错方法，其中可以在有限的数据速率下实现有界失真。最后，为了克服不可靠信道中传输错误的累积，我们提出了一种基于周期性反馈的纠错机制。然后采用扩散近似来确定最佳反馈率和间隔。