一类连续Markov过程的最优因果率约束抽样
Optimal Causal Rate-Constrained Sampling for a Class of Continuous Markov Processes
摘要：
考虑以下通信场景。编码器观察一个随机过程，并在每秒传输的预期比特数的约束下，因果地决定何时传输以及传输什么。解码器使用接收到的码字来实时的因果地估计该过程。编码器和解码器在时间上是同步的。对于一类满足正则性条件的连续Markov过程，在速率约束下，找到了使端到端估计均方误差最小的最优编码和解码策略.我们表明，最佳的编码策略传输一个$1$位码字的过程中的创新，一旦通过两个阈值之一。最佳解码器从1位码字和码字生成时间戳中无噪声地恢复最后一个样本，并使用它来决定当前过程的运行估计，直到下一个码字到达。特别是，我们显示的最佳因果码的Ornstein-Uhlenbeck过程，并计算其失真率函数。此外，我们表明，最佳因果码也最小化的连续时间控制系统的均方成本由一个连续的马尔可夫过程和控制的附加控制信号。
英文摘要：
Consider the following communication scenario. An encoder observes a stochastic process and causally decides when and what to transmit about it, under a constraint on the expected number of bits transmitted per second. A decoder uses the received codewords to causally estimate the process in real time. The encoder and the decoder are synchronized in time. For a class of continuous Markov processes satisfying regularity conditions, we find the optimal encoding and decoding policies that minimize the end-to-end estimation mean-square error under the rate constraint. We show that the optimal encoding policy transmits a $1$-bit codeword once the process innovation passes one of two thresholds. The optimal decoder noiselessly recovers the last sample from the 1-bit codewords and codeword-generating time stamps, and uses it to decide the running estimate of the current process, until the next codeword arrives. In particular, we show the optimal causal code for the Ornstein-Uhlenbeck process and calculate its distortion-rate function. Furthermore, we show that the optimal causal code also minimizes the mean-square cost of a continuous-time control system driven by a continuous Markov process and controlled by an additive control signal.
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