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 bits transmitted per second. A decoder uses the received codewords to causally estimate the process in real time. We aim to find the optimal encoding and decoding policies that minimize the end-to-end estimation mean-square error under the rate constraint. For a class of continuous Markov processes satisfying regularity conditions, 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 as 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.
翻译:考虑以下通信情景。 编码器观察一个随机过程, 并在对每秒传输的位数的限制下, 以因果方式决定它的时间和传输方式。 编码器使用收到的编码词对过程进行实时的因果估计。 我们的目标是找到最佳编码和解码政策, 最大限度地减少利率限制下的端到端估计平均方差错误。 对于符合常规条件的连续马可夫进程类别, 我们显示, 最佳编码政策在进程创新通过两个阈值之一后, 传输一个 $$- bit 的编码字。 最佳解码器无噪音从 1 位编码词和代码生成时间印章中回收最后的样本, 并用作当前进程的运行估计, 直到下一个编码出现。 特别是, 我们展示Ornstein- Uhlenbeck 进程的最佳因果代码, 并计算其扭曲率函数 。