Optimal Causal Rate-Constrained Sampling of the Wiener Process

We consider the following communication scenario. An encoder causally observes the Wiener process and decides when and what to transmit about it. A decoder estimates the process using causally received codewords in real time. We determine the causal encoding and decoding policies that jointly minimize the mean-square estimation error, under the long-term communication rate constraint of R bits per second. We show that an optimal encoding policy can be implemented as a causal sampling policy followed by a causal compressing policy. We prove that the optimal encoding policy samples the Wiener process once the innovation passes either root 1/R or -root 1/R and compresses the sign of innovation (SOI) using a 1-bit codeword. The SOI coding scheme achieves the operational distortion-rate function, which is equal to D-op(R) = 1/6R Surprisingly, this is significantly better than the distortion-rate tradeoff achieved in the limit of infinite delay by the best noncausal code. This is because the SOI coding scheme leverages the free timing information supplied by the zero-delay channel between the encoder and the decoder. The key to unlocking that gain is the event-triggered nature of the SOI sampling policy. In contrast, the distortion-rate tradeoffs achieved with deterministic sampling policies are much worse: we prove that the causal informational distortion-rate function in that scenario is as high as D-DET (R) = 5/6R It is achieved by the uniform sampling policy with the sampling interval 1/R. In either case, the optimal strategy is to sample the process as fast as possible and to transmit 1-bit codewords to the decoder without delay. We show that the SOI coding scheme also minimizes the mean-square cost of a continuous-time control system driven by the Wiener process and controlled via rate-constrained impulses.

Automated summary

This research investigates a communication scenario where an encoder observes and transmits information about a Wiener process in a causal manner, and a decoder estimates the process in real time. The study determines the optimal causal encoding and decoding policies that minimize estimation error while considering the communication rate constraint. Moreover, it shows that the optimal encoding policy involves sampling the Wiener process and compressing the sign of innovation using 1-bit codewords.