Optimal Causal Rate-Constrained Sampling of the Wiener Process

An encoder causally observes the Wiener process and decides when what to transmit about it. A decoder makes real-time estimation of using received codewords. We determine causal encoding decoding policies that minimize mean-square error, under long-term communication rate constraint R bits/s. show an optimal policy can be implemented as a sampling followed by compressing policy. prove samples once innovation passes some threshold, compresses sign (SOI) 1-bit codeword. The SOI coding scheme achieves operational distortion-rate function. This is better than tradeoff achieved best non-causal code, because uses free timing information supplied zero-delay channel between decoder. tradeoffs with deterministic are much worse. strategy sample fast possible codewords without delay.