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Optimal Causal RateConstrained 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 makes realtime estimation of the process using causally received codewords. We determine the causal encoding and decoding policies that jointly minimize the meansquare estimation error, under the longterm 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 √(1/R) or √(1/R), and compresses the sign of the innovation (SOI) using a 1bit codeword. The SOI coding scheme achieves the operational distortionrate function, which is equal to D^op(R)=1/6R. Surprisingly, this is significantly better than the distortionrate 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 zerodelay channel between the encoder and the decoder. The key to unlock that gain is the eventtriggered nature of the SOI sampling policy. In contrast, the distortionrate tradeoffs achieved with deterministic sampling policies are much worse: we prove that the causal informational distortionrate 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 1bit codewords to the decoder without delay.
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