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Optimal Causal Rate-Constrained Sampling of the Wiener Process

Guo, Nian and Kostina, Victoria (2021) Optimal Causal Rate-Constrained Sampling of the Wiener Process. IEEE Transactions on Automatic Control . ISSN 0018-9286. doi:10.1109/tac.2021.3071953. (In Press)

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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 √1/R or −√1/R and compresses the sign of the 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 non-causal 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 1R. 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.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper ItemConference Paper
Guo, Nian0000-0003-4490-328X
Kostina, Victoria0000-0002-2406-7440
Additional Information:© 2021 IEEE. This work was supported in part by the National Science Foundation (NSF) under grant CCF-1751356. A part of this work is presented at the 57th Annual Allerton Conference [36]; the conference version does not contain Section VI or any proofs.
Funding AgencyGrant Number
Subject Keywords:Causal lossy source coding, sequential estimation, sampling, rate-distortion theory, continuous-time tracking
Record Number:CaltechAUTHORS:20210412-071230573
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Official Citation:N. Guo and V. Kostina, "Optimal Causal Rate-Constrained Sampling of the Wiener Process," in IEEE Transactions on Automatic Control, doi: 10.1109/TAC.2021.3071953
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108684
Deposited By: Tony Diaz
Deposited On:13 Apr 2021 23:16
Last Modified:19 Apr 2021 18:01

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