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 minimi...

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Veröffentlicht in:IEEE transactions on automatic control Jg. 67; H. 4; S. 1776 - 1791
Hauptverfasser: Guo, Nian, Kostina, Victoria
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Causal lossy source coding
Coders
Codes
Coding
Constraints
Continuous time systems
continuous-time tracking
Decoding
Delay
Delays
Distortion
Encoding
Estimation
Innovations
Linear systems
Policies
Process control
rate-distortion theory
Sampling
sequential estimation
Tradeoffs
ISSN:0018-9286, 1558-2523
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Zusammenfassung: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 <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">\sqrt{\frac{1}{R}}</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">-\sqrt{\frac{1}{R}}</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">D^{\mathrm{op}}(R)=\frac{1}{6R}</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">D_{\mathrm{DET}}(R) = \frac{5}{6R}</tex-math></inline-formula>. It is achieved by the uniform sampling policy with the sampling interval <inline-formula><tex-math notation="LaTeX">\frac{1}{R}</tex-math></inline-formula>. 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.
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2021.3071953