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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| Published in: | IEEE transactions on automatic control Vol. 67; no. 4; pp. 1776 - 1791 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9286, 1558-2523 |
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| Abstract | 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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| AbstractList | 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 [Formula Omitted] 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 [Formula Omitted] or [Formula Omitted] 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 [Formula Omitted]. 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 [Formula Omitted]. It is achieved by the uniform sampling policy with the sampling interval [Formula Omitted]. 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. 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. |
| Author | Guo, Nian Kostina, Victoria |
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| SubjectTerms | 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 |
| Title | Optimal Causal Rate-Constrained Sampling of the Wiener Process |
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