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 makes real-time estimation of the process using causally received codewords. 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 unlock 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 1/R. 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

Sampling of the Wiener Process for Remote Estimation over a Channel with Random Delay

In this paper, we consider a problem of sampling a Wiener process, with samples forwarded to a remote estimator over a channel that is modeled as a queue. The estimator reconstructs an estimate of the real-time signal value from causally received samples. We study the optimal online sampling strategy that minimizes the mean square estimation error subject to a sampling rate constraint. We prove that the optimal sampling strategy is a threshold policy, and find the optimal threshold. This threshold is determined by how much the Wiener process varies during the random service time and the maximum allowed sampling rate. Further, if the sampling times are independent of the observed Wiener process, the above sampling problem for minimizing the estimation error is equivalent to a sampling problem for minimizing the age of information. This reveals an interesting connection between the age of information and remote estimation error. Our comparisons show that the estimation error achieved by the optimal sampling policy can be much smaller than those of age-optimal sampling, zero-wait sampling, and periodic sampling.Comment: Accepted by IEEE Transactions on Information Theor

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 makes real-time estimation of the process using causally received codewords. 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 unlock 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 1/R. 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