Optimal Causal Rate-Constrained Sampling of the Wiener Process

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 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 DDET(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. 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.

Value of Information in Feedback Control: Global Optimality

“…Guo and Kostina [9] addressed the estimation of the scalar Wiener and scalar Ornstein-Uhlenbeck processes in the presence of the signaling effect, and obtained a result that matches with that in [8]. They also looked at the estimation of the scalar Wiener process with fixed communication delay in the presence of the signaling effect in [10], and obtained a similar structural result. Furthermore, Sun et al [11] studied the estimation of the scalar Wiener process with random communication delay by discarding the signaling effect, and showed that the optimal triggering policy is symmetric threshold.…”

“…This leads to the employment of the linear-quadratic regulator with a state estimate at the controller without the signaling residual. More specifically, we have u k = −L k xk , where L k = (B T S k+1 B + R) −1 B T S k+1 A is the linear-quadratic regulator gain, xk is a state estimate at the controller satisfying (10) with ı k = 0 for all k ∈ K, and S k 0 is a matrix that satisfies the algebraic Riccati equation…”

“…There are multiple works that are closely related to our study [5][6][7][8][9][10][11]. In these works, optimal triggering policies were characterized in settings that are different from ours.…”

State Estimation over Delayed and Lossy Channels: An Encoder-Decoder Synthesis

“…packet-erasure channel in [5], [6], remote estimation of a scalar Markov process with symmetric noise distribution over an ideal channel in [9], remote estimation of a scalar autoregressive Markov process with symmetric noise distribution over an ideal channel, an i.i.d. packet-erasure channel, and a Gilbert-Elliott packeterasure channel in [10]- [12], remote estimation of the scalar Wiener and scalar Ornstein-Uhlenbeck processes over an ideal channel and a fixed-delay channel in [13], [14], remote estimation of the scalar Wiener process over a random-delay channel in [15], remote estimation of multiple random variables with arbitrary distributions over a collision channel in [16], [17], remote estimation of multiple random variables with arbitrary distributions over unicast and broadcast channels in [18], and remote estimation of two Gaussian random variables over a multi-access channel in [19]. These studies established certain characteristics such as a symmetric, asymmetric, or threshold structure of the optimal scheduling policy with respect to the estimation discrepancy.…”