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Timely Tracking of a Wiener Process With Single Bit Quantization

Ismail Cosandal, Sahan Liyanaarachchi, Sennur Ulukus

TL;DR

This work addresses timely tracking of a Wiener process over a delay channel using periodic sampling and a three-region, single-bit quantizer that includes an empty symbol to denote no transmission. The authors develop an optimum Lloyd-Max quantization by tracking the distributions of the quantization error and propose two low-complexity alternatives—Last-Bit Aware and Gaussian Approximation quantization—that achieve near-identical mean-squared error and transmission cost. They derive a closed-form long-run MSE that decomposes into a process term and the steady-state quantization error $\mathbb{E}[Q^2]$, and demonstrate robustness to random delays, which is especially beneficial for energy-constrained sensors. The results show that practical, low-complexity schemes can closely match the performance of the computationally demanding optimum quantizer, enabling energy-efficient timely tracking in IoT settings.

Abstract

We consider the problem of timely tracking of a Wiener process via an energy-conserving sensor by utilizing a single bit quantization strategy under periodic sampling. Contrary to conventional single bit quantizers which only utilize the transmitted bit to convey information, in our codebook, we use an additional `$\emptyset$' symbol to encode the event of \emph{not transmitting}. Thus, our quantization functions are composed of three decision regions as opposed to the conventional two regions. First, we propose an optimum quantization method in which the optimum quantization functions are obtained by tracking the distributions of the quantization error. However, this method requires a high computational cost and might not be applicable for energy-conserving sensors. Thus, we propose two additional low complexity methods. In the last-bit aware method, three predefined quantization functions are available to both devices, and they switch the quantization function based on the last transmitted bit. With the Gaussian approximation method, we calculate a single quantization function by assuming that the quantization error can be approximated as Gaussian. While previous methods require a constant delay assumption, this method also works for random delay. We observe that all three methods perform similarly in terms of mean-squared error and transmission cost.

Timely Tracking of a Wiener Process With Single Bit Quantization

TL;DR

This work addresses timely tracking of a Wiener process over a delay channel using periodic sampling and a three-region, single-bit quantizer that includes an empty symbol to denote no transmission. The authors develop an optimum Lloyd-Max quantization by tracking the distributions of the quantization error and propose two low-complexity alternatives—Last-Bit Aware and Gaussian Approximation quantization—that achieve near-identical mean-squared error and transmission cost. They derive a closed-form long-run MSE that decomposes into a process term and the steady-state quantization error , and demonstrate robustness to random delays, which is especially beneficial for energy-constrained sensors. The results show that practical, low-complexity schemes can closely match the performance of the computationally demanding optimum quantizer, enabling energy-efficient timely tracking in IoT settings.

Abstract

We consider the problem of timely tracking of a Wiener process via an energy-conserving sensor by utilizing a single bit quantization strategy under periodic sampling. Contrary to conventional single bit quantizers which only utilize the transmitted bit to convey information, in our codebook, we use an additional `' symbol to encode the event of \emph{not transmitting}. Thus, our quantization functions are composed of three decision regions as opposed to the conventional two regions. First, we propose an optimum quantization method in which the optimum quantization functions are obtained by tracking the distributions of the quantization error. However, this method requires a high computational cost and might not be applicable for energy-conserving sensors. Thus, we propose two additional low complexity methods. In the last-bit aware method, three predefined quantization functions are available to both devices, and they switch the quantization function based on the last transmitted bit. With the Gaussian approximation method, we calculate a single quantization function by assuming that the quantization error can be approximated as Gaussian. While previous methods require a constant delay assumption, this method also works for random delay. We observe that all three methods perform similarly in terms of mean-squared error and transmission cost.
Paper Structure (6 sections, 1 theorem, 13 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 6 sections, 1 theorem, 13 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Single bit Periodic Sampling
  3. Optimum Quantization
  4. Numerical Results
  5. Last-Bit Aware Quantization
  6. Gaussian Approximation Based Quantization

Key Result

Lemma 1

If the $k$th sample taken at time $kT$ has a quantization error $Q_k$ for a normalized Wiener process $(\sigma^2=1)$, the MSE$(t)$ at time $t$ is equivalent to the AoI as if the sample was taken at $kT-Q_k^2$ instead of $kT$. In essence, this is as if the quantization error induces an additional del

Figures (5)

  • Figure 1: A general system model for timely tracking of a Wiener process. The quantization function generator updates the quantization functions based on the transmitted bit, and the new quantization function is available for both the encoder and the decoder.
  • Figure 2: a) A sample path for $W(t)$ and $\hat{W}(t)$ for $\sigma^2=1$ and $d=0.3$. Quantization parameters for b) source and c) monitor within a single period.
  • Figure 3: The analogy between MSE$(t)$ and AoI$(t)$.
  • Figure 4: Obtaining $f_{Y_2}(y)$ from $f_{Q_1}(q)$ when the corresponding bit for the symbol '$+$' is transmitted in the first period.
  • Figure 5: MSE comparison of different quantization functions. Lines correspond to the analytical results obtained from \ref{['eq:mse_gen']} using average distortions in Table \ref{['tab:quant']}.

Theorems & Definitions (1)

  • Lemma 1