Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Source Coding for a Wiener Process

Sahan Liyanaarachchi, Ismail Cosandal, Sennur Ulukus

TL;DR

The paper tackles remotely monitoring a Wiener process under practical transmission constraints by introducing an event-driven sampling scheme using monotone function thresholds that yield four threshold-based events and no quantization error. It derives an analytical MSE expression based on hitting times and offers a method to optimize the associated source-code lengths under a sampling-rate constraint, using Dinkelbach and KKT techniques. A key finding is that zero-wait sampling is optimal without rate constraints, while a nonzero, monotone-threshold scheme is optimal when a sampling-rate constraint is active; the framework provides a principled path to quantization-free, rate-constrained Wiener process monitoring with validated numerical results. This approach has practical impact for real-time monitoring systems where transmitting full samples is impractical and quantization must be avoided while respecting bandwidth limits.

Abstract

We develop a novel source coding strategy for sampling and monitoring of a Wiener process. For the encoding process, we employ a four level ``quantization'' scheme, which employs monotone function thresholds as opposed to fixed constant thresholds. Leveraging the hitting times of the Wiener process with these thresholds, we devise a sampling and encoding strategy which does not incur any quantization errors. We give analytical expressions for the mean squared error (MSE) and find the optimal source code lengths to minimize the MSE under this monotone function threshold scheme, subject to a sampling rate constraint.

Source Coding for a Wiener Process

TL;DR

The paper tackles remotely monitoring a Wiener process under practical transmission constraints by introducing an event-driven sampling scheme using monotone function thresholds that yield four threshold-based events and no quantization error. It derives an analytical MSE expression based on hitting times and offers a method to optimize the associated source-code lengths under a sampling-rate constraint, using Dinkelbach and KKT techniques. A key finding is that zero-wait sampling is optimal without rate constraints, while a nonzero, monotone-threshold scheme is optimal when a sampling-rate constraint is active; the framework provides a principled path to quantization-free, rate-constrained Wiener process monitoring with validated numerical results. This approach has practical impact for real-time monitoring systems where transmitting full samples is impractical and quantization must be avoided while respecting bandwidth limits.

Abstract

We develop a novel source coding strategy for sampling and monitoring of a Wiener process. For the encoding process, we employ a four level ``quantization'' scheme, which employs monotone function thresholds as opposed to fixed constant thresholds. Leveraging the hitting times of the Wiener process with these thresholds, we devise a sampling and encoding strategy which does not incur any quantization errors. We give analytical expressions for the mean squared error (MSE) and find the optimal source code lengths to minimize the MSE under this monotone function threshold scheme, subject to a sampling rate constraint.

Paper Structure

This paper contains 11 sections, 7 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Monotone Function Thresholds
  4. Main Results
  5. Optimal Source Codes
  6. Numerical Results
  7. Conclusion
  8. Proof of Lemma \ref{['lem:iid']}
  9. Proof of Theorem \ref{['thrm:mse_main']}
  10. Proof of Theorem \ref{['thrm:mse_main_2']}
  11. Proof of Theorem \ref{['thrm:Kraft']}

Key Result

Lemma 1

The source coding lengths are generated independently and identically.

Figures (5)

  • Figure 1: System model.
  • Figure 2: Monotone function thresholding scheme.
  • Figure 3: A sample path with $a=b=\mu=1$ and $L_n=L=2$.
  • Figure 4: Comparison of optimum source code scheme under no sample rate constraint with benchmark schemes. Curves are obtained from analytical expressions, and dots represent simulation results.
  • Figure 5: a) MSE and b) sampling rate under different sampling rate constraints.

Theorems & Definitions (14)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 3
  • ...and 4 more