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Rate Distortion Approach to Joint Communication and Sensing With Markov States: Open Loop Case

Colton P. Lindstrom, Matthieu R. Bloch

TL;DR

This work extends the rate-distortion framework for joint communication and sensing to settings where the state evolves as a Markov process and past estimates inform current sensing. It shows that Bayesian filtering remains optimal for state estimation and derives an open-loop capacity–distortion region $C^{(open)}(D)=\lim_{n\to\infty} \max_{P_{X^n}\in\overrightarrow{\mathcal{P}}_D^{(n)}} \frac{1}{n}\sum_{i=1}^n I(X_i;Y_i|S_i)$, with a detailed construction of the cost-constrained input set. The beam pointing specialization illustrates how beam-switching and multi-beam strategies impact the tradeoff, using Kalman filtering for sensing with Gauss–Markov dynamics and two open-loop strategies, including a formal bounding framework and numerical comparisons. The results offer practical insights for designing JCAS systems that jointly optimize communication rate and state tracking in dynamic environments, informing beamforming and resource allocation choices in future networks.

Abstract

We investigate a joint communication and sensing (JCAS) framework in which a transmitter concurrently transmits information to a receiver and estimates a state of interest based on noisy observations. The state is assumed to evolve according to a known dynamical model. Past state estimates may then be used to inform current state estimates. We show that Bayesian filtering constitutes the optimal sensing strategy. We analyze JCAS performance under an open loop encoding strategy with results presented in terms of the tradeoff between asymptotic communication rate and expected per-block distortion of the state. We illustrate the general result by specializing the analysis to a beam-pointing model with mobile state tracking. Our results shed light on the relative performance of two beam control strategies, beam-switching and multi-beam.

Rate Distortion Approach to Joint Communication and Sensing With Markov States: Open Loop Case

TL;DR

This work extends the rate-distortion framework for joint communication and sensing to settings where the state evolves as a Markov process and past estimates inform current sensing. It shows that Bayesian filtering remains optimal for state estimation and derives an open-loop capacity–distortion region , with a detailed construction of the cost-constrained input set. The beam pointing specialization illustrates how beam-switching and multi-beam strategies impact the tradeoff, using Kalman filtering for sensing with Gauss–Markov dynamics and two open-loop strategies, including a formal bounding framework and numerical comparisons. The results offer practical insights for designing JCAS systems that jointly optimize communication rate and state tracking in dynamic environments, informing beamforming and resource allocation choices in future networks.

Abstract

We investigate a joint communication and sensing (JCAS) framework in which a transmitter concurrently transmits information to a receiver and estimates a state of interest based on noisy observations. The state is assumed to evolve according to a known dynamical model. Past state estimates may then be used to inform current state estimates. We show that Bayesian filtering constitutes the optimal sensing strategy. We analyze JCAS performance under an open loop encoding strategy with results presented in terms of the tradeoff between asymptotic communication rate and expected per-block distortion of the state. We illustrate the general result by specializing the analysis to a beam-pointing model with mobile state tracking. Our results shed light on the relative performance of two beam control strategies, beam-switching and multi-beam.
Paper Structure (21 sections, 6 theorems, 52 equations, 4 figures)

This paper contains 21 sections, 6 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. System Model
  4. Main Results
  5. Beam Pointing Example
  6. Mobility Model and Kalman Filtering
  7. Beam Switching Strategy
  8. Multi-Beam Strategy
  9. Numerical Results
  10. Proof of Lemma \ref{['lem:optimalEstimator']} (Optimal estimator)
  11. Proof of Theorem \ref{['thm:generalCapacity']} (General capacity)
  12. Converse Proof
  13. Achievability Proof
  14. Proof of Lemma \ref{['lem:bsExpectedARE']} (Beam switching expected error)
  15. Proof of Lemma \ref{['lem:boundingExpectedCovariance']} (Bounding expected covariance)
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\hat{S}^{*n}=g^*(X^n,Z^n)$ denote the optimal state estimate sequence that minimizes the average per-block distortion $\Delta^{(n)}$. The optimal causal state estimate is, where each symbol estimator $g_i^*(X^i,Z^i)$ is .

Figures (4)

  • Figure 1: Block diagram for the proposed JCAS model with dynamic state.
  • Figure 2: Illustration of a beam pointing JCAS system with a fixed receiver and separate mobile target.
  • Figure 3: Rate-distortion regions for the unstable (left) and stable (right) system for a noiseless channel.
  • Figure 4: Comparison of strategies over Gaussian channel with SNR of 1.75 dB (top row) and SNR of 20 dB (bottom row) for the unstable (left) and stable (right) system.

Theorems & Definitions (18)

  • Remark 1
  • Definition 1: Achievability
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • ...and 8 more