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A Rate-Distortion Bound for ISAC

Mohammadreza Bakhshizadeh Mohajer, Alex Dytso, Daniela Tuninetti, Luca Barletta

TL;DR

This work introduces a rate-distortion bound (RDB) as a universal converse for Integrated Sensing and Communication (ISAC) systems, enabling performance characterizations under arbitrary priors and distortion measures. By leveraging the rate-distortion function $R_{\mathsf{A},d}(D)$ and a generic bound that couples sensing distortion to mutual information through $I(\mathsf{A};\mathsf{Y}_s|\mathsf{X})$, the authors derive an outer bound on the achievable sensing–rate region and show how it subsumes or tightens classical CRB-based analyses. The bound is proved tight in the high-sensing-noise regime and, under MMSE fidelity, can be strictly tighter than the Bayesian Cramér–Rao bound in low-noise conditions, with a conditional Stam-type argument supporting the scalar case. The framework is validated through Nakagami fading and occupancy-detection case studies, illustrating its applicability to both continuous and discrete sensing problems and highlighting its potential as a general tool for guiding ISAC system design in 6G and beyond.

Abstract

This paper addresses the fundamental performance limits of Integrated Sensing and Communication (ISAC) systems by introducing a novel converse bound based on rate-distortion theory. This rate-distortion bound (RDB) overcomes the restrictive regularity conditions of classical estimation theory, such as the Bayesian Cramér-Rao Bound (BCRB). The proposed framework is broadly applicable, holding for arbitrary parameter distributions and distortion measures, including mean-squared error and probability of error. The bound is proved to be tight in the high sensing noise regime and can be strictly tighter than the BCRB in the low sensing noise regime. The RDB's utility is demonstrated on two challenging scenarios: Nakagami fading channel estimation, where it provides a valid bound even when the BCRB is inapplicable, and a binary occupancy detection task, showcasing its versatility for discrete sensing problems. This work provides a powerful and general tool for characterizing the ultimate performance tradeoffs in ISAC systems.

A Rate-Distortion Bound for ISAC

TL;DR

This work introduces a rate-distortion bound (RDB) as a universal converse for Integrated Sensing and Communication (ISAC) systems, enabling performance characterizations under arbitrary priors and distortion measures. By leveraging the rate-distortion function and a generic bound that couples sensing distortion to mutual information through , the authors derive an outer bound on the achievable sensing–rate region and show how it subsumes or tightens classical CRB-based analyses. The bound is proved tight in the high-sensing-noise regime and, under MMSE fidelity, can be strictly tighter than the Bayesian Cramér–Rao bound in low-noise conditions, with a conditional Stam-type argument supporting the scalar case. The framework is validated through Nakagami fading and occupancy-detection case studies, illustrating its applicability to both continuous and discrete sensing problems and highlighting its potential as a general tool for guiding ISAC system design in 6G and beyond.

Abstract

This paper addresses the fundamental performance limits of Integrated Sensing and Communication (ISAC) systems by introducing a novel converse bound based on rate-distortion theory. This rate-distortion bound (RDB) overcomes the restrictive regularity conditions of classical estimation theory, such as the Bayesian Cramér-Rao Bound (BCRB). The proposed framework is broadly applicable, holding for arbitrary parameter distributions and distortion measures, including mean-squared error and probability of error. The bound is proved to be tight in the high sensing noise regime and can be strictly tighter than the BCRB in the low sensing noise regime. The RDB's utility is demonstrated on two challenging scenarios: Nakagami fading channel estimation, where it provides a valid bound even when the BCRB is inapplicable, and a binary occupancy detection task, showcasing its versatility for discrete sensing problems. This work provides a powerful and general tool for characterizing the ultimate performance tradeoffs in ISAC systems.

Paper Structure

This paper contains 20 sections, 14 theorems, 84 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Contribution and Outline
  4. Notation
  5. General ISAC Model
  6. Generic Converse
  7. Rate-Distortion Preliminaries
  8. Generic Converse
  9. Tightness of Generic Converse in the High Sensing-Noise Regime
  10. Applying Shannon Lower Bound
  11. Comparison to the Cramér-Rao Bound
  12. Case Study: Nakagami Fading
  13. ISAC Model
  14. Converse RDB
  15. Bayesian Cramér-Rao Bound
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathsf{V} \in \mathcal{V}$ and $\mathsf{W} \in \mathcal{W}$. Then, for any $g: \mathcal{W} \to \mathbb{R}^n$, we have that

Figures (3)

  • Figure 1: General ISAC model.
  • Figure 2: Converse bounds on $\mathcal{C}$ obtained by RDB and BCRB for Nakagami-$m$ fading with $m_{\mathrm{s}} = m_{\mathrm{c}} \in \{0.5, 1, 2\}$. Solid curves denote the RDB converse for $m=0.5$ (severe), $m=1$ (Rayleigh), $m=2$ (moderate). Dashed and dotted lines denote the BCRB converse for $m=1$ and $m=2$, respectively. System parameters are listed in Table \ref{['tab:params']}.
  • Figure 3: RDB converse for the joint occupancy detection and communication problem. System parameters are listed in Table \ref{['tab:parameters_occupancy']}.

Theorems & Definitions (32)

  • Definition 1: (DM–ISAC channel with receiver state information for communication)
  • Definition 2: (Distortion or Estimation Error)
  • Definition 3: (Achievable Rate)
  • Definition 4: (Rate-distortion Capacity Region)
  • Definition 5: (Rate-distortion function)
  • Theorem 1
  • proof
  • Corollary 1: (Rate-distortion bound)
  • proof
  • Theorem 2
  • ...and 22 more