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Integrated Communication and Bayesian Estimation of Fixed Channel States

Daewon Seo

TL;DR

ISAC systems must simultaneously support data transmission and estimation of a fixed continuous state $S$. The paper introduces an exact rate–MSE tradeoff characterization using the asymptotically tight Bayesian Cramér-Rao bound (ATBCRB), valid for general channels under regularity, and proves achievability with constant composition codes and ML/MAP estimation. The main result provides a tight region defined by $R \le \min_{s\in\mathcal{S}} I(X;\tilde{Y})$ and $\alpha \ge \mathbb{E}_S\left[ \mathbb{E}_X^{-1}\left[ J_{D,X}(S) \right]\right]$, with MSE decaying as $n\,\mathrm{MSE} \to \mathbb{E}_S\left[ \mathbb{E}_X^{-1}\left[ J_{D,X}(S) \right]\right]$. Numerical example in spectrum sensing shows ATBCRB-based tradeoffs are significantly tighter than BCRB-based proxies, underscoring the suboptimality of BCRB in general Bayesian ISAC models. This work thus provides the first information-theoretic rate–MSE framework for ISAC with exact MSE metrics and actionable code-design insights.

Abstract

This work studies an information-theoretic performance limit of an integrated sensing and communication (ISAC) system where the goal of sensing is to estimate a random continuous state. Considering the mean-squared error (MSE) for estimation performance metric, the Bayesian Cramér-Rao lower bound (BCRB) is widely used in literature as a proxy of the MSE; however, the BCRB is not generally tight even asymptotically except for restrictive distributions. Instead, we characterize the full tradeoff between information rate and the exact MSE using the asymptotically tight BCRB (ATBCRB) analysis, a recent variant of the BCRB. Our characterization is applicable for general channels as long as the regularity conditions are met, and the proof relies on constant composition codes and ATBCRB analysis with the codes. We also perform a numerical evaluation of the tradeoff in a variance estimation example, which commonly arises in spectrum sensing scenarios.

Integrated Communication and Bayesian Estimation of Fixed Channel States

TL;DR

ISAC systems must simultaneously support data transmission and estimation of a fixed continuous state . The paper introduces an exact rate–MSE tradeoff characterization using the asymptotically tight Bayesian Cramér-Rao bound (ATBCRB), valid for general channels under regularity, and proves achievability with constant composition codes and ML/MAP estimation. The main result provides a tight region defined by and , with MSE decaying as . Numerical example in spectrum sensing shows ATBCRB-based tradeoffs are significantly tighter than BCRB-based proxies, underscoring the suboptimality of BCRB in general Bayesian ISAC models. This work thus provides the first information-theoretic rate–MSE framework for ISAC with exact MSE metrics and actionable code-design insights.

Abstract

This work studies an information-theoretic performance limit of an integrated sensing and communication (ISAC) system where the goal of sensing is to estimate a random continuous state. Considering the mean-squared error (MSE) for estimation performance metric, the Bayesian Cramér-Rao lower bound (BCRB) is widely used in literature as a proxy of the MSE; however, the BCRB is not generally tight even asymptotically except for restrictive distributions. Instead, we characterize the full tradeoff between information rate and the exact MSE using the asymptotically tight BCRB (ATBCRB) analysis, a recent variant of the BCRB. Our characterization is applicable for general channels as long as the regularity conditions are met, and the proof relies on constant composition codes and ATBCRB analysis with the codes. We also perform a numerical evaluation of the tradeoff in a variance estimation example, which commonly arises in spectrum sensing scenarios.
Paper Structure (7 sections, 2 theorems, 31 equations, 2 figures)

This paper contains 7 sections, 2 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulation & Preliminary
  3. Formulation
  4. Preliminary
  5. Main Result
  6. Numerical Evaluation
  7. Conclusion

Key Result

Theorem 1

The rate-MSE decay region $\mathcal{R}$ is the set of pairs $(R, \alpha)$ such that for some $p_X$,

Figures (2)

  • Figure 1: Problem model.
  • Figure 2: The optimal tradeoff region $\mathcal{R}$ in Thm. \ref{['thm:ATBCRB_ISAC']} and the outer bound $\mathcal{R}_{\textsf{out}}$ using the BCRB in Cor. \ref{['cor:BCRB_outer']} are plotted together. Note that $\mathcal{R}_{\textsf{out}}$ is strictly far from the optimal tradeoff, i.e., not achievable.

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1