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Sensing Mutual Information with Random Signals in Gaussian Channels: Bridging Sensing and Communication Metrics

Lei Xie, Fan Liu, Jiajin Luo, Shenghui Song

TL;DR

This work addresses the gap of evaluating sensing performance with random signals in ISAC systems by deriving an explicit SMI expression using random matrix theory. It reveals strong links between SMI and traditional sensing metrics, including ELMMSE, EMMSE, and EBCRB, establishing SMI as a bridge for unified performance analysis. The paper then develops SMI-oriented precoding methods for both sensing-only and ISAC scenarios and validates them via simulations, showing that MI-based design can effectively balance sensing accuracy and communication requirements. The results offer a principled, information-theoretic framework for joint sensing and communication system design with random signaling.

Abstract

Sensing performance is typically evaluated by classical radar metrics, such as Cramer-Rao bound and signal-to-clutter-plus-noise ratio. The recent development of the integrated sensing and communication (ISAC) framework motivated the efforts to unify the performance metric for sensing and communication, where mutual information (MI) was proposed as a sensing performance metric with deterministic signals. However, the need of communication in ISAC systems necessitates the transmission of random signals for sensing applications, whereas an explicit evaluation for the sensing mutual information (SMI) with random signals is not yet available in the literature. This paper aims to fill the research gap and investigate the unification of sensing and communication performance metrics. For that purpose, we first derive the explicit expression for the SMI with random signals utilizing random matrix theory. On top of that, we further build up the connections between SMI and traditional sensing metrics, such as ergodic minimum mean square error (EMMSE), ergodic linear minimum mean square error (ELMMSE), and ergodic Bayesian Cramér-Rao bound (EBCRB). Such connections open up the opportunity to unify sensing and communication performance metrics, which facilitates the analysis and design for ISAC systems. Finally, SMI is utilized to optimize the precoder for both sensing-only and ISAC applications. Simulation results validate the accuracy of the theoretical results and the effectiveness of the proposed precoding designs.

Sensing Mutual Information with Random Signals in Gaussian Channels: Bridging Sensing and Communication Metrics

TL;DR

This work addresses the gap of evaluating sensing performance with random signals in ISAC systems by deriving an explicit SMI expression using random matrix theory. It reveals strong links between SMI and traditional sensing metrics, including ELMMSE, EMMSE, and EBCRB, establishing SMI as a bridge for unified performance analysis. The paper then develops SMI-oriented precoding methods for both sensing-only and ISAC scenarios and validates them via simulations, showing that MI-based design can effectively balance sensing accuracy and communication requirements. The results offer a principled, information-theoretic framework for joint sensing and communication system design with random signaling.

Abstract

Sensing performance is typically evaluated by classical radar metrics, such as Cramer-Rao bound and signal-to-clutter-plus-noise ratio. The recent development of the integrated sensing and communication (ISAC) framework motivated the efforts to unify the performance metric for sensing and communication, where mutual information (MI) was proposed as a sensing performance metric with deterministic signals. However, the need of communication in ISAC systems necessitates the transmission of random signals for sensing applications, whereas an explicit evaluation for the sensing mutual information (SMI) with random signals is not yet available in the literature. This paper aims to fill the research gap and investigate the unification of sensing and communication performance metrics. For that purpose, we first derive the explicit expression for the SMI with random signals utilizing random matrix theory. On top of that, we further build up the connections between SMI and traditional sensing metrics, such as ergodic minimum mean square error (EMMSE), ergodic linear minimum mean square error (ELMMSE), and ergodic Bayesian Cramér-Rao bound (EBCRB). Such connections open up the opportunity to unify sensing and communication performance metrics, which facilitates the analysis and design for ISAC systems. Finally, SMI is utilized to optimize the precoder for both sensing-only and ISAC applications. Simulation results validate the accuracy of the theoretical results and the effectiveness of the proposed precoding designs.
Paper Structure (29 sections, 10 theorems, 100 equations, 11 figures)

This paper contains 29 sections, 10 theorems, 100 equations, 11 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Communication model
  4. Sensing model
  5. Sensing Mutual Information with Random Signals
  6. Upper bound of SMI
  7. Asymptotic approximation of SMI
  8. Effect of the number of frames
  9. Sensing DoF loss
  10. Relation between SMI and Other Sensing Performance Metrics
  11. ELMMSE and EMMSE for estimating $\mathbf{h}_s$
  12. Ergodic Bayesian CRB for estimating $\mathbf{h}_s$
  13. Relation between different metrics
  14. SMI-Oriented Precoding Design
  15. Sensing-only scenario
  16. ...and 14 more sections

Key Result

Proposition 1

As $N_S,N_T \to \infty$ with a finite constant ratio $c$, i.e., $N_S/N_T = c$, we have where with $\{\lambda_{R,j}\}_{j=1}^{K}$ denotes the eigenvalues of $\sigma_s^{-2} \mathbf{R}_{R}$, and $\delta(\rho)$ is the solution to the following equation

Figures (11)

  • Figure 1: Illustration of the connection between SMI and other communication/sensing metrics.
  • Figure 2: Illustration of the considered ISAC system.
  • Figure 3: Sensing MI versus the number of frames $N_S$.
  • Figure 4: Sensing MI versus the number of targets $K$.
  • Figure 5: Normalized ELMMSE versus the number of frames $N_S$.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Lemma 1
  • Proposition 5
  • Corollary 1
  • Lemma 2: hachem2008new
  • Lemma 3: hachem2008new
  • ...and 1 more