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Cramér-Rao Bound Analysis and Beamforming Design for Integrated Sensing and Communication with Extended Targets

Yiqiu Wang, Meixia Tao, Shu Sun

TL;DR

This work analyzes a MU-MIMO ISAC system that senses an extended Target (ET) modeled by a truncated Fourier contour while serving multiple downlink users. It derives closed-form Cramér-Rao Bound (CRB) expressions for the ET’s central range $d_o$, direction $\phi_o$, and orientation $\varphi$, and frames a CRB-minimization beamforming problem under SINR and 3-dB contour-coverage constraints. The authors propose two solutions: a semidefinite relaxation (SDR) method with a rank-one extraction step and a low-complexity zero-forcing (ZF) approach that leverages the communication null space to perform sensing. Numerical results show the SDR design achieves lower CRB and mean-squared error (MSE) for ET parameter estimation than benchmark designs, while the ZF design substantially reduces computation with only modest sensing performance loss. Overall, the paper provides a practical CRB-guided beamforming framework for ISAC systems with extended targets and demonstrates its effectiveness across differently shaped ETs.

Abstract

This paper studies an integrated sensing and communication (ISAC) system, where a multi-antenna base station transmits beamformed signals for joint downlink multi-user communication and radar sensing of an extended target (ET). By considering echo signals as reflections from valid elements on the ET contour, a set of novel Cramér-Rao bounds (CRBs) is derived for parameter estimation of the ET, including central range, direction, and orientation. The ISAC transmit beamforming design is then formulated as an optimization problem, aiming to minimize the CRB associated with radar sensing, while satisfying a minimum signal-to-interference-pulse-noise ratio requirement for each communication user, along with a 3-dB beam coverage constraint tailored for the ET. To solve this non-convex problem, we utilize semidefinite relaxation (SDR) and propose a rank-one solution extraction scheme for non-tight relaxation circumstances. To reduce the computation complexity, we further employ an efficient zero-forcing (ZF) based beamforming design, where the sensing task is performed in the null space of communication channels. Numerical results validate the effectiveness of the obtained CRB, revealing the diverse features of CRB for differently shaped ETs. The proposed SDR beamforming design outperforms benchmark designs with lower estimation error and CRB, while the ZF beamforming design greatly improves computation efficiency with minor sensing performance loss.

Cramér-Rao Bound Analysis and Beamforming Design for Integrated Sensing and Communication with Extended Targets

TL;DR

This work analyzes a MU-MIMO ISAC system that senses an extended Target (ET) modeled by a truncated Fourier contour while serving multiple downlink users. It derives closed-form Cramér-Rao Bound (CRB) expressions for the ET’s central range , direction , and orientation , and frames a CRB-minimization beamforming problem under SINR and 3-dB contour-coverage constraints. The authors propose two solutions: a semidefinite relaxation (SDR) method with a rank-one extraction step and a low-complexity zero-forcing (ZF) approach that leverages the communication null space to perform sensing. Numerical results show the SDR design achieves lower CRB and mean-squared error (MSE) for ET parameter estimation than benchmark designs, while the ZF design substantially reduces computation with only modest sensing performance loss. Overall, the paper provides a practical CRB-guided beamforming framework for ISAC systems with extended targets and demonstrates its effectiveness across differently shaped ETs.

Abstract

This paper studies an integrated sensing and communication (ISAC) system, where a multi-antenna base station transmits beamformed signals for joint downlink multi-user communication and radar sensing of an extended target (ET). By considering echo signals as reflections from valid elements on the ET contour, a set of novel Cramér-Rao bounds (CRBs) is derived for parameter estimation of the ET, including central range, direction, and orientation. The ISAC transmit beamforming design is then formulated as an optimization problem, aiming to minimize the CRB associated with radar sensing, while satisfying a minimum signal-to-interference-pulse-noise ratio requirement for each communication user, along with a 3-dB beam coverage constraint tailored for the ET. To solve this non-convex problem, we utilize semidefinite relaxation (SDR) and propose a rank-one solution extraction scheme for non-tight relaxation circumstances. To reduce the computation complexity, we further employ an efficient zero-forcing (ZF) based beamforming design, where the sensing task is performed in the null space of communication channels. Numerical results validate the effectiveness of the obtained CRB, revealing the diverse features of CRB for differently shaped ETs. The proposed SDR beamforming design outperforms benchmark designs with lower estimation error and CRB, while the ZF beamforming design greatly improves computation efficiency with minor sensing performance loss.
Paper Structure (23 sections, 2 theorems, 59 equations, 9 figures, 1 algorithm)

This paper contains 23 sections, 2 theorems, 59 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Communication Signal Model
  4. Extended Target Contour Model
  5. Received Sensing Signal Model
  6. CRB analysis
  7. CRB for Extended Target
  8. CRB for Point Target
  9. Joint Beamforming Design
  10. General Beamforming Optimization Problem
  11. Joint Transmit Beamforming Design via SDR
  12. Joint Transmit Beamforming Design via ZF
  13. Complexity Analysis
  14. Numerical Results
  15. Numerical analysis of CRB
  16. ...and 8 more sections

Key Result

Theorem 1

The CRBs on ${{d}_{o}}$, ${{\phi }_{o}}$ and $\varphi$ can be expressed as $(CRB d_o)-(CRB varphi)$, respectively, where ${{Z}_{1,k}}={{\pi }^{2}}\left( N_{r}^{2}-1 \right){{\cos }^{2}}\phi_{k} /12$, ${{Z}_{2}}={{\left( 4\pi B/c \right)}^{2}}$, ${{X}_{k}}={{- \boldsymbol{\nu} _{k}^{T}\mathbf{m}\cos

Figures (9)

  • Figure 1: Monostatic ISAC system in which one BS performs communication and sensing tasks simultaneously.
  • Figure 2: Bird-eye view of an ET with TFS contour Garcia22TSP.
  • Figure 3: CRBs of target parameters versus distance. (a) range; (b) direction; (c) orientation.
  • Figure 4: The MF-based estimation results of vehicle contour target. The general parameters of ET are $d_o=30m$, $\phi_o=0^{\circ}$, $\varphi=0^{\circ}$. (a) The normalized MF output at various directions. The ET direction estimation results of (b) CRB-min Design; (c) Average-Null Design; (d) Average Design. The colors in (b)-(d) corresponds to the normalized MF value projected onto the contour element with all considered directions. The estimated direction is a straight line extended from the BS to the contour element with maximal MF value.
  • Figure 5: The MF-based estimation results of UAV contour target. The general parameters of ET are $d_o=26m$, $\phi_o=0^{\circ}$, $\varphi=5^{\circ}$. (a) The normalized MF output at various directions. The ET direction estimation results of (b) CRB-min Design; (c) Average-Null Design; (d) Average Design.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2