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Multi-Active-IRS-Assisted Cooperative Sensing: Cramér-Rao Bound and Joint Beamforming Design

Yuan Fang, Xianghao Yu, Jie Xu, Ying-Jun Angela Zhang

Abstract

This paper studies the multi-intelligent reflecting surface (IRS)-assisted cooperative sensing, in which multiple active IRSs are deployed in a distributed manner to facilitate multi-view target sensing at the non-line-of-sight (NLoS) area of the base station (BS). Different from prior works employing passive IRSs, we leverage active IRSs with the capability of amplifying the reflected signals to overcome the severe multi-hop-reflection path loss in NLoS sensing. In particular, we consider two sensing setups without and with dedicated sensors equipped at active IRSs. In the first case without dedicated sensors at IRSs, we investigate the cooperative sensing at the BS, where the target's direction-of-arrival (DoA) with respect to each IRS is estimated based on the echo signals received at the BS. In the other case with dedicated sensors at IRSs, we consider that each IRS is able to receive echo signals and estimate the target's DoA with respect to itself. For both sensing setups, we first derive the closed-form Cramér-Rao bound (CRB) for estimating target DoA. Then, the (maximum) CRB is minimized by jointly optimizing the transmit beamforming at the BS and the reflective beamforming at the multiple IRSs, subject to the constraints on the maximum transmit power at the BS, as well as the maximum amplification power and the maximum power amplification gain constraints at individual active IRSs. To tackle the resulting highly non-convex (max-)CRB minimization problems, we propose two efficient algorithms to obtain high-quality solutions for the two cases with sensing at the BS and at the IRSs, respectively, based on alternating optimization, successive convex approximation, and semi-definite relaxation.

Multi-Active-IRS-Assisted Cooperative Sensing: Cramér-Rao Bound and Joint Beamforming Design

Abstract

This paper studies the multi-intelligent reflecting surface (IRS)-assisted cooperative sensing, in which multiple active IRSs are deployed in a distributed manner to facilitate multi-view target sensing at the non-line-of-sight (NLoS) area of the base station (BS). Different from prior works employing passive IRSs, we leverage active IRSs with the capability of amplifying the reflected signals to overcome the severe multi-hop-reflection path loss in NLoS sensing. In particular, we consider two sensing setups without and with dedicated sensors equipped at active IRSs. In the first case without dedicated sensors at IRSs, we investigate the cooperative sensing at the BS, where the target's direction-of-arrival (DoA) with respect to each IRS is estimated based on the echo signals received at the BS. In the other case with dedicated sensors at IRSs, we consider that each IRS is able to receive echo signals and estimate the target's DoA with respect to itself. For both sensing setups, we first derive the closed-form Cramér-Rao bound (CRB) for estimating target DoA. Then, the (maximum) CRB is minimized by jointly optimizing the transmit beamforming at the BS and the reflective beamforming at the multiple IRSs, subject to the constraints on the maximum transmit power at the BS, as well as the maximum amplification power and the maximum power amplification gain constraints at individual active IRSs. To tackle the resulting highly non-convex (max-)CRB minimization problems, we propose two efficient algorithms to obtain high-quality solutions for the two cases with sensing at the BS and at the IRSs, respectively, based on alternating optimization, successive convex approximation, and semi-definite relaxation.
Paper Structure (23 sections, 4 theorems, 73 equations, 7 figures)

This paper contains 23 sections, 4 theorems, 73 equations, 7 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Sensing at BS: Active IRSs without Dedicated Sensors
  4. Received Echo Signal Model
  5. Estimation CRB
  6. Joint Transmit and Reflective Beamforming Design
  7. Optimal Transmit Signal Covariance under Given $\{\mathbf{\Psi}_{l}\}$
  8. Reflective Beamforming Optimization under Given $\{\mathbf{R}_{s,l}\}$
  9. Sensing at IRSs: Active IRSs with Dedicated Sensors
  10. Received Echo Signal Model
  11. Estimation CRB
  12. Joint Transmit and Reflective Beamforming Design
  13. Optimal Transmit Signal Covariance under Given $\{\mathbf{\Psi}_{l}\}$
  14. Reflective Beamforming Optimization under Given $\{\mathbf{R}_{s,l}\}$
  15. Numerical Results
  16. ...and 8 more sections

Key Result

Proposition 1

We define the derivatives of ${{\mathbf{a}}}$ with respect to $\theta_{l}$ and $\phi_{l}$ as $\dot{{\mathbf{a}}}_{\theta_{l}}$ and $\dot{{\mathbf{a}}}_{\phi_{l}}$, respectively. The FIM for estimating ${\mathbf{F}}_{l}$ is given by where with $\varrho,\varpi \in \{\theta_{l},\phi_{l}\}$, $\mathbf{H}_{l} =\mathbf{G}_{l}^{T}\mathbf{\Psi}_{l} {{\mathbf{a}}}_{l} {\mathbf{a}} _l^T {{\mathbf{\Psi }}_

Figures (7)

  • Figure 1: The multi-active-IRS cooperative sensing system, (a): Sensing at the BS; (b): Sensing at the IRSs.
  • Figure 2: The location topology.
  • Figure 3: The achieved max-CRB versus the maximum transmit power $P_{\text{t}}$ at the BS with $P_{\text{s}} = 0.1$ W, $M=16$, and $N_v = N_h = 4$.
  • Figure 4: The achieved max-CRB versus the maximum transmit power $P_{\text{s}}$ at the IRSs with $M=16$, $N_v = N_h = 4$, and $a_{\text{max}}=8$.
  • Figure 5: The achieved max-CRB versus the maximum transmit power $P_{\text{t}}$ at the BS with $P_{\text{s}} = 0.1$ W, $M=16$, and $N_v = N_h = 4$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Remark 1
  • Lemma 1
  • Proposition 2
  • Remark 2
  • Lemma 2