Multi-Active-IRS-Assisted Cooperative Sensing: Cram\'er-Rao Bound and Joint Beamforming Design

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 Cram\'er-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.