Multi-Target Estimation via Tensor Decomposition for Beyond Diagonal RIS-Aided Bistatic Sensing

We investigate the performance of beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) for bistatic MIMO multi-target sensing using a two-stage tensor Doppler-delay-angle estimation (TenDAE). The first stage solves a Kronecker sum approximation (KSA) with a rank equal to the number of targets. The second stage employs a nested tensor factorization estimation (NTFE) that exploits the inherent multidimensional structure via two tensor decompositions that are solved in parallel. The first employs a PARAFAC decomposition to extract the targets' angles, and the second uses a nested PARAFAC decomposition to find the targets' delay and Doppler parameters. This two-stage approach decouples acquisition of the angles and delays/Dopplers using either alternating least squares or a higher-order singular value decomposition, followed by a high-resolution subspace technique, such as ESPRIT. We further compare the performance of a BD-RIS with a classical diagonal RIS. For the latter, we solve a Khatri-Rao sum approximation problem rather than the KSA due to the specific structure of the received signal. Notably, our NTFE framework remains blind to the underlying RIS architecture while simultaneously estimating all targets with minimal sensing resources. Additionally, we show that employing a nested-PARAFAC decomposition enables the decoupling of the delay-Doppler and angle domains. We also derive the Cramér-Rao lower bound to further assess the performance of the TenDAE framework. Finally, we numerically evaluate the solutions presented in this paper and demonstrate their efficiency in terms of RMSE compared with state-of-the-art approaches.