Sum Rate Maximization Over Riemannian Manifolds for RIS-Assisted Coexistence of Radar and Communication Systems

Radar and communication coexistence (RCC) aims at improving the spectral efficiency of next generation wireless systems. One of the challenges facing RCC, however, is the mutual interference between the two coexisting subsystems. A reconfigurable intelligent surface (RIS) can thus be used to address such a challenge by optimizing its phase shifts along with the beamforming weights at the base station (BS). In this article, we aim to maximize the communication sum rate (SR), while limiting the interference toward the radar to a certain limit. Motivated by the unexplored fact that covariance matrices of RCC signals are Hermitian positive definite (HPD) and hence, can be represented over Riemannian manifolds (i.e., curved surfaces), the RCC SR maximization problem is reformulated as minimization of a Riemannian metric, which is the geodesic distance between the covariance matrices of the radar and the RIS-relayed signals. Such geometric reformulation paves the road for a low-complexity optimization approach over Riemannian manifolds, which simultaneously optimizes the beamforming weights and phase shifts at the BS and RIS, respectively. Simulation results demonstrate that the proposed solution significantly increases the communication SR, while meeting the constraint on the interference toward radar. Equally important, the proposed optimization approach over Riemannian manifolds exhibits a reduced complexity compared to state-of-the-art algorithms over Euclidean spaces.