Hybrid SMI Realization via Matrix Completion and Riemannian Manifold Optimization on Narrowband Sub-Array Based Architectures

Hybrid beamforming architectures reduce hardware complexity but restrict access to full array observations, rendering direct implementation of classical covariance based methods such as minimum variance distortionless response (MVDR) and sample matrix inversion (SMI) infeasible. This work introduces a structured covariance completion framework, termed Rock Road to Dublin (RR2D), which estimates the unobservable analytical covariance matrix (ACM) from a partially observed sample covariance matrix (SCM). RR2D exploits signal stationarity across the array and enforces physical measurement consistency using Dykstra's alternating projection algorithm with positive semidefinite, Toeplitz, and block constraints. The reconstructed virtual ACM enables a realizable hybrid SMI (HSMI) formulation that remains fully compatible with existing hybrid MVDR optimization frameworks. Empirical results for a 32 element hybrid array demonstrate both the expected degradation of HSMI implemented directly under prior HMVDR formulations and the performance gains achieved through RR2D. The proposed HSMI consistently outperforms previous hybrid SMI and partial digital baselines, achieving performance close to the HMVDR reference. Overall, RR2D bridges the gap between theoretical HMVDR formulations and practical hybrid hardware by enabling structured covariance reconstruction from incomplete observations.