HQE-MUSIC for Direction-of-Arrival Estimation Using Vector-Sensor Arrays: Modeling, Analysis, and Optimization

For vector sensors (VSs) comprising multiple components, quaternion embedding (QE) is a promising modeling approach that provides an elegant algebraic representation and captures cross-component correlations. In particular, the quaternion formalism facilitates diverse signal models. By coupling this advantage with the spatial flexibility provided by multiple VSs within an array, we propose the hybrid QE (HQE) methodology. Unlike conventional fixed representations of array data, we adopt different embedding types across VSs, yielding a novel family of signal models. This motivates us to investigate the following issues in a unified manner. First, by leveraging quaternion orthogonality, we develop the HQE-MUSIC algorithm for VS-array direction-of-arrival estimation and show that mutually complementary HQE patterns yield identical spatial spectra. Furthermore, we derive a closed-form expression for the asymptotic mean square error (MSE) of HQE-MUSIC and reveal its tight lower bound. Building on this, we formulate an MSE-based optimization problem to identify satisfactory HQE models. Considering the numerous candidates resulting from a large array, we introduce a tailored genetic algorithm to alleviate the computational burden. We demonstrate that HQE serves as a remedy for conventional QE schemes, mitigating their potential performance compromise in terms of MSE. Moreover, the optimized HQE-MUSIC, in contrast to its complex-valued counterpart, provides improved estimation accuracy, achieves higher angular resolution, especially in snapshot-limited cases, and exhibits robustness to cross-component correlated noise.