Covariance Eigenanalysis for Direction Finding with Analytic Ray-Traced Steering Vectors

Abstract

In our setup of multiple phased array antennas, the goal was to replicate current techniques for object location. For this, we incorporated Multiple Signal Classification, a subspace direction of arrival estimator for multiple narrowband sources impinging on our phased arrays. This method relies on an eigendecomposition of the sample spatial covariance matrix and the resulting partition of the complex space into orthogonal signal and noise subspaces. Candidate directions are evaluated by their energy in the noise subspace, and direction of arrival hypotheses are taken from peaks in the reciprocal of this noise-subspace energy, often referred to as the pseudospectrum.

In our implementation, the steering vector is generated by an analytic ray tracer rather than by a conventional ideal plane-wave model. For each scan direction, the ray tracer computes the complex field at each array element by integrating refractive optical path length and attenuation along the propagation path to each sensor, with optional instrument phase and gain terms. The Multiple Signal Classification algorithm is then applied as usual, but the mismatch between the assumed and actual steering vector due to refraction and loss is shifted from an unmodeled error term into the forward model used inside the noise-subspace energy calculation.

This combination provides several operational advantages. Multiple Signal Classification preserves super-resolution angular discrimination by exploiting covariance eigenstructure rather than relying on steering limited by beamwidth. Ray tracing reduces deterministic bias by ensuring the steering vector includes atmospheric refraction and absorption, preventing these effects from being absorbed into leakage between the signal and noise subspaces. The result is a direction of arrival estimator that remains mathematically identical to standard Multiple Signal Classification, but produces bearings that are physically consistent under non-ideal propagation conditions, improving downstream fusion stability and track repeatability.

The development proceeds from the narrowband array model through finite-snapshot covariance estimation and Hermitian eigenanalysis to an explicit summation-form master equation that maps directly to loop-based implementations. The same algebraic structure is retained when the steering vector is replaced by ray-integrated propagation, enabling a propagation-aware pseudospectrum evaluation under refracting and lossy atmospheres.