Asymptotically efficient estimators for multidimensional harmonic retrieval based on the geometry of the Stiefel manifold

A variety of signal processing applications require multidimensional harmonic retrieval on regular arrays and R-dimensional subspace-based methods (e.g; R-D Unitary ESPRIT ) are often used for this task. The conventional subspace estimation step via an SVD is the MLE for the unconstrained assumption but is suboptimal here since the harmonic signal structure or constraint is not exploited. Subspace estimation methods such as F/B averaging, HO-SVD, and SLS, which make use of the structure to varying degrees, yield improved performance but remain suboptimal in the sense that they do not satisfy a maximum likelihood criterion. Using a modified complex Stiefel manifold for the domain of the likelihood function we derive a quadratic ML criterion with a geometric constraint for the R- dimensional problem. This constraint is expressed in terms of the tangent space of an appropriate submanifold. For the special case when the submanifold satisfies a shift-invariance condition, we present a stand-alone estimation algorithm that computes the submanifold tangent space as the null space of a matrix that represents the linearized form of a geometric constraint. The estimator's performance is compared to existing approaches and to the intrinsic subspace CRB for highly stressful scenarios of closely spaced and highly correlated sources.