Stochastic Maximum Likelihood Array Processing for Rank Deficient Array Manifolds

Maximum likelihood estimation of emitter source parameters in array signal processing for stochastic source signals is well established in the literature. Currently available results in the literature however, have relied on the assumption that the array response matrix has full column rank. In certain models used in for example chemistry, telecommunications, oceanography, and biomedical engineering this is not necessarily the case. In this paper we focus on magnetoencephalography, where the method has been previously proposed in brain function connectivity analysis. Here rank deficiency of the response matrix occurs if spherically symmetric head models for MEG signals are used. In this paper we show that the assumption of a full rank response matrix is unnecessary, which has the advantage that complicating reparameterization is unnecessary. We show that the method of concentrating the likelihood remains valid, including statistical inference from the concentrated likelihood, and generalize a well known algorithm to this case. We exemplify the derived results in simulations.