On Maximum-Likelihood Methods for Localizing More Sources Than Sensors

This letter offers several new insights into the maximum-likelihood direction-of-arrival (DOA) estimation problem, when the number of sources exceeds the number of sensors. Two maximum-likelihood problems are studied: one for estimating the Toeplitz-structured coarray covariance matrix from the measurements, and the other for estimating the DOAs directly from the measurements. We establish the equivalence of both problems when the number of sources is assumed to be unknown and can potentially exceed the number of sensors. Additionally, it is shown that when the source waveforms satisfy certain orthogonality conditions, the Toeplitz-constrained maximum-likelihood covariance estimation framework provably produces the true DOAs without requiring to know the number of sources. When the number of sources exceeds the number of sensors, the maximum-likelihood algorithms studied in this letter outperform other recently studied methods, as demonstrated through numerical experiments.