Weichao Zheng, Zai Yang
Direction-of-arrival (DOA) estimation is a fundamental problem in array signal processing, for which the Cramer-Rao bound (CRB) serves as a standard performance benchmark under both stochastic and deterministic source models. In multipath environments, the source matrix is of low-rank and, more specifically, exhibits a within-group proportionality structure. Although stochastic CRBs for coherent DOA estimation have been studied, theoretical results for their deterministic counterparts remain limited. This paper aims to bridge this gap. We first derive a tangent-space condition characterizing when structural constraints on the source matrix can strictly reduce the frequency CRB. We then prove that, for uniform linear arrays, imposing only the low-rankness of the source matrix leaves the frequency CRB unchanged, although it reduces the source-matrix CRB whenever the rank constraint is nontrivial, whereas fully exploiting the coherence structure yields a strictly smaller frequency CRB for almost all parameter values under mild conditions. These conclusions are further extended to analytic array manifolds and multidimensional DOA models, with componentwise results established for uniform planar arrays. The results demonstrate that performance gains in DOA estimation arise from the detailed coherence structure rather than low-rankness alone. Numerical experiments are finally provided to validate the theoretical findings.