AI 中文总结
该研究针对相干DOA估计的确定性CRB，推导切空间条件，证明仅低秩性不降低频率CRB、相干结构可严格减小CRB，扩展结论到多类阵列模型，通过数值实验验证，明确性能提升源于相干结构而非仅低秩性。
AI 中文摘要
波达方向（DOA）估计是阵列信号处理中的基础问题，克拉美-罗界（CRB）是随机源模型和确定源模型下的标准性能基准。在多径环境中，源矩阵为低秩，且更具体地呈现组内比例结构。尽管已有相干DOA估计的随机CRB研究，但确定性对应结果仍有限，本文旨在填补这一空白。我们首先推导了一个切空间条件，用于表征源矩阵的结构约束何时能严格降低频率CRB。随后证明，对于均匀线性阵列，仅施加源矩阵的低秩性会使频率CRB保持不变，但当秩约束非平凡时，该约束会降低源矩阵CRB；而在温和条件下，充分利用相干结构会对几乎所有参数值得到严格更小的频率CRB。这些结论进一步扩展到解析阵列流形和多维DOA模型，并针对均匀平面阵列建立了分量级结果。结果表明，DOA估计的性能提升源于详细的相干结构，而非仅低秩性。最后提供数值实验以验证理论发现。
英文摘要
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.
