This paper considers recovering a continuous angular power spectrum (APS) from the channel covariance. Building on the projection-onto-linear-variety (PLV) algorithm, an affine-projection approach introduced by Miretti \emph{et. al.}, we analyze PLV in a well-defined \emph{weighted} Fourier-domain to emphasize its geometric interpretability. This yields an explicit fixed-dimensional trigonometric-polynomial representation and a closed-form solution via a positive-definite matrix, which directly implies uniqueness. We further establish an exact energy identity that yields the APS reconstruction error and leads to a sharp identifiability/resolution characterization: PLV achieves perfect recovery if and only if the ground-truth APS lies in the identified trigonometric-polynomial subspace; otherwise it returns the minimum-energy APS among all covariance-consistent spectra.

翻译：本文研究从信道协方差中恢复连续角功率谱（APS）的问题。基于Miretti等人提出的投影到线性簇（PLV）算法及其仿射投影方法，我们在一个定义良好的加权傅里叶域中分析PLV算法，以突出其几何可解释性。这导出了一个显式的固定维三角多项式表示，以及通过正定矩阵获得的闭式解，该解直接蕴含了唯一性。我们进一步建立了一个精确的能量恒等式，该恒等式给出了APS重构误差，并导出了一个尖锐的可辨识性/分辨率刻画：当且仅当真实APS位于所识别的三角多项式子空间内时，PLV算法可实现完美恢复；否则，算法将返回所有协方差一致谱中能量最小的APS。