This paper proposes a Chebyshev polynomial expansion framework for the recovery of a continuous angular power spectrum (APS) from channel covariance. By exploiting the orthogonality of Chebyshev polynomials in a transformed domain, we derive an exact series representation of the covariance and reformulate the inherently ill-posed APS inversion as a finite-dimensional linear regression problem via truncation. The associated approximation error is directly controlled by the tail of the APS's Chebyshev series and decays rapidly with increasing angular smoothness. Building on this representation, we derive an exact semidefinite characterization of nonnegative APS and introduce a derivative-based regularizer that promotes smoothly varying APS profiles while preserving transitions of clusters. Simulation results show that the proposed Chebyshev-based framework yields accurate APS reconstruction, and enables reliable downlink (DL) covariance prediction from uplink (UL) measurements in a frequency division duplex (FDD) setting. These findings indicate that jointly exploiting smoothness and nonnegativity in a Chebyshev domain provides an effective tool for covariance-domain processing in multi-antenna systems.

翻译：本文提出了一种基于切比雪夫多项式展开的框架，用于从信道协方差中恢复连续的角功率谱。通过利用切比雪夫多项式在变换域中的正交性，我们推导了协方差的一个精确级数表示，并通过截断将固有的不适定APS反演问题重新表述为一个有限维线性回归问题。相关的近似误差直接由APS切比雪夫级数的尾部控制，并随着角度平滑度的增加而迅速衰减。基于此表示，我们推导了非负APS的一个精确半定特征描述，并引入了一种基于导数的正则化器，该正则化器在保持簇间过渡的同时，促进了平滑变化的APS剖面。仿真结果表明，所提出的基于切比雪夫多项式的框架能够实现精确的APS重建，并在频分双工设置下，能够从上行链路测量中可靠地预测下行链路协方差。这些发现表明，在切比雪夫域中联合利用平滑性和非负性，为多天线系统中的协方差域处理提供了一种有效的工具。