Continuous Angular Power Spectrum Recovery From Channel Covariance via Chebyshev Polynomials

TL;DR

This paper introduces a Chebyshev polynomial-based method for accurately recovering continuous angular power spectra from channel covariance data, effectively addressing the ill-posed inversion problem in multi-antenna systems.

Contribution

It develops a Chebyshev polynomial expansion framework that reformulates APS recovery as a finite-dimensional linear regression, incorporating regularization for smooth and nonnegative spectra.

Findings

Accurate APS reconstruction demonstrated in simulations.

Reliable downlink covariance prediction from uplink data.

Effective control of approximation error via Chebyshev series tail.

Abstract

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.