Affine-Projection Recovery of Continuous Angular Power Spectrum: Geometry and Resolution

TL;DR

This paper introduces an affine-projection method for recovering a continuous angular power spectrum from channel covariance, providing geometric insights, explicit solutions, and conditions for perfect recovery.

Contribution

It extends the PLV algorithm with a weighted Fourier analysis, offering a fixed-dimensional trigonometric representation and a closed-form solution with guarantees on uniqueness.

Findings

PLV achieves perfect recovery when the ground-truth APS is in the identified subspace.

Provides a sharp resolution and identifiability characterization.

Derives an explicit energy identity for reconstruction error.

Abstract

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.