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

2025-12-29

AI Summary

This work addresses the channel covariance-driven continuous angular power spectrum (APS) recovery problem. We propose a weighted Fourier-domain affine projection reconstruction method. By establishing an exact energy identity, we derive, for the first time, an analytical expression for the APS reconstruction error and rigorously characterize identifiability: perfect recovery is achievable if and only if the true APS lies in a specific trigonometric polynomial subspace; otherwise, the method returns the minimum-energy consistent solution. The approach integrates weighted Fourier analysis, projection onto linear manifolds (PLM), closed-form solutions for positive-definite matrices, and trigonometric polynomial approximation, yielding a unique closed-form solution with an explicit error bound. Experiments demonstrate that the method achieves optimal APS estimation under covariance consistency constraints, significantly enhancing angular resolution and geometric interpretability.

Technology Category

• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Intelligent Robots: State Estimation
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Recover continuous angular power spectrum from channel covariance
• Analyze PLV algorithm in weighted Fourier domain for geometric interpretability
• Establish exact energy identity for reconstruction error and identifiability characterization

Innovation

• Affine-projection approach in weighted Fourier-domain
• Closed-form solution via positive-definite matrix
• Minimum-energy APS for covariance-consistent spectra