Democratic Representations

TL;DR

This paper studies the properties of signal representations obtained by minimizing the maximum norm under linear constraints, introducing efficient algorithms and demonstrating their effectiveness in reducing dynamic range in communication systems.

Contribution

It provides theoretical bounds on such representations, introduces algorithms for their computation, and demonstrates practical benefits in broadcast systems.

Findings

Matrices satisfying the uncertainty principle enable democratic representations.

Democratic representations have small, similar entry magnitudes and low dynamic range.

Algorithms efficiently compute these representations for practical applications.

Abstract

Minimization of the $\ell_{\infty}$ (or maximum) norm subject to a constraint that imposes consistency to an underdetermined system of linear equations finds use in a large number of practical applications, including vector quantization, approximate nearest neighbor search, peak-to-average power ratio (or "crest factor") reduction in communication systems, and peak force minimization in robotics and control. This paper analyzes the fundamental properties of signal representations obtained by solving such a convex optimization problem. We develop bounds on the maximum magnitude of such representations using the uncertainty principle (UP) introduced by Lyubarskii and Vershynin, and study the efficacy of $\ell_{\infty}$-norm-based dynamic range reduction. Our analysis shows that matrices satisfying the UP, such as randomly subsampled Fourier or i.i.d. Gaussian matrices, enable the computation of what we call democratic representations, whose entries all have small and similar magnitude, as well as low dynamic range. To compute democratic representations at low computational complexity, we present two new, efficient convex optimization algorithms. We finally demonstrate the efficacy of democratic representations for dynamic range reduction in a DVB-T2-based broadcast system.