Thursday, June 11, 2015

Democratic Representations - implementation -

Christoph just sent me the following:

Hi Igor,

We finally managed to prepare a simple software package that includes three efficient solvers that compute democratic representations via infinity-norm minimization. The MATLAB software package can be found here:

The papers describing our algorithms can be found—as always—on arXiv:


Christoph Studer
Assistant Professor
School of ECE, Rhodes Hall 331
Cornell University
Ithaca, NY 14853, USA

Thanks Christoph ! I think this is the first time we have a phase transition diagram for these anti-sparse decompositions.

Democratic Representations  by Christoph Studer, Tom Goldstein, Wotao Yin, Richard G. Baraniuk
Minimization of the (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 -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.

PAR-Aware Large-Scale Multi-User MIMO-OFDM Downlink  by Christoph Studer, Erik G. Larsson
We investigate an orthogonal frequency-division multiplexing (OFDM)-based downlink transmission scheme for large-scale multi-user (MU) multiple-input multiple-output (MIMO) wireless systems. The use of OFDM causes a high peak-to-average (power) ratio (PAR), which necessitates expensive and power-inefficient radio-frequency (RF) components at the base station. In this paper, we present a novel downlink transmission scheme, which exploits the massive degrees-of-freedom available in large-scale MU-MIMO-OFDM systems to achieve low PAR. Specifically, we propose to jointly perform MU precoding, OFDM modulation, and PAR reduction by solving a convex optimization problem. We develop a corresponding fast iterative truncation algorithm (FITRA) and show numerical results to demonstrate tremendous PAR-reduction capabilities. The significantly reduced linearity requirements eventually enable the use of low-cost RF components for the large-scale MU-MIMO-OFDM downlink.
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