On the Minimax Capacity Loss under Sub-Nyquist Universal Sampling

TL;DR

This paper analyzes the worst-case capacity loss in multiband Gaussian channels under universal sub-Nyquist sampling, showing that randomized sampling methods can nearly achieve minimal capacity loss in high SNR and large subband regimes.

Contribution

It characterizes the minimax capacity loss for universal sub-Nyquist sampling in multiband channels, revealing the effectiveness of randomized sampling strategies in high-dimensional settings.

Findings

Minimax capacity loss depends mainly on band sparsity and undersampling factors.

Randomized sampling methods approach the minimax capacity loss exponentially fast.

Results hold in large SNR and large number of subbands regimes.

Abstract

This paper investigates the information rate loss in analog channels when the sampler is designed to operate independent of the instantaneous channel occupancy. Specifically, a multiband linear time-invariant Gaussian channel under universal sub-Nyquist sampling is considered. The entire channel bandwidth is divided into $n$ subbands of equal bandwidth. At each time only $k$ constant-gain subbands are active, where the instantaneous subband occupancy is not known at the receiver and the sampler. We study the information loss through a capacity loss metric, that is, the capacity gap caused by the lack of instantaneous subband occupancy information. We characterize the minimax capacity loss for the entire sub-Nyquist rate regime, provided that the number $n$ of subbands and the SNR are both large. The minimax limits depend almost solely on the band sparsity factor and the undersampling factor, modulo some residual terms that vanish as $n$ and SNR grow. Our results highlight the power of randomized sampling methods (i.e. the samplers that consist of random periodic modulation and low-pass filters), which are able to approach the minimax capacity loss with exponentially high probability.