On the Minimax Capacity Loss Under Sub-Nyquist Universal Sampling

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 an information , that is, the gap of achievable rates caused by the lack of instantaneous subband occupancy information. We characterize the minimax information rate loss for the sub-Nyquist 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 information rate loss with exponentially high probability.