On sparse sensing and sparse sampling of coded signals at sub-Landau rates

TL;DR

This paper explores the possibility of reconstructing and detecting support of coded signals sampled below the Landau rate, showing that coding allows for reduced sampling rates and providing bounds on performance.

Contribution

It demonstrates that coding enables sub-Landau sampling for signal reconstruction and support detection, relaxing the classical Landau condition.

Findings

Support detection performance bounds are derived for Gaussian and frequency-sparse channels.

Sampling rates below the Landau rate can suffice for accurate reconstruction with coding.

High-dimensional systems require high SNR, which can be mitigated by reducing prior support uncertainty.

Abstract

Advances of information-theoretic understanding of sparse sampling of continuous uncoded signals at sampling rates exceeding the Landau rate were reported in recent works. This work examines sparse sampling of coded signals at sub-Landau sampling rates. It is shown that with coded signals the Landau condition may be relaxed and the sampling rate required for signal reconstruction and for support detection can be lower than the effective bandwidth. Equivalently, the number of measurements in the corresponding sparse sensing problem can be smaller than the support size. Tight bounds on information rates and on signal and support detection performance are derived for the Gaussian sparsely sampled channel and for the frequency-sparse channel using the context of state dependent channels. Support detection results are verified by a simulation. When the system is high-dimensional the required SNR is shown to be finite but high and rising with decreasing sampling rate, in some practical applications it can be lowered by reducing the a-priory uncertainty about the support e.g. by concentrating the frequency support into a finite number of subbands.