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

Abstract

ABSTRACTAdvances in information‐theoretic analysis 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 and with discrete signals, the Landau condition may be relaxed, and the sampling rates 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 signal to noise ratio 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, for example, by concentrating the frequency support into a finite number of subbands. The sub‐Landau sampling rates are analysed also in large systems with discrete signal constellations replacing the coding. Copyright © 2013 John Wiley & Sons, Ltd.