Xampling--Part I: Practice

We introduce Xampling, a design methodology for sub-Nyquist sampling of continuous-time analog signals. The main principles underlying this framework are the ability to capture a broad signal model, low sampling rate, efficient analog and digital implementation and lowrate baseband processing. The main hypothesis of Xampling is that in order to break through the Nyquist barrier, one has to combine classic methods and results from sampling theory together with recent developments from the literature of compressed sensing. In this paper, we present the Xampling framework and examine several subNyquist approaches in light of the four Xampling principles. It is shown that previous methods suffer from analog implementation issues, large computational loads in the digital domain, and have no baseband processing capabilities. An exception is the recently proposed modulated wideband converter (MWC) which satisfies the model, rate and implementation criteria, though lacking the baseband processing capability. Here, we extend the MWC by proposing a digital algorithm which extracts each band of the signal from the compressed measurements, thus enabling lowrate (baseband) processing. The converter with the proposed algorithm conforms with the Xampling desiderata. In addition, we describe two configurations of the converter for efficient spectrum sensing in wideband cognitive radio receivers. In the second part of this work we study theoretical aspects of rate and stability of sub-Nyquist systems, following the pragmatic theme of the Xampling methodology.