3
Xampling: Compressed Sensing of
Analog Signals
Moshe Mishali and Yonina C. Eldar
Department of Electrical Engineering, Technion, Haifa 32000, Israel.
This chapter generalizes compressed sensing (CS) to reduced-rate sampling of
analog signals. It introduces Xampling, a unified framework for low rate sam-
pling and processing of signals lying in a union of subspaces. Xampling consists
of two main blocks: Analog compression that narrows down the input bandwidth
prior to sampling with commercial devices followed by a nonlinear algorithm that
detects the input subspace prior to conventional signal processing. A variety of
analog CS applications are reviewed within the unified Xampling framework
including a general filter-bank scheme for sparse shift-invariant spaces, periodic
nonuniform sampling and modulated wideband conversion for multiband com-
munications with unknown carrier frequencies, acquisition techniques for finite
rate of innovation signals with applications to medical and radar imaging, and
random demodulation of sparse harmonic tones. A hardware-oriented viewpoint
is advocated throughout, addressing practical constraints and exemplifying hard-
ware realizations where relevant.
3.1 Introduction
Analog-to-digital conversion (ADC) technology constantly advances along the
route that was delineated in the last century by the celebrated Shannon-Nyquist
[1, 2] theorem, essentially requiring the sampling rate to be at least twice the
highest frequency in the signal. This basic principle underlies almost all digital
signal processing (DSP) applications such as audio, video, radio receivers, wire-
less communications, radar applications, medical devices, optical systems and
more. The ever growing demand for data, as well as advances in radio frequency
(RF) technology, have promoted the use of high-bandwidth signals, for which the
rates dictated by the Shannon-Nyquist theorem impose demanding challenges on
the acquisition hardware and on the subsequent storage and DSP processors. A
holy grail of compressed sensing (CS) is to build acquisition devices that exploit
signal structure in order to reduce the sampling rate, and subsequent demands
on storage and DSP. In such an approach, the actual information contents should
dictate the sampling rate, rather than the ambient signal bandwidth. Indeed, CS
was motivated in part by the desire to sample wideband signals at rates far below
1
arXiv:1103.2960v1 [cs.IT] 15 Mar 2011
