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beyondny-tit-2009 - 1 Beyond Nyquist Efficient Sampling of...

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1 Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals Joel A. Tropp, Member, IEEE , Jason N. Laska, Student Member, IEEE , Marco F. Duarte, Member, IEEE , Justin K. Romberg, Member, IEEE , and Richard G. Baraniuk, Fellow, IEEE Abstract — Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the ban- dlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator , that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O( K log( W/K )) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations. Index Terms — analog-to-digital conversion, compressive sam- pling, sampling theory, signal recovery, sparse approximation Dedicated to the memory of Dennis M. Healy. I. I NTRODUCTION T HE Shannon sampling theorem is one of the founda- tions of modern signal processing. For a continuous-time signal f whose highest frequency is less than W/ 2 Hz, the theorem suggests that we sample the signal uniformly at a rate of W Hz. The values of the signal at intermediate points in time are determined completely by the cardinal series f ( t ) = X n Z f n W sinc ( Wt - n ) . (1) In practice, one typically samples the signal at a somewhat higher rate and reconstructs with a kernel that decays faster than the sinc function [1, Ch. 4]. This well-known approach becomes impractical when the bandlimit W is large because it is challenging to build sam- pling hardware that operates at a sufficient rate. The demands of many modern applications already exceed the capabilities of current technology. Even though recent developments in Submitted: 30 January 2009. Revised: 12 September 2009. A preliminary report on this work was presented by the first author at SampTA 2007 in Thessaloniki. JAT was supported by ONR N00014-08-1-0883, DARPA/ONR N66001- 06-1-2011 and N66001-08-1-2065, and NSF DMS-0503299. JNL, MFD, and RGB were supported by DARPA/ONR N66001-06-1-2011 and N66001-08- 1-2065, ONR N00014-07-1-0936, AFOSR FA9550-04-1-0148, NSF CCF- 0431150, and the Texas Instruments Leadership University Program. JR was supported by NSF CCF-515632.
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