Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
Wideband analog signals push contemporary analog- to-digital conversion (ADC) 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 band limit, 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 band limit in hertz. 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 $W$ hertz. In contrast to 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.
© 2010 IEEE. Current Version Published [online]: 2009-12-28. Manuscript received January 31, 2009; revised September 18, 2009. Current version published December 23, 2009. The work of J. A. Tropp was supported by ONR under Grant N00014-08-1-0883, DARPA/ONR under Grants N66001-06-1-2011 and N66001-08-1-2065, and NSF under Grant DMS-0503299. The work of J. N. Laska, M. F. Duarte, and R. G. Baraniuk was supported by DARPA/ONR under Grants N66001-06-1-2011 and N66001-08-1-2065, ONR under Grant N00014-07-1-0936, AFOSR under Grant FA9550-04-1-0148, NSF under Grant CCF-0431150, and the Texas Instruments Leadership University Program. The work of J. K. Romberg was supported by NSF under Grant CCF-515632. The material in this paper was presented in part at SampTA 2007, Thessaloniki, Greece, June 2007.
Submitted - 0902.0026.pdf
Published - Tropp2010p6850Ieee_T_Inform_Theory.pdf