## Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals (2009)

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### BibTeX

@MISC{Tropp09beyondnyquist:,

author = {Joel A. Tropp and Jason N. Laska and Marco F. Duarte and Justin K. Romberg and Richard G. Baraniuk},

title = { Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals},

year = {2009}

}

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### 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 bandlimit, 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.