Beyond Nyquist: efficient sampling of sparse bandlimited signals

  • Authors:

  • Joel A. Tropp;Jason N. Laska;Marco F. Duarte;Justin K. Romberg;Richard G. Baraniuk

  • Affiliations:

  • California Institute of Technology, Pasadena, CA;Rice University, Houston, TX;Princeton University, Princeton, NJ and Rice University, Houston, TX;Georgia Institute of Technology, Atlanta, GA;Rice University, Houston, TX

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

Wideband analog signals push contemporary analogto-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(Klog(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.