DART: Discrete Prolate Spheroidal Sequences in Dictionaries for Sampled Multiband Signals

Mike Wakin just sent me the following:

Hi Igor,

Mark Davenport and I have just completed a paper that aims to help bridge the gap between the problem of analog signal acquisition and the discrete, finite framework of Compressive Sensing (CS). To do this, we exploit the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of timelimiting and bandlimiting operations. Very few of these classical ideas have appeared in the CS literature, however, and so one goal of this paper is to carefully explain--from a CS perspective--the natural role that these ideas can indeed play in CS.

To be more specific, we use DPSS's to construct a dictionary that provides sparse approximations of sampled multiband signals. This dictionary is far more efficient than the conventional DFT basis. In other words, this is one way of overcoming the "DFT leakage" problem that one encounters when dealing with finite-length sample vectors from analog signals.

If you are interested in learning more, our paper is available here:

• http://arxiv.org/abs/1109.3649

A companion software package is available here:

• http://www-stat.stanford.edu/~markad/software/

And a recent talk on this topic is available here:

• http://sahd.pratt.duke.edu/Videos/sessionK3.html (video)
• http://sahd.pratt.duke.edu/Videos/wakin-duke-sahd-2011-dpss-split.pdf (slides)

We welcome your comments or questions. Best regards,
-Mike

Thanks Mike. The DPSS Approximation and Recovery Toolbox (DART) is here.

The paper:

Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences by Mark A. Davenport, Michael B. Wakin. The abstract reads:

Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements.

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