scs-tsp - Spectral Compressive Sensing Marco F. Duarte,...

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Unformatted text preview: Spectral Compressive Sensing Marco F. Duarte, Member, IEEE, and Richard G. Baraniuk, Fellow, IEEE Abstract Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency domain. In practical applications, the standard frequency domain signal representation is the discrete Fourier transform (DFT). Unfortunately, the DFT coefficients of a frequency-sparse signal are themselves sparse only in the contrived case where the sinusoid frequencies are integer multiples of the DFT’s fundamental frequency. As a result, practical DFT-based CS acquisition and recovery of smooth signals does not perform nearly as well as one might expect. In this paper, we develop a new spectral compressive sensing (SCS) theory for general frequency-sparse signals. The key ingredients are an over-sampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectrum estimation. Using peridogram and eigen-analysis based spectrum estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current state-of-the-art CS algorithms while providing provable bounds on the number of measurements required for stable recovery. I. INTRODUCTION The emerging theory of compressive sensing (CS) combines digital data acquisition with digital data compression to enable a new generation of signal acquisition systems that operate at sub-Nyquist rates. Rather than acquiring N samples x = [ x [1] x [2] ... x [ N ]] T of an analog signal at the Nyquist rate, a CS system acquires M < N measurements via the linear dimensionality reduction y = Φ x , where Φ Date: February 4, 2010. MFD is with the Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544. RGB is with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005. Email: [email protected], [email protected] MFD and RGB were supported by grants NSF CCF-0431150 and CCF-0728867, DARPA/ONR N66001-08-1-2065, ONR N00014-07-1-0936 and N00014-08-1-1112, AFOSR FA9550-07-1-0301, ARO MURIs W911NF-07-1-0185 and W911NF-09-1-0383, and the Texas Instruments Leadership Program. MFD was also supported by NSF Supplemental Funding DMS-0439872 to UCLA-IPAM, P.I. R. Caflisch. is an M × N measurement matrix. When the signal x has a sparse representation x = Ψ θ in terms of an N × N orthonormal basis matrix Ψ , meaning that only K N out of N signal coefficients θ are nonzero, then the number of measurements required to ensure that y retains all of the information in x is just M = O ( K log( N/K )) [1–3]. Moreover, a sparse signal x can be recovered from its compressive measurements y via a convex optimization or iterative greedy algorithm. Random matrices play a centralvia a convex optimization or iterative greedy algorithm....
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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scs-tsp - Spectral Compressive Sensing Marco F. Duarte,...

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