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dbwb-jstsp-2010 - IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL...

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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 2, APRIL 2010 445 Signal Processing With Compressive Measurements Mark A. Davenport , Student Member, IEEE , Petros T. Boufounos , Member, IEEE , Michael B. Wakin , Member, IEEE , and Richard G. Baraniuk , Fellow, IEEE Abstract— The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, de- spite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems—such as detection , classification , or estimation —and filtering problems using only compressive mea- surements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results. Index Terms— Compressive sensing (CS), compressive signal processing, estimation, filtering, pattern classification, random projections, signal detection, universal measurements. I. I NTRODUCTION A. From DSP to CSP I N recent decades, the digital signal processing (DSP) community has enjoyed enormous success in developing algorithms for capturing and extracting information from sig- nals. Capitalizing on the early work of Whitaker, Nyquist, and Shannon on sampling and representation of continuous signals, signal processing has moved from the analog to the digital domain and ridden the wave of Moore’s law. Digitization has enabled the creation of sensing and processing systems that are more robust, flexible, cheaper and, therefore, more ubiquitous than their analog counterparts. Manuscript received February 28, 2009; revised November 12, 2009. Current version published March 17, 2010. The work of M. A. Davenport and R. G. Baraniuk was supported by the Grants NSF CCF-0431150, CCF-0728867, CNS-0435425, and CNS-0520280, DARPA/ONR N66001-08-1-2065, ONR N00014-07-1-0936, N00014-08-1-1067, N00014-08-1-1112, and N00014-08-1-1066, AFOSR FA9550-07-1-0301, ARO MURI W311NF-07-1- 0185, ARO MURI W911NF-09-1-0383, and by the Texas Instruments Leadership University Program. The work of M. B. Wakin was supported by NSF Grants DMS-0603606 and CCF-0830320, and DARPA Grant HR0011-08-1-0078. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Rick Chartrand. M. A. Davenport and R. G. Baraniuk are with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005 USA (e-mail: [email protected]; [email protected]).
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