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Adaptive Signal Processing

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Excursions in Harmonic Analysis, Volume 4

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Adaptive frequency band (AFB) and ultra-wideband (UWB) systems require either rapidly changing or very high sampling rates. Conventional analog-to-digital devices are nonadaptive and have limited dynamic range. We investigate AFB and UWB sampling via a basis projection method. The method decomposes the signal into a basis over time segments via a continuous-time inner product operation and then samples the basis coefficients in parallel. The signal may then be reconstructed from the basis coefficients to recover the signal in the time domain. The overarching goal of the theory developed in this chapter is to develop a computable atomic decomposition of time-frequency space. The idea is to come up with a way of nonuniformly tiling time and frequency so that if the signal has a burst of high-frequency information, we tile quickly and efficiently in time and broadly in frequency, whereas if the signal has a relatively low-frequency segment, we can tile broadly in time and efficiently in frequency. Computability is key; systems are designed so that they can be constructed using splines and implemented in circuitry.

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Notes

  1. 1.

    Meyer [28] gives an excellent overview of these different points in Chapter 1 of his book.

  2. 2.

    Computations throughout the chapter have been adjusted to compensate for our definition of the Fourier transform.

  3. 3.

    Given two functions f and g, we say that \(f = \mathcal{O}(g)\) if there exists a positive constant K such that f(ω) < Kg(ω) for ω sufficiently large.

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Acknowledgments:

The author’s research was partially supported by US Army Research Office Scientific Services program, administered by Battelle (TCN 06150, Contract DAAD19-02-D-0001) and US Air Force Office of Scientific Research Grant Number FA9550-12-1-0430.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, American University, Washington, DC, 20016-8050, USA

    Stephen D. Casey

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  1. Stephen D. Casey
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Correspondence to Stephen D. Casey .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Matthew Begué

  3. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Kasso A. Okoudjou

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Casey, S.D. (2015). Adaptive Signal Processing. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 4. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20188-7_11

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