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Signal and System Approximation from General Measurements

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New Perspectives on Approximation and Sampling Theory

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

Abstract

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley–Wiener space \(\mathcal{P}\mathcal{W}_{\pi }^{1}\). We study approximation processes, where the input signal is not directly used to generate the system output but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.

Dedicated to Professor Paul Butzer on his 85th birthday

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Acknowledgements

The authors would like to thank Ingrid Daubechies for valuable discussions of Conjectures 1 and 2 and for pointing out connections to frame theory at the Strobl’11 conference and the “Applied Harmonic Analysis and Sparse Approximation” workshop at the Mathematisches Forschungsinstitut Oberwolfach in 2012. Further, the authors are thankful to Przemysław Wojtaszczyk and Yurii Lyubarskii for valuable discussions of Conjecture 1 at the Strobl’11 conference and Joachim Hagenauer and Sergio Verdú for drawing our attention to [16] and for discussions of related topics. We would also like to thank Mario Goldenbaum for carefully reading the manuscript and providing helpful comments. H. Boche was supported by the German Research Foundation (DFG) under grant BO 1734/13-2. U. Mönich was supported by the German Research Foundation (DFG) under grant MO 2572/1-1.

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Authors and Affiliations

  1. Technische Universität München, Lehrstuhl für Theoretische Informationstechnik, Arcisstr. 21, 80290, München, Germany

    Holger Boche

  2. Massachusetts Institute of Technology, Research Laboratory of Electronics, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Ullrich J. Mönich

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  1. Holger Boche
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Correspondence to Holger Boche .

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  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Boche, H., Mönich, U.J. (2014). Signal and System Approximation from General Measurements. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_6

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