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Non-Translation-Invariance and the Synchronization Problem in Wavelet Sampling

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Abstract

One of the major differences between Paley-Wiener spaces of bandlimited signals and the principal shift-invariant (PSI) spaces of wavelet theory is that the latter, although shift-invariant, are in general not translation-invariant. In this paper we study the extra difficulties non-translation-invariance creates for the sampling theory of PSI and multiresolution spaces. In particular it is shown that sampling in PSI spaces requires an extra initialization step to determine the times at which sampled data is acquired. An algorithm is developed to provide this initialization and its effectiveness shown theoretically and demonstrated on a synthetic data set.

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, University of Newcastle, NSW, 2308, Newcastle, Australia

    Jeffrey Hogan

  2. Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001 Department 3MB, Las Cruces, NM, 88003-8001, USA

    Joseph Lakey

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  1. Jeffrey Hogan
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  2. Joseph Lakey
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Correspondence to Jeffrey Hogan.

Additional information

This paper was partly written while Jeff Hogan was on sabbatical at the University of Vienna, where he was supported by the European Union EUCETIFA grant MEXT-CT-2004-517154 and a University of Arkansas Fulbright College of Arts and Sciences Research Incentive Grant.

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Hogan, J., Lakey, J. Non-Translation-Invariance and the Synchronization Problem in Wavelet Sampling. Acta Appl Math 107, 373–398 (2009). https://doi.org/10.1007/s10440-009-9480-y

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  • DOI: https://doi.org/10.1007/s10440-009-9480-y

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