A density theorem for time-continuous filter banks

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Abstract

We show in this paper the following result. When a > 0 and gnm = gm(· - na), n, m e Z, is a frame for L2(), where each gm e L2() is localized in the frequency-domain around a point b m e , then ( = 1. Here is the asymptotic lower bound of the average number of bm in a symmetric interval around 0 as the interval length tends to 8. As a particular case it is found that a multi-window Gabor-type system gp(t - na) exp(2pimbt) with n, m e Z, p = 0, …, P - 1, can generate a frame for L2() only if P(ab)-1 = 1. The main result of this paperis based on the Ron-Shen criterion in the frequency-domain for duality of two shift- invariant systems of functions, combined with an amplification of Janssen's elementary proof of the fact that a Weyl-Heisenberg system g(t - na)exp(2pimbt) can generate a framefor L2() only if (ab)-1 = 1.
Original languageEnglish
Pages (from-to)513-523
JournalWavelet Analysis and its Applications
Volume7
DOIs
Publication statusPublished - 1998

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