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Time and Band Limiting of Multiband Signals

  • Jeffrey A. Hogan
  • Joseph D. Lakey
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

When \( {\mathcal a} \) = 2ΩT, the operator PΩQ T corresponding to single time and frequency intervals has an eigenvalue λ \( {}_{\mathcal a} \) 1/2, as Theorem 4.1.2 below will show. The norm λ0(\( {\mathcal a} \) = 1) of the operator PQ1/2 satisfies λ0(\( {\mathcal a} \) = 1) ≥ ∥sinc [1/2, 1/2]∥ > 0.88. The trace of PQ1/2 is equal to \( {\mathcal a} \) = 1, on the one hand and to Σλ n on the other, so λ1(\( {\mathcal a} \) = 1) ≤ 1λ0(\( {\mathcal a} \) = 1) < 1/2. Suppose that T = 1 and Σ is a finite, pairwise disjoint union of \( {\mathcal a} \) frequency intervals I1, … , I\(_{} {\mathcal a} \) each of unit length. Then PΣQ should have on the order of \( {\mathcal a} \) eigenvalues of magnitude at least 1/2. Consider now the limiting case in which the frequency intervals become separated at infinity. Any function ψ j that is concentrated in frequency on I j will be almost orthogonal over [−T,T], in the separation limit, to any function ψ k that is frequency-concentrated on I k when j ≠ k.

Keywords

Pairwise Disjoint Partition Element Stirling Number Frequency Product Frequency Support 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  2. 2.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

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