Title

An Inhomogeneous Uncertainty Principle For Digital Low-Pass Filters

Keywords

Digital low-pass filter; Filter efficiency; Trigonometric polynomials; Uncertainty principle

Abstract

This article introduces an inhomogeneous uncertainty principle for digital low-pass filters. The measure for uncertainty is a product of two factors evaluating the frequency selectivity in comparison with the ideal filter and the effective length of the filter in the digital domain, respectively. We derive a sharp lower bound for this product in the class of filters with so-called finite effective length and show the absence of minimizers. We find necessary and certain sufficient conditions to identify minimizing sequences. When the class of filters is restricted to a given maximal length, we show the existence of an uncertainty minimizer. The uncertainty product of such minimizing filters approaches the unrestricted infimum as the filter length increases. We examine the asymplotics and explicitly construct a sequence of finite-length filters with the same asymptotics as the sequence of finite-length minimizers. © 2006 Birkhäuser Boston. All rights reserved.

Publication Date

4-1-2006

Publication Title

Journal of Fourier Analysis and Applications

Volume

12

Issue

2

Number of Pages

181-211

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00041-005-5018-5

Socpus ID

33646504148 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/33646504148

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