An inhomogeneous uncertainty principle for digital low-pass filters
Abbreviated Journal Title
J. Fourier Anal. Appl.
digital low-pass filter; filter efficiency; uncertainty principle; trigonometric polynomials; WAVELETS; Mathematics, Applied
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 asymptotics and explicitly construct a sequence of finite-length filters with the same asymptotics as the sequence of finite-length minimizers.
Journal of Fourier Analysis and Applications
"An inhomogeneous uncertainty principle for digital low-pass filters" (2006). Faculty Bibliography 2000s. 5967.