Digital Frequency-Selective Filters Based on Spectra of Atomic Functions

Abstract

A method for synthesis of digital low-pass filter (LPF) based on spectra of atomic functions \({{{\text{h}}}_{a}}(x)\) is first proposed. Compactly supported function \({{{\text{h}}}_{a}}(x)\) is a solution to the problem of partition of unity, which makes it possible to consider a sum of its \(S\) shifts \({{h}_{{S,a}}}(x)\) as a perfect frequency response of LPF. Truncation of the spectrum of function \({{{\text{h}}}_{{S,a}}}(x)\) is used to obtain digital filters with rapidly decaying coefficients. Formulas for estimation of deviations of frequency responses in the passband and stopband are derived. Application of new filters in multirate signal processing is considered.