The asymptotics of optimal (equiripple) filters

Author

Shen, Jianhong ; Strang, Gilbert

Author_Institution

Dept. of Comput. & Appl. Math., California Univ., Los Angeles, CA, USA

Volume

47

Issue

4

fYear

1999

fDate

4/1/1999 12:00:00 AM

Firstpage

1087

Lastpage

1098

Abstract

For equiripple filters, the relation among the filter length N+1, the transition bandwidth Δω, and the optimal passband and stopband errors δ_p and δ_s has been a secret for more than 20 years. This paper is aimed at solving this mystery. We derive the exact asymptotic results in the weight-free case δ_p=δ_s=δ, which enables us to interpret and improve the existing empirical formulas. Our main results are finally combined into a formula. In the transition band, the filter response is discovered to be asymptotically close to a scaled error function. The main tools are potential theory in the complex plane and asymptotic analysis

Keywords

approximation theory; circuit optimisation; equiripple filters; error analysis; filtering theory; low-pass filters; approximation theory; asymptotic analysis; complex plane; empirical formulas; exact asymptotic results; filter length; filter response; low pass filter; optimal equiripple filters; optimal passband error; optimal stopband error; scaled error function; transition band; transition bandwidth; weight-free case; Algorithm design and analysis; Band pass filters; Bandwidth; Frequency; Iterative algorithms; MATLAB; Mathematics; Minimax techniques; Passband; Polynomials;

fLanguage

English

Journal_Title

Signal Processing, IEEE Transactions on

Publisher

ieee

ISSN

1053-587X

Type

jour

DOI

10.1109/78.752606

Filename

752606