Title :
The asymptotics of optimal (equiripple) filters
Author :
Shen, Jianhong ; Strang, Gilbert
Author_Institution :
Dept. of Comput. & Appl. Math., California Univ., Los Angeles, CA, USA
fDate :
4/1/1999 12:00:00 AM
Abstract :
For equiripple filters, the relation among the filter length N+1, the transition bandwidth Δω, and the optimal passband and stopband errors δp and δs 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 δp=δs=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;
Journal_Title :
Signal Processing, IEEE Transactions on
