The asymptotics of optimal (equiripple) filters

For equiripple filters, the relation among the filter length N + 1, the tra nsition bandwidth Delta omega, and the optimal passband and stopband errors delta(p) and delta(s) has been a secret for more than 20 years. This paper is aimed at solving this mystery. We derive the exact asymptotic results i n the weight-free case delta(p) = delta(s) = delta, which enables us to int erpret and improve the existing empirical formulas. Our main results are fi nally combined into formula (17), In the transition band, the filter respon se is discovered to be asymptotically close to a scaled error function. The main tools are potential theory in the complex plane and asymptotic analys is.