The asymptotics of optimal (equiripple) filters

For equiripple filters, the relation among the filter length N + 1, the transition bandwidth !, and the optimal passband and stopband errors p and 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 weightfree case p = s = , which enables us to interpret and improve the existing empirical formulas. Our main results are finally combined into formula (17). 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.