The asymptotics of optimal (equiripple) filters

Jianhong Shen, G. Strang
1999 IEEE Transactions on Signal Processing  
For equiripple filters, the relation among the filter length N + 1, the transition bandwidth 1!, 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
more » ... o be asymptotically close to a scaled error function. The main tools are potential theory in the complex plane and asymptotic analysis.
doi:10.1109/78.752606 fatcat:zplzgogyhje4tlyy22qltffs5a