Filters with maximally flat amplitude and controlled delay responses

Closed-form expressions are given for two classes of nonminimum-phase transfer functions in lumped-element, microwave, and digital filters. The first exhibits simultaneous maximally flat amplitude and delay characteristics about the origin . The second maintains the maximally flat amplitude response while possessing one delay condition less than the maximum possible number. The functions have general numerators (i.e., containing odd as well as even powers), thus allowing considerable freedom for control of both amplitude and delay. The results indicate that the amplitude response of the second class of functions possesses sharper cutoff than the first. On the other hand, functions belonging to the first class are more suitable for cases where the delay specifications are very stringent and/or the passbands are relatively wide.