Maximally Flat Filter Functions With the Maximum Number of Transmission Zeros Having Maximal Multiplicity

The subject of the maximally flat continuous-time filter transfer function with the maximum number of transmission zeros with maximal multiplicity (MFMM) is revisited. It is found that in previous research a simple fact was overlooked: the numerator and denominator polynomials for even-order functions may be of equal orders. New solutions are presented for even-order transfer functions. It is shown that closed-form expressions may be produced for all properties of the amplitude characteristic and the design equations. The function thus obtained is compared with some existing monotonic functions having maximum number, but distinct, transmission zeroes. Starting with a low-pass prototype and using usual transformations, bandpass, bandstop, and high-pass MFMM filter characteristic functions were generated and described for the first time. A design procedure is recommended and design examples are given.