Response of maximally flat lowpass filters to polynomial signals

The response of lowpass maximally at digital filters to polynomial signals is analyzed by means of a formal algebraic method in the z domain. The general class of maximally flat FIR filters, encompassing linear-phase, non-linear-phase, and half-band filters, is studied. A characterization of polynomial signals in the z domain is introduced, and it is shown that for a given member of the class, all polynomial signals of a certain degree pass through the filter unaltered after a possibly fractional delay. It is then proved that using appropriate maximally flat filters from the family, it is possible to upsample and then fractionally delay any polynomial signal by a factor of 2 in an exact manner.