A simple method to derive a closed-form expression for the transfer functio
n of linear-phase half-band filters with maximally Bat amplitude-response c
haracteristics is presented. The method is based on the binomial series. It
results in hierarchical and modular structures with low hard ware complexi
ty for low-pass and high-pass filters. For a filter of a given order, the s
tructures provide access to all maximally Bat filters of lon er orders. Two
types of structures are presented. The first enjoys a set of multiplier co
efficients with reduced dynamic range, and the second can be realized free
of multiplier coefficients in a modular manner. Extension of the order of t
he filter can be achieved by cascading additional building blocks for the f
ormer structure or by adding an extra layer of modules for the latter struc
ture. The proposed structures and formulas are applicable to maximally flat
Hilbert transformers as well.