Multiplierless and hierarchical structures for maximally flat half-band FIR filters

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.