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

A simple method to derive a closed-form expression for the transfer function of linear-phase half-band filters with maximally flat amplitude-response characteristics is presented. The method is based on the binomial series. It results in hierarchical and modular structures with low hardware complexity for low-pass and high-pass filters. For a filter of a given order, the structures provide access to ail maximally flat filters of lower orders. Two types of structures are presented. The first enjoys a set of multiplier coefficients with reduced dynamic range, and the second can be realized free of multiplier coefficients in a modular manner. Extension of the order of the filter can be achieved by cascading additional building blocks for the former structure or by adding an extra layer of modules for the latter structure. The proposed structures and formulas are applicable to maximally flat Hilbert transformers as well