1. OVERVIEW OF POLYPHASE IIR FILTERS

The idea of a polyphase structure can be easily derived from an FIR filter if all coefficients are substituted by appropriate allpass subfilters with constant unity gain and frequency dependent phase shifts leading to the structure as presented in Figure 1 1(a).

Figure 1 1: Polyphase filter: structure, (a), and a rule of filtering, (b) [1]

Each allpass (A ₀ A _{N ?1} ) has a different, carefully designed phase response. This is why the structure is called polyphase. The frequency selective characteristics of the filters are due to the phase shifts between consecutive branches (Figure 1 1(b)). This book concentrates on the special class of polyphase structure in which AllPass Filters (APF) are made of N ^th-order one-coefficient allpass sections (2), where N is the number of branches in the structure. It is worth noting that their impulse responses are sequences of samples with non-zero values at i N locations, where i (0, ?). The step response changes its value every each N samples. The impulse and step responses, and the average time delay of ?= N indicate that allpass filters may be very useful for decimation with factor N. The general transfer function of the polyphase structure presented in Figure 1 1 is [2]:

Allpass filters A _i,k (z ^?N ) have N poles at z=( ? ? _i,k ) ^1/N and N zeros at z=( ? ?