Polyphase structures

The polyphase structure for FIR Filters was developed for the efficient implementation of sampling rate converters; however, it can be used in other applications. Further, the polyphase structure can be developed for any filter, FIR or IIR. We give below an introduction. 

Polyphase Structure for FIR Filters The impulse response of the FIR filter h(n) is of finite length, N. The system function with N coefficients is

We shall use another parameter M: we shall divide the number of coefficients into M groups (or branches or phases), modulo M. In other words the N terms in H(z) are arranged into M branches with each branch containing at most (Int (N -1/ M )+1) terms.

Type 1 polyphase decomposition For illustration, let N = 11 and M =2 so that one group contains 6 coefficients and the other 5 as developed below:

In The spacific case we have

This decomposition of H(z) is known as type 1 polyphase decomposition. The corresponding structure is shown below. The functions P₀(z²) and P₁(z²) can each be implemented as a direct form.

By observing the expressions for P₀(z) and P₁(z) we can further generalize the functions Pm(z) for any m as

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