# Polyphase decomposition of IIR filter

Is there a way to decompose IIR filter into set of poly phase components, that are All pass. E.g. a an IIR filter of order M, can be decomposed into a cascade of All zero filter of order 2*M( which can be implemented as polyphase) and all pole of same order( as Direct form), by rationalizing the Denominator. But, I have not been able to find a way to do it directly as sum of all pass polyphase components.

any help?

• very little of what you write makes sense to me. yes, an IIR filter transfer function can be factored into 2nd-order components (or 1st-order components, if you don't mind possibly having complex coefficients). certainly, you can separate the denominator (that has the poles) from the numerator (that depicts the zeros) and, in both, replacing the missing poles and zeros (that were separated) with poles and zeros at the origin, $z=0$. then you will have an all-pole filter and an all-zero filter, the latter is an FIR filter. dunno what the concept of "polyphase" has to do with this. – robert bristow-johnson Nov 14 '14 at 20:56

Using matlabs tf2cl, it may require the DSP toolbox, you can turn the transsfer function (numerator and denominator coefficients) in to a cascade allpass lattice coefficients, which is a two phase filter.

% Generate some IIR filter coefficients
[b,a]=cheby1(9,.5,.4);

% Coupled allpass decomposition
[k1,k2]=tf2cl(b,a);


I assume that you are trying to rewrite your transfer function $H(z)$ as

$$H(z) = \sum_{i=0}^{N-1} A_i(z^N),$$ where $A_i(z^N)$ are allpass filters.

In general, no, as the pole and zero locations are related when you implement it as parallel all pass filters. It typically works for the standard filter approximations and $N=2$ filters though.

What you may be looking for are the papers Recursive Nth-band digital filters - Parts I and II by Renfors and Saramäki. These describe how to design filters which consists of N parallel allpass filters, each allpass filter being a polynomial in $z^N$, and are useful for decimation/interpolation as one can change order of filtering and compression/expansion.