Polyphase structure for a decimator

The decimator block diagram is shown below: it consists of an anti-aliasing filter, H(z), which could be an FIR or an IIR filter, followed by an M-fold down sampler.

We replace the filter H(z) by its M-component polyphase decomposition

The sub filters P₀(z^M), P₁(z^M), ..., P_M-1(z^M) could be FIR or IIR depending on H(z). The block diagram then appears as below.

 We can use identity #1 to move the down-sampler to the immediate right of Pm (z^M) in every branch, and then use identity #3 to change the down-sampler from the immediate right to the immediate left of Pm (.) while at the same time modifing Pm (z^M) to m(z). The result seems as below. It may be seen from that diagram that in this structure the number of multiplications is replaced by a factor of M.

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