# 1.9 Polyphase decimation filter

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Implementation of polyphase decimation filters.

## Polyphase decimation

Recall the standard decimation method in [link] .

Note that this procedure is computationally inefficient because it discards the majority of the computed filteroutputs. Through the use of the Noble identities, it is possible to rearrange [link] so that filter outputs are not discarded.

In order to apply the Noble identity for decimation, we must transform $H(z)$ into its upsampled polyphase components ${H}_{p}(z^{M})$ , $p=\{0, \dots , M-1\}$ , defined previously in the context of polyphase interpolation .

$H(z)=\sum h(n)z^{-n}=\sum \sum_{p=0}^{M-1} h(kM+p)z^{(-(kM)-p)}$
via $≔(k, \lfloor \frac{n}{M}\rfloor )$ , $≔(p, n\mod M)$
$H(z)=\sum_{p=0}^{M-1} \sum_{k} {h}_{p}(k)z^{-(kM)}z^{-p}$
via $≔({h}_{p}(k), h(kM+p))$
$H(z)=\sum_{p=0}^{M-1} {H}_{p}(z^{M})z^{-p}$
Using these unsampled polyphase components, the preceding block diagram can be redrawn as [link] .

Applying the Noble identity for decimation to [link] yields [link] . The ladder of delays and downsamplers on the left below accomplishes a form ofserial-to-parallel conversion.

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