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Filtering and Downsampling, Revisited

Let's return to the example of §11.1.3, but this time have the FIR lowpass filter h(n) be length $ M=LN$ , $ L\in{\bf Z}$ . In this case, the $ N$ polyphase filters, $ e_l(n)$ , are each length $ L$ .12.2 Recall that

$\displaystyle H(z) \eqsp E_0(z^N) + z^{-1}E_1(z^N) + \cdots + z^{-(N-1)}E_{N-1}(z^N)$ (12.15)

leading to the result shown in Fig.11.11.

Figure: Polyphase decomposition of a length $ M=LN$ FIR filter followed by a downsampler.
\includegraphics[width=0.7\twidth]{eps/down_FIR_poly}

Figure: Polyphase decomposition of a length $ M=LN$ FIR filter followed by a downsampler.
\includegraphics[width=0.7\twidth]{eps/down_FIR_poly_com}

Next, we commute the $ N$ :$ 1$ downsampler through the adders and upsampled (stretched) polyphase filters $ E_l(z^N)$ to obtain Fig.11.12. Commuting the downsampler through the subphase filters $ E_l(z^N)$ to obtain $ E_l(z)$ is an example of a multirate noble identity.

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Multirate Noble Identities
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Type II Polyphase Decomposition