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 $M=LN$ , $L\in\mathbb{Z}$ . In this case, the $N$ polyphase filters, $e_l(n)$ , are each length $L$ .^12.2 Recall that

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

Figure: Polyphase decomposition of a length $M=LN$ FIR filter followed by a downsampler.

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.