Filtering and Downsampling, Revisited

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Filtering and Downsampling, Revisited

As another example of polyphase filtering, let's return to the example §11.1.3. This time, however, let the FIR lowpass filter h(n) be of length $M=LN, L\in{\bf Z}$ . The polyphase filters, , are each length . Recall that

$\displaystyle H(z) = E_0(z^N) + z^{-1}E_1(z^N) + z^{-2}E_2(z^N) + \cdots + z^{-(N-1)}E_{N-1}(z^N)$

leading to the result shown in Fig.11.12.

**Figure:** Polyphase decomposition of a length FIR filter followed by a downsampler.
$\includegraphics[scale=0.8]{eps/down_FIR_poly}$

Next, we commute the downsampler through the adders and through the upsampled polyphase filters, to obtain Fig.11.13.

**Figure:** Polyphase decomposition of a length FIR filter followed by a downsampler.
$\includegraphics[scale=0.8]{eps/down_FIR_poly_com}$

Commuting the downsampler through the subphase filters to get is an example of a multirate noble identity.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Polyphase Filtering
          Filtering and Downsampling, Revisited