Sliding Polyphase Filter Bank

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Sliding Polyphase Filter Bank

**Figure:** Simplified filter bank when inverts and there are no downsamplers or upsamplers ().
$\begin{figure}\input fig/polyNchanIR1.pstex_t \end{figure}$

When , there is no downsampling or upsampling, and the system further reduces to the case shown in Fig.11.25. Working backward along the output delay chain, the output sum can be written as

$\begin{eqnarray*} \hat{X}(z) &=& \left[z^{-0}z^{-(N-1)} + z^{-1}z^{-(N-2)} + z^{... ...z^{-1} + z^{-(N-1)}z^{-0} \right] X(z)\\ &=& N z^{-(N-1)} X(z) \end{eqnarray*}$

Thus, when , the output is

$\displaystyle {\hat x}(n) = N x(n-N+1)$

and we again have perfect reconstruction.

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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
       Perfect Reconstruction Filter Banks
          Sliding Polyphase Filter Bank