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Sliding Polyphase Filter Bank

Figure: Simplified filter bank when $ R(z)$ inverts $ E(z)$ and there are no downsamplers or upsamplers ($ R=1$ ).
\includegraphics{eps/polyNchanIR1}

When $ R=1$ , there is no downsampling or upsampling, and the system further reduces to the case shown in Fig.11.24. 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^{-2}z^{-(N-3)} + \cdots \right.\\
& & \left. + z^{-(N-2)}z^{-1} + z^{-(N-1)}z^{-0} \right] X(z)\\
&=& N z^{-(N-1)} X(z).
\end{eqnarray*}

Thus, when $ R=1$ , the output is

$\displaystyle {\hat x}(n) \eqsp N x(n-N+1)$ (12.57)

and we again have perfect reconstruction.


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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