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N-Channel Polyphase Decomposition

Figure 11.9: Schematic illustration of three interleaved polyphase signal components.
\includegraphics[scale=0.8]{eps/polytime}

For the general case of arbitrary $ N$ , the basic idea is to decompose $ x(n)$ into its periodically interleaved subsequences, as indicated schematically in Fig.11.9. The polyphase decomposition into $ N$ channels is given by

$\displaystyle H(z) \eqsp \sum_{l=0}^{N-1} z^{-l}E_l(z^N) \protect$ (12.11)

where the subphase filters are defined by

$\displaystyle E_l(z) \eqsp \sum_{n=-\infty}^{\infty}e_l(n)z^{-n},\; l=0,1,\ldots,N-1,$ (12.12)

with

$\displaystyle e_l(n) \isdefs h(Nn+l). \qquad\hbox{($l$th subphase filter)}.$ (12.13)

The signal $ e_l(n)$ can be obtained by passing $ h(n)$ through an advance of $ l$ samples, followed by downsampling by the factor $ N$ . as shown in Fig.11.10.


\begin{psfrags}
% latex2html id marker 30222\psfrag{M}{{\normalsize $N$}}\psfrag{ztl}{{\Large $z^l$}}\psfrag{h[n]}{{\Large $h(n)$}}\psfrag{eln}{{\Large $e_l(n)$}}\begin{figure}[htbp]
\includegraphics[width=0.5\twidth]{eps/polypick}
\caption{Advance by $l$\ samples followed by a downsampling by the factor $N$.}
\end{figure}
\end{psfrags}


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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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