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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       Polyphase Filtering
          N-Channel Polyphase Decomposition

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N-Channel Polyphase Decomposition

Figure 10.9: 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.10.9. The polyphase decomposition into $ N$ channels is given by

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

where the subphase filters are defined by

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

with

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

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.10.10.


\begin{psfrags} % latex2html id marker 26744\psfrag{M}{{\normalsize $N$}}\p... ...mples followed by a downsampling by the factor $N$.} \end{figure} \end{psfrags}

For $ N=3$, we have the system diagram shown in Fig.10.11.


\begin{psfrags} % latex2html id marker 26760\psfrag{X}{{\large $H(z)\quad{}$}}... ...-Channel Polyphase Decomposition and Reconstruction} \end{figure} \end{psfrags}

Previous: Two-Channel Case
Next: Type II Polyphase Decomposition

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About the Author: 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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