N-Channel Polyphase Decomposition

Figure 11.9: Schematic illustration of three interleaved polyphase signal components.

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

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

where the subphase filters are defined by

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

with

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