Polyphase Decomposition into N Channels

For the general case of arbitrary $N$ , the basic idea is to decompose $x[n]$ into its periodically interleaved subsequences:

The polyphase decomposition into $N$ channels is given by

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

where the subphase filters are

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

with

$$e_l(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}h(Nn+l). \qquad\hbox{($l$th subphase filter)}$$

The signal $e_l(n)$ can be obtained by passing $h(n)$ through an advance of $l$ samples, followed by downsampling by the factor $N$ :