Sliding Polyphase Filter Bank
When
, 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*}](http://www.dsprelated.com/josimages_new/sasp2/img2121.png)
Thus, when
, the output is
![]() |
(12.57) |
and we again have perfect reconstruction.
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Hopping Polyphase Filter Bank
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Simple Examples of Perfect Reconstruction