Sliding Polyphase Filter Bank

When $R=1$ , there is no downsampling or upsampling, and the system further reduces to the case below:

Working backward along the output delay chain, the output sum can be written as

$$\begin{eqnarray*} {\hat X}(z) &=& \left[z^{-0}z^{-(N-1)} + z^{-1}z^{-(N-2)} + z^{-2}z^{-(N-3)} + \cdots \right.\\ & & \left. + z^{-(N-2)}z^{-1} + z^{-0}z^{-(N-1)} \right] X(z)\\ &=& N z^{-(N-1)} X(z) \end{eqnarray*}$$

Thus, when $R=1$ , the output is

$${\hat x}(n) = N x(n-N+1)$$

and we again have perfect reconstruction.