Example: Downsampling by 2

As an example, when $N=2$ , $y[n] = x[2n]$ , and
(since $W_2\mathrel{\stackrel{\mathrm{\Delta}}{=}}e^{-j2\pi/2}=-1$ )

$$\begin{eqnarray*} Y(z) &=& \frac{1}{2}\left[X\left(W^0_2 z^{1/2}\right) + X\left(W^1_2 z^{1/2}\right)\right] \\ [0.1in] &=& \frac{1}{2}\left[X\left(e^{-j2\pi 0/2} z^{1/2}\right) + X\left(e^{-j2\pi 1/2}z^{1/2}\right)\right] \\ [0.1in] &=& \frac{1}{2}\left[X\left(z^{1/2}\right) + X\left(-z^{1/2}\right)\right] \\ [0.1in] &=& \frac{1}{2}\left[\hbox{\sc Stretch}_2(X) + \hbox{\sc Stretch}_2\left(\hbox{\sc Shift}_\pi(X)\right)\right] \end{eqnarray*}$$