If the $Z$-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the $Z$ transform of the signal $y[n]=x[2 n]$ is
- $Y(z)=X\left(z^{2}\right)$
- $Y(z)=\frac{1}{2}\left[X\left(z^{-1 / 2}\right)+X\left(-z^{-1 / 2}\right)\right]$
- $Y(z)=\frac{1}{2}\left[X\left(z^{1 / 2}\right)+X\left(-z^{1 / 2}\right)\right]$
- $Y(z)=\frac{1}{2}\left[X\left(z^{2}\right)+X\left(-z^{2}\right)\right]$