The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is
- $Y(z)=z(X(z)-X(-z)) / 2$
- $Y(z)=\sqrt{z}(X(\sqrt{z})-X(-\sqrt{z})) / 2$
- $Y(z)=z^{2}\left(X\left(z^{2}\right)-X\left(-z^{2}\right)\right) / 2$
- $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$
- $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$