If the [[discrete-time signal]] $x(n)$ is shifted by $n_{0}$ samples in the time domain, its z-transform becomes: $ x(n - n_0) \xrightarrow[Z]{} z^{-n_0} X(z) $ This is a direct result of the application of the [[z-transform]] equation to the signal $y(n)=x(n-n_{0})$: $ \begin{align} Y(z) &= \sum_{n = -\infty}^{+\infty} y(n) z^{-n} \\ &= \sum_{n = -\infty}^{+\infty} x(n-n_{0}) z^{-n} \\ &= \sum_{m = -\infty}^{+\infty} x(m) z^{-(m+n_{0})} \\ &= z^{-n_{0}} \sum_{m = -\infty}^{+\infty} x(m) z^{-m} \\ &= z^{-n_{0}} X(z) \end{align} $