Shift Theorem

The shift theorem says that a delay of $\Delta$ samples in the time domain corresponds to a multiplication by $z^{-\Delta}$ in the frequency domain:

$${\cal Z}_z\{$$SHIFT$$_\Delta\{x\}\} = z^{-\Delta} X(z), \quad \Delta\ge 0$$

or, using more common notation,

$$\zbox {x(n-\Delta) \leftrightarrow z^{-\Delta} X(z), \quad \Delta\ge 0.}$$

Thus, $x(\cdot - \Delta)$, which is the waveform $x(\cdot)$ delayed by $\Delta$ samples, has the z transform $z^{-\Delta}X(z)$.

Proof:

$$\begin{eqnarray*} {\cal Z}_z\{\mbox{{\sc Shift}}_\Delta\{x\}\} &\isdef & \sum_{n... ... } \sum_{m=0}^{\infty}x(m) z^{-m} \\ &\isdef & z^{-\Delta} X(z) \end{eqnarray*}$$

where we used the causality assumption $x(m)=0$ for $m<0$.