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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:

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

or, using more common notation,

$\displaystyle \zbox {x(n-\Delta) \;\leftrightarrow\; z^{-\Delta} X(z), \; \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=0}^{\infty}x(n-\Delta) z^{-n} \\
&=& \sum_{m=-\Delta}^{\infty} x(m) z^{-(m+\Delta)}
\qquad(m\;\isdef \; n-\Delta) \\
&=& \sum_{m=0}^{\infty}x(m) z^{-m} z^{-\Delta} \qquad \hbox{($\Delta\ge 0$, $x(n)=0$\ for $n<0$)}\\ \\
&=& z^{-\Delta } \sum_{m=0}^{\infty}x(m) z^{-m} \\
&\isdef & z^{-\Delta} X(z), % \quad\pfendmath
\end{eqnarray*}

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


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA