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Introduction to Digital Filters
    Transfer Function Analysis
       Shift and Convolution Theorems
          Shift Theorem

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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... ...ty}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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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