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The frequency function of a pulse at time
is
.The factor
occurs so often in applied work
that it has a name:
| ![\begin{displaymath}
Z \eq e^{i\omega \, \Delta t}\end{displaymath}](img46.gif) |
(20) |
With this Z, the pulse at time tn is compactly represented as Zn.
The variable Z makes
Fourier transforms
look like polynomials,
the subject of a literature called
``Z-transforms.''
The Z-transform is a variant form of the Fourier transform
that is particularly useful for time-discretized (sampled) functions.
From the definition (20), we have
,
,etc.
Using these equivalencies,
equation (19) becomes
| ![\begin{displaymath}
B(\omega) \eq
B(\omega (Z)) \eq
\sum_n \ b_n \ Z^n\end{displaymath}](img49.gif) |
(21) |
Next: Unit circle
Up: Convolution and Spectra
Previous: Fourier sum
Stanford Exploration Project
10/21/1998