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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:

| |
(20) |

With this *Z*, the pulse at time *t*_{n} is compactly represented as *Z*^{n}.
The variable *Z* makes
**Fourier transform**s
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

| |
(21) |

** Next:** Unit circle
** Up:** Convolution and Spectra
** Previous:** Fourier sum
Stanford Exploration Project

10/21/1998