Editing Discrete-time Fourier transform (section)

== Relationship to the Z-transform ==
<math>S_{2\pi}(\omega)</math> is a [[Fourier series]] that can also be expressed in terms of the bilateral [[Z-transform]].&nbsp; I.e.''':'''

:<math>S_{2\pi}(\omega) = \left. S_z(z) \, \right|_{z = e^{i \omega}} = S_z(e^{i \omega}),</math>

where the <math>S_z</math> notation distinguishes the Z-transform from the Fourier transform.  Therefore, we can also express a portion of the Z-transform in terms of the Fourier transform''':'''

:<math>
\begin{align}
S_z(e^{i \omega}) &= \ S_{1/T}\left(\tfrac{\omega}{2\pi T}\right)
 \ = \ \sum_{k=-\infty}^{\infty} S\left(\tfrac{\omega}{2\pi T} - k/T\right)\\
&= \sum_{k=-\infty}^{\infty} S\left(\tfrac{\omega - 2\pi k}{2\pi T} \right).
\end{align}
</math>

Note that when parameter {{mvar|T}} changes, the terms of <math>S_{2\pi}(\omega)</math> remain a constant separation <math>2 \pi</math> apart, and their width scales up or down. The terms of {{math|''S''<sub>1/''T''</sub>(''f'')}} remain a constant width and their separation {{math|1/''T''}} scales up or down.