Editing Laplace transform (section)

=== Z-transform ===
{{further|Z-transform#Relationship to Laplace transform}}

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of
<math display=block> z \stackrel{\mathrm{def} }{ {}={} } e^{sT} ,</math>
where {{math|1=''T'' = 1/''f<sub>s</sub>''}} is the [[sampling interval]] (in units of time e.g., seconds) and  {{math|''f<sub>s</sub>''}} is the [[sampling rate]] (in [[samples per second]] or [[hertz]]).

Let
<math display=block> \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\  \sum_{n=0}^{\infty}  \delta(t - n T) </math>
be a sampling impulse train (also called a [[Dirac comb]]) and
<math display=block>\begin{align}
  x_q(t)   &\stackrel{\mathrm{def} }{ {}={} }  x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty}  \delta(t - n T) \\
           &= \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T)
\end{align}</math>
be the sampled representation of the continuous-time {{math|''x''(''t'')}}
<math display=block> x[n] \stackrel{\mathrm{def} }{ {}={} }  x(nT) ~.</math>

The Laplace transform of the sampled signal {{math|''x''<sub>''q''</sub>(''t'') }} is
<math display=block>\begin{align}
  X_q(s) &= \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\
         &= \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] e^{-n s T}~.
\end{align}</math>

This is the precise definition of the unilateral Z-transform of the discrete function {{math|''x''[''n'']}}

<math display=block> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>
with the substitution of {{math|''z'' → ''e''<sup>''sT''</sup>}}.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
<math display=block>X_q(s) = X(z) \Big|_{z=e^{sT}}.</math>

The similarity between the Z- and Laplace transforms is expanded upon in the theory of [[time scale calculus]].