Relation to the z Transform
The Laplace transform is used to analyze continuous-time
systems. Its discrete-time counterpart is the
transform:
If we define
![$ z=e^{sT}$](http://www.dsprelated.com/josimages_new/filters/img1708.png)
, the
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
transform becomes proportional to the
Laplace transform of a sampled continuous-time
signal:
As the
sampling interval ![$ T$](http://www.dsprelated.com/josimages_new/filters/img96.png)
goes to zero, we have
where
![$ t_n\isdef nT$](http://www.dsprelated.com/josimages_new/filters/img1711.png)
and
![$ \Delta t \isdef t_{n+1} - t_n = T$](http://www.dsprelated.com/josimages_new/filters/img1712.png)
.
In summary,
Note that the
plane and
plane are generally related by
In particular, the discrete-time frequency axis
![$ \omega_d \in(-\pi/T,\pi/T)$](http://www.dsprelated.com/josimages_new/filters/img1715.png)
and
continuous-time frequency axis
![$ \omega_a \in(-\infty,\infty)$](http://www.dsprelated.com/josimages_new/filters/img1716.png)
are related
by
For the mapping
![$ z=e^{sT}$](http://www.dsprelated.com/josimages_new/filters/img1708.png)
from the
![$ s$](http://www.dsprelated.com/josimages_new/filters/img1471.png)
plane to the
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
plane to be invertible, it is necessary that
![$ X(j\omega_a )$](http://www.dsprelated.com/josimages_new/filters/img1718.png)
be zero for all
![$ \vert\omega_a \vert\geq \pi/T$](http://www.dsprelated.com/josimages_new/filters/img1719.png)
. If this is true, we say
![$ x(t)$](http://www.dsprelated.com/josimages_new/filters/img1659.png)
is
bandlimited to half the sampling rate. As is well known, this
condition is necessary to prevent
aliasing when
sampling the
continuous-time signal
![$ x(t)$](http://www.dsprelated.com/josimages_new/filters/img1659.png)
at the rate
![$ f_s=1/T$](http://www.dsprelated.com/josimages_new/filters/img265.png)
to produce
![$ x(nT)$](http://www.dsprelated.com/josimages_new/filters/img393.png)
,
![$ n=0,1,2,\ldots\,$](http://www.dsprelated.com/josimages_new/filters/img1720.png)
(see [
84, Appendix G]).
Next Section: Laplace Transform TheoremsPrevious Section: Analytic Continuation