When dealing with
linear time-invariant systems the z-transform is often of the form
This can also expressed as
where
represents the nonzero zeros of
and
represents the nonzero poles.
If
then
can be represented as
This form allows for easy inversions of each term of the sum
using the
inspection
method and the
transform table . If the numerator is
a polynomial, however, then it becomes necessary to use
partial-fraction
expansion to put
in the above form. If
then
can be expressed as
Find the inverse z-transform of
where the ROC is
.
In this case
,
so we have to use long division to get
Next factor the denominator.
Now do partial-fraction expansion.
Now each term can be inverted using the inspection method
and the Laplace-transform table. Thus, since the ROC is
,
Interactive experiment illustrating how the Partial Fraction Expansion method is used to solve a variety of numerator and denominator problems. (To view and interact with
the simulation, download the free Mathematica player athttp://www.wolfram.com/products/player/download.cgi)
One of the advantages of the power series expansion method is
that many functions encountered in engineering problems havetheir power series' tabulated. Thus functions such as log,
sin, exponent, sinh, etc, can be easily inverted.
where
is a counter-clockwise contour in the ROC of
encircling the origin of the s-plane. To further expand on
this method of finding the inverse requires the knowledge ofcomplex variable theory and thus will not be addressed in this
module.
Demonstration of contour integration
Interactive experiment illustrating how the contour integral is applied on a simple example. For a more in-depth discussion of this method, some background in complex analysis is required. (To view and interact with
the simulation, download the free Mathematica player athttp://www.wolfram.com/products/player/download.cgi)
Conclusion
The Inverse Laplace-transform is very useful to know for the purposes of designing a filter, and there are many ways in which to calculate it, drawing from many disparate areas of mathematics. All nevertheless assist the user in reaching the desired time-domain signal that can then be synthesized in hardware(or software) for implementation in a real-world filter.