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
Demonstration of Partial Fraction Expansion
Khan Lecture on Partial Fraction Expansion
Power Series Expansion Method
When the z-transform is defined as a power series in the form
then each term of the sequence
can be determined by looking at the coefficients of the
respective power of
Now look at the Laplace-transform of a finite-length
In this case, since there were no poles, we multiplied the
Now, by inspection, it is clear that
One of the advantages of the power series expansion method is
that many functions encountered in engineering problems have
their power series' tabulated. Thus functions such as log,
sin, exponent, sinh, etc, can be easily inverted.
Contour Integration Method
Without going in to much detail
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 of
complex variable theory and thus will not be addressed in this
Demonstration of Contour Integration
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.