Z Transform, Inversion Integral.
Similar to the inversion integral for the Laplace Transform, there is an inversion
integral for the z transform. It takes the form of a contour integral
shown below. The contour includes the poles of X(z) within it.
Because of the properties of complex functions, the above integral can be rewritten
as a sum of residues as shown below.
Consider the following Z Transform X(z).
Divide X(z) by z to expand in partial fractions.
Substituting into the inversion integral gives:-
and expanding into partial fractions:-
The first integral can be evaluated
and the total integral gives the inverse Z
Note that for this X(z) the inverse is the
same as that obtained by
other methods. The inversion integral
is the "mathematically
correct" way of finding the inverse
Z transform, however partial fraction expansion
is usually preferable.
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© 2000 Cuthbert A. Nyack.