Z Transform, Inversion Integral.

Cuthbert Nyack
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 Transform.
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.
Return to main page
Return to page index
© 2000 Cuthbert A. Nyack.