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.

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