[PDF] determine the z transform including the region of convergence

When does a Z-transform converge?

Since the z-transform is a power series, it converges when x[n]z?n x [ n] z ? n is absolutely summable. Stated differently, must be satisfied for convergence. The Region of Convergence has a number of properties that are dependent on the characteristics of the signal, x[n] x [ n]. The ROC cannot contain any poles.

Why is the region of convergence important?

The region of convergence, known as the ROC, is important to understand because it defines the region where the z-transform exists. The z-transform of a sequence is defined as The ROC for a given x[n] x [ n], is defined as the range of z z for which the z-transform converges.

Is Z-transform a rational function of Z?

We can see that the fact that the z-transform is a rational function of z, excludes the points at the circle of convergence as possible points of the ROC7. This fact allows us to easily determine the possible ROC’s for a rational z-transform F (z) by just finding its poles and subsequently the open annulus delimited by them .

Does a Z-transform converge in an open annulus?

From our previous discussion is easy to see that in the cases when the ROC of a z-transform includes an open annulus, the z-transform converges absolutely in that open annulus and what’s more, it converges uniformly. From this it can be proved to be analytic in that open annulus.