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# Region of Convergence and Examples

Whether the z-transform of a signal exists depends on the complex variable as well as the signal itself. exists if and only if the argument is inside the region of convergence (ROC) in the z-plane, which is composed of all values for the summation of the Z-transform to converge. The ROC of the Z-transform is determined by (a circle), the magnitude of variable , while the ROC for the Laplace transform is determined by , (a vertical line), the real part of . This formula is always needed in the examples:

Example 1: The Z transform of a right sided signal is

For this summation to converge, i.e., for to exist, it is necessary to have , i.e., the ROC is . As a special case when , and we have

Example 2: The Z-transform of a left sided signal is:

For the summation above to converge, it is required that , i.e., the ROC is . Comparing the two examples above we see that two different signals can have identical z-transform, but with different ROCs.

Example 3: Find the inverse of the given z-transform . Comparing this with the definition of z-transform:

we get

In general, we can use the time shifting property

to inverse transform the given above to directly.

Example 4: Sometimes the inverse transform of a given can be obtained by long division.

By a long division, we get

which converges if the ROC is , i.e., and we get

. Alternatively, the long division can also be carried out as:

which converges if the ROC is , i.e., and we get

Next: Zeros and Poles of Up: Z_Transform Previous: Conformal Mapping between S-Plane
Ruye Wang 2014-10-28