If *X*(*z*) represents the rational *z*-transform, it can be expressed as

The poles and zeros are the values of *z* for which the value of *X*(*z*) are ∞ and 0. In ROC *X*(*z*) has the finite value whereas *X*(*z*) has the value ∞ at poles. Therefore, the poles does not lie in the ROC of *X*(*z*). Again, *X*(*z*) has the finite value at zeros. Hence zeros lie in the ROC of *X*(*z*). We can write the Eq. (6.77) in the short hand form as

From Eq. (6.77), we have

From Eq. (6.79), we conclude that

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