Page 192 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 181

4.5. Show that if x[n] is a right-sided sequence and X(z) converges for some value of z,
then the ROC of X(z) is of the form

where r,, is the maximum magnitude of any of the poles of X(z).
Consider a right-sided sequence x[nl so that

and X(z) converges for (zl = r,. Then from Eq. (4.3)

Now if r, > r,, then

since (r, /r,)-" is a decaying sequence. Thus, X(z) converges for r = r, and the ROC of X(z)
is of the form

Since the ROC of X(z) cannot contain the poles of X(z), we conclude that the ROC of X(z)
is of the form

where r,, is the maximum magnitude of any of the poles of X(z).
If N, < 0, then

That is, X(z) contains the positive powers of z and becomes unbounded at z = m. In this case
the ROC is of the form

From the above result we can tell that a sequence x[n] is causal (not just right-sided) from the
ROC of XCz) if z = oo is included. Note that this is not the case for the Laplace transform.

4.6. Find the z-transform X(z) and sketch the pole-zero plot with the ROC for each of the
following sequences: