Page 180 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 169

transform, specification of the z-transform requires both the algebraic expression and the
ROC.

C. Properties of the ROC:
As we saw in Examples 4.1 and 4.2, the ROC of X(z) depends on the nature of x[n].
The properties of the ROC are summarized below. We assume that X(Z) is a rational
function of z.

Property 1: The ROC does not contain any poles.
Property 2: If x[n] is a finite sequence (that is, x[n] = 0 except in a finite interval Nl ~n s N,,
where N, and N, are finite) and X(z) converges for some value of z, then the ROC is
the entire z-plane except possibly z = 0 or z = co.
Property 3: If x[n] is a right-sided sequence (that is, x[n] = 0 for n < N, < 03) and X(z) converges
for some value of z, then the ROC is of the form

where r,, equals the largest magnitude of any of the poles of X(z). Thus, the ROC is
the exterior of the circle lzl= r,, in the z-plane with the possible exception of z = m.
Property 4: If x[n] is a left-sided sequence (that is, x[nl = 0 for n > N, > - 03) and X(z) converges
for some value of z, then the ROC is of the form

where r,, is the smallest magnitude of any of the poles of X(z). Thus, the ROC is the
interior of the circle lzlE rmin in the z-plane with the possible exception of z = 0.
Property 5: If x[n] is a two-sided sequence (that is, x[n] is an infinite-duration sequence that is
neither right-sided nor left-sided) and X(z) converges for some value of z, then the
ROC is of the form

where r, and r, are the magnitudes of the two poles of X(z). Thus, the ROC is an
annular ring in the z-plane between the circles lzl= r, and lzl = r2 not containing any
poles.
Note that Property 1 follows immediately from the definition of poles; that is, X(z)
is infinite at a pole. For verification of the other properties, see Probs. 4.2 and 4.5.

4.3 z-TRANSFORMS OF SOME COMMON SEQUENCES
A. Unit Impulse Sequence 61 nl:
From definition (1.45) and (4.3)

m
X(z) = 6[n] z-" = z-O = 1 all z
n= -m