Page 190 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Thus,

1 -a-'z z 1
-
X(z) = 1 - - =-- - Izl < I4 (4.52)
1-a-'z 1-a-'z z-a 1-az-'
(b) Similarly,

Again by Eq. (1.91)

Thus,

4.2. A finite sequence x[n] is defined as

N, In IN,
=O otherwise

where N, and N, are finite. Show that the ROC of X(z) is the entire z-plane except
possibly z = 0 or z = m.
From Eq. (4.3)

For z not equal to zero or infinity, each term in Eq. (4.54) will be finite and thus X(z) will
converge. If N, < 0 and N2 > 0, then Eq. (4.54) includes terms with both positive powers of z
and negative powers of z. As lzl- 0, terms with negative powers of z become unbounded,
and as lzl+ m, terms with positive powers of z become unbounded. Hence, the ROC is the
entire z-plane except for z = 0 and z = co. If N, 2 0, Eq. (4.54) contains only negative powers
of z, and hence the ROC includes z = m. If N, I 0, Eq. (4.54) contains only positive powers of
z, and hence the ROC includes z = 0.

4.3. A finite sequence x[n] is defined as

Find X(z) and its ROC.