Page 217 - Schaum's Outline of Theory and Problems of Signals and Systems

THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

If we now let z ---, 1, then from Eq. (4.96) we have

lim (I -2-')X(z) = lim (x[n] -x[n - I]} = lim x[N]
z-+ 1 N-+m N+w
n =O

Supplementary Problems

4.41. Find the z-transform of the following x[n]:
(a) x[nl= (;,I, - $1
(b) x[n] = 2S[n + 21 - 3S[n - 21
(c) x[n] = 3(- f)"u[n] - 2(3)"u[-n - I]
(d) x[nl= 3(i)"u[nl- 2(a)"u[-n - 11
Am. (a) X(z) = f + z-' - $z-', 0 < lzl

(d) X( z) does not exist.

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

IzI<rmin or O<IzI<rmin
where rmin is the smallest magnitude of any of the poles of X(z).
Hint: Proceed in a manner similar to Prob. 4.5.

4.43. Given

(a) State all the possible regions of convergence.
(b) For which ROC is X(z) the z-transform of a causal sequence?
Ans. (a) 0 < lzl < 1,l < lzI< 2,2 < Izk 3, lzl> 3
(b) lz1>3

4.44. Verify the time-reversal property (4.23), that is,

Hint: Change n to -n in definition (4.3).