Page 216 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Substituting y[- 11 = 1, y[-2]= 2, and X,(z) into the above expression, we get

Thus,

Hence,
y[n] = $ - (f)" + +(f)" nr -2

439. Let x[n] be a causal sequence and
xbl -X(z)
Show that

x[O] = lim X(z)
Z'W
Equation (4.94) is called the initial value theorem for the z-transform.
Since x[n] = 0 for n < 0, we have

As z -+ oa, z-" -+ 0 for n > 0. Thus, we get
lim X( z) = x[O]
2--.m

4.40. Let x[n] be a causal sequence and

Show that if X(z) is a rational function with all its poles strictly inside the unit circle
except possibly for a first-order pole at z = 1, then
lim x[N] = lim (1 -2-')X(z) (4.95)
N-tm 1-1
Equation (4.95) is called the final value theorem for the z-transform.
From the time-shifting property (4.19) we have

The left-hand side of Eq. (4.96) can be written as