Page 113 - Schaum's Outline of Theory and Problems of Signals and Systems

102 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

where a is a constant. Find y[n] with the auxiliary condition y[- l] = y- , and

x[n] = Kbnu[n] (2.150)
Let yLn1 =ypLn1 +yhLn1
where y,[n] is the particular solution satisfying Eq. (2.149) and yh[n] is the homogeneous
solution which satisfies
y[n] -ay[n - 11 =O
Assume that
y,[n] = Abn nrO

Substituting Eq. (2.1521 into Eq. (2.1491, we obtain
Abn -aAbn-' =Kbn
from which we obtain A = Kb/(b - a), and

To obtain yh["], we assume
y,[n] = Bzn
Substituting this into Eq. (2.151) gives

from which we have z = a and
=Ban
~~[n]
Combining yp[n] and yh[n], we get
K
b"+l
y[n] =Ban + - n20
b-a
In order to determine B in Eq. (2.155) we need the value of y[O]. Setting n = 0 in Eqs. (2.149)
and (2.1501, we have
Y[O] -ay[-11 =y[O] -ay-, =x[O] =K
or y[O] =K+ay-,
Setting n = 0 in Eq. (2.155), we obtain
b
y[O] =B +K-
b-a
Therefore, equating Eqs. (2.156) and (2.1571, we have

from which we obtain

B =ay-, - K-
b-a
Hence, Eq. (2.155) becomes