
where
~
g
k,1
¼
0, k ¼ 0
(2)
k1
, k ¼ 1, 2, 3, ...
(4:396)
~
g
k,2
¼
0, k ¼ 0, 1
(2)
k2
, k ¼ 2, 3, 4, ...
(4:397)
Combining Equations 4.395 and 4.396, the inverse z-transform is
f
k
¼
0, k ¼ 0
1, k ¼ 1
(2)
k1
þ (2)
k2
, k ¼ 2, 3, 4, ...
8
<
:
(4:398)
Simplifying the expression in Equation 4.398 when k ¼2, 3, 4, . . . gives
f
k
¼
0, k ¼ 0
1, k ¼ 1
( 2)
k2
, k ¼ 2, 3, 4, ...
8
<
:
(4:399)
(b) Long division of the denominator in Equation 4.392 into the numerator results in an infinite
series. The first few terms are
z þ 1
z
2
þ 2z
¼ z
1
z
2
þ 2z
3
4z
4
þ 8z
5
(4:400)
Looking at Equation 4.400, it is possible to recognize a pattern in the coefficients starting with the
z
2
term. This ...