Z-Transforms and Solution of Difference Equations13-11
9. Evaluate z (〈na
n
〉) using the definition of Z-transform.
Ans:
2
()
a
z
za−
Recurrence Formula
10. Using the recurrence formula
Z ( 〈n
p
〉) =
(/)
z
ddz
1
[()]
p
Zn
−
〈〉
, fi nd Z (〈n
3
〉) and Z ( 〈n
4
〉) given that
Z ( 〈n
2
〉) =
2
3
2
(1)
z
z
+
−
.
Ans:
32
3
4
432
4
5
4
();
(1)
1111
()
(1)
zzz
Zn
z
zzzz
Zn
z
++
〈〉=
−
+++
〈〉=
−
13.6 INVERSEZ-TRANSFORM
We have already defined the inverse Z-transforms of
u
(z), a function of a complex variable
z, by
Z
−1
(
u
(z)) =〈u
n
〉 (13.61)
which exists provided the series,
0
n
n
n
uz
∞
−
=
∑
con-
verges and 〈u
n
〉 is the sequence generating the series.
We will now consider methods for finding the
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