Solution of ECE 316 Test #9 S03 3/26/03
( ) =
1

2
1.

 z
−
2 z


1
3


then, by finding the partial fraction expansion of this improper fraction in z two different ways, its inverse z transform,
[ ]
can be written in two different forms, h
[ ] =



A


1
2
 n
 +
B


−
1
3

 n


 u
[ ]
and h
[ ] = δ [ ] +



C


1
2

 n
−
1
+
D


−
1
3

 n
−
1


 u
[ n
−
1
]
.
Find A, B, C, and D (2 pts each).
H
( )
= z

 z
− z
1
2
 

3
1
3


= z
5
−
1
2
2
+ z
5
+
1
3
⇒
H
( ) = z
3 z
5
−
1
2
+ z
2 z
5
+
1
3
⇒ h
[ ] =



3
5


1
2
 n
 +
2
5


−
1
3

 n


 u
[ ]
A
= =
5
0 6
2
5
( ) =

 z
− z
2
1
2
 

1
3


= z
2 z
2 −
1 z
−
6
1
6 h
[ ] = δ [ ] +



3
10


1
2

 n
−
1
−
2
15


−
1
3

 n
−
1


 u
[ n
−
1
]
C
=
3
=
10
0 3
= −
2
= −
15
0 1333
K
1
6

 z
− z
+
1
1
2
 

1
3


1
6



9
5 z
−
1
2
− z
4
5
+
1



3
Circle two correct answers.
2.
The
ω
(2 pts) Which of these lines in the s plane maps into the unit circle in the z plane?
axis and The line segment 0
2
π σ
=
0
T s
1