Digital Signal Processing Reference
In-Depth Information
For the unit impulse function occurring at
n = 0(n
0
= 0,),
d[n]
Î
z
"
1.
(11.20)
Additional transform pairs will now be derived. Consider the
z
-transform pair
from Table 11.1:
z
z - a
.
a
n
Î
z
"
(11.21)
a
n
Recall that
is exponential and can be expressed as
a
n
= (e
b
)
n
= e
bn
, b = ln a.
Pair (11.21) can then be expressed as
z
z - a
=
z
z - e
b
.
= e
bn
Î
z
"
a
n
(11.22)
We now consider sinusoidal functions. By Euler's identity,
e
jbn
+ e
-jbn
2
cos bn =
.
Hence,
1
2
[
z
[e
jbn
] +
z
[e
-jbn
]],
z
[cos bn] =
by the linearity property (11.8). Then, from (11.22), with
b = jb,
z - e
-jb
+ z - e
jb
(z - e
jb
)(z - e
-jb
)
1
2
z
z - e
jb
+
z
z - e
-jb
z
2
B
R
B
R
z
[cos bn] =
=
2z - (e
jb
+ e
-jb
)
z(z - cos b)
z
2
B
R
=
=
- 2z cos b + 1
,
z
2
- (e
jb
+ e
-jb
)z + 1
z
2
where Euler's relation was used in the last step. By the same procedure, since
sin bn = (e
jbn
- e
-jbn
)/2 j,
z - e
-jb
- z + e
jb
(z - e
jb
)(z - e
-jb
)
1
2j
z
z - e
jb
-
z
z - e
-jb
z
2j
B
R
B
R
z
[sin bn] =
=
e
jb
- e
-jb
z
2j
z sin b
B
R
=
=
- 2z cos b + 1
.
z
2
- (e
jb
+ e
-jb
)z + 1
z
2
