Analog and Digital Filters
591
Observe that at W = 0.
H 3 (W = 0) = 0. Therefore, this function cannot be used to implement an LPF (H 3
(W = π) = 0). As a result, this function can be used to implement either a BS or an
HP fi lter.
The following sequence illustrates a type-3 FIR fi lter:
h(n)
(n)
(n
)
(n
)
(n
)
ϭ
ϩ
ϩ
ϩ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
0.019
0.3593
1 0.023
2
0.3593
3
0.01
9 9
4
(n
)
Ϫ
R.6.123 The type-4 FIR fi lter is defi ned by
h(n) = −h (N − n) for 1 ≤ n ≤ N, with N = even
Then
H(W)
a (n)sin[W(n 1/2)]
ϭ
Ϫ
Ϫ
ր
ր
ր
e
e
jNW
j
n
N
2
2
1
2
4
ϭ
∑
where
a (n)
h
n
4
2
for
1, 2,
ϭ
ϩ Ϫ
ϭ
ϩ
N
n
N
1
2
1
2

 

 
…
Observe that at W = 0, H(W = 0) = 0; therefore, this transfer function cannot be
used to implement an LPF.
The following sequence illustrates a type-4 FIR fi lter:
h(n) = −0.019(n) + 0.3593(n − 1) − 0.3593(n − 3) + 0.019(n − 3)
Analytical examples of each FIR fi lter type are presented in the following points.
R.6.124 For example, analyze by hand a type-1 FIR fi lter with N = 5.
ANALYTICAL Solution
Then
H(z) = h(0) + h(1)z
−1
+ h(2)z
−2
+ h(3)z
−3
+ h(4)z
−4
Note that h(0) = h(4) and h(1) = h(3).
Then
H(z) = [h(0) + h(4)z
−4
] + [h(1)z
−1
+ h(3)z
−3
] + h(2)z
−2
H(z) = h(0)z
−2 (z
2
+ z
−2 ) + h(1)z
−2 [z
1
+ z
−1
] + h(2)z
−2
substituting z by e
jW (z → e
jW )
H(e
jW ) = h(0)e
−2jW (e
jW2
+ e
−jW2 ) + h(1)e
−jW2 (e
−jW
+ e
jW ) + h(2)e
−2jW
CRC_47760_CH006.indd 591
CRC_47760_CH006.indd 591
7/23/2008 2:26:39 PM
7/23/2008 2:26:39 PM