590 Practical MATLAB® Applications for Engineers
R.6.120 Type-1 FIR fi lter is defi ned by the following relation:
h(n) = h(N − n) for 0 ≤ n ≤ N − 1, with N = odd.
Then the amplitude response is given by
H(W) 11 a(n)cos(W)n
e jNW
n
N
2
0
2
∑
where a 1 ( 0 ) = h(N/ 2 ) is the mid-point and a 1 (n) = 2 h((N/ 2 ) − n) for 1 ≤ n ≤ N/ 2.
Type-1 FIR fi lters can be used to design any fi lter type. The following sequence
illustrates a type-1 FIR fi lter:
h(n) 0.09(n) 0.30(n )^1 0.3593(n^2 )0.30(n 3)0.09(nn)^4
R.6.121 The type-2 FIR fi lter is defi ned by
h(n) = h(N − n) for 0 ≤ n ≤ N − 1, with N = even
then
H(W) 22 a(n)cosW(n 12 )
e jNW
n
N
2
0
12
[]
()
∑
where
an h
N
2 () nn (N)
2
1
2
12 1 2
for = , ,...
Since the response is defi ned for −π ≤ W ≤ π, then
H(W^2 )^0
Therefore, type-2 FIR fi lters cannot be used to implement an HP or a BP fi lter.
The following sequence illustrates a type-2 FIR fi lter:
h(n) = −0.09δ(n) + 0.30δ(n − 1 ) + 0.3593δ(n − 2 ) + 0.3593δ(n − 3 )
- 0.30δ(n − 4 ) −0.09δ(n − 5 )
R.6.122 The type-3 FIR fi lter is defi ned by
h(n) = −h(N − n) for 0 ≤ n ≤ N, with N = odd
H (W) 33 [a (n)sin(Wn)]
eejNW j
n
N
22
1
2
∑
where a 3 (n) = 2 h (^) [ N__
2
− n] for n = 1, 2, ... N/2)