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312 Chapter 7 FIR FILTER DESIGN
Type-2 linear-phase FIR filter: Symmetrical impulse response,
M even In this case again β =0,h(n) =h(M−1−n), 0 ≤ n ≤ M−1,
but α =(M −1)/2isnot an integer. Then we can show (see Problem P7.3)
that
⎡
M/2
∑
{ (
H(e jω )= ⎣ b(n) cos ω n − 1 2)} ⎤ ⎦ e −jω(M−1)/2 (7.9)
where
n=1
( ) M
b(n) =2h
2 − n , n =1, 2,..., M 2
(7.10)
Hence
Note:
At ω = π we get
M/2
∑
H r (ω) =
M/2
n=1
∑
H r (π) =
n=1
{ (
b(n) cos ω n − 1 )}
2
{ (
b(n) cos π n − 1 )}
=0
2
(7.11)
regardless of b(n) orh(n). Hence we cannot use this type (i.e., symmetric
h(n), M even) for highpass or bandstop filters.
Type-3 linear-phase FIR filter: Antisymmetric impulse response,
M odd In this case β = π/2, α =(M − 1)/2 isaninteger, h(n) =
−h(M − 1 − n), 0 ≤ n ≤ M − 1, and h((M − 1)/2) = 0. Then we can
show (see Problem P7.4) that
⎡
⎤
(M−1)/2
∑
H(e jω )= ⎣ c(n) sin ωn⎦ e j[ π M−1
2
−(
2 )ω] (7.12)
where
and
n=1
( )
M − 1
c(n) =2h − n , n =1, 2,..., M − 1
2
2
H r (ω) =
(M−1)/2
∑
n=1
(7.13)
c(n) sin ωn (7.14)
Note: At ω =0and ω = π we have H r (ω) =0, regardless of c(n) or
h(n). Furthermore, e jπ/2 = j, which means that jH r (ω) ispurely imaginary.
Hence this type of filter is not suitable for designing a lowpass filter
or a highpass filter. However, this behavior is suitable for approximating
ideal digital Hilbert transformers and differentiators. An ideal Hilbert
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