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Properties of Linear-phase FIR Filters 311
Magnitude Response
Amplitude Response
|H|
Hr
0
0
0 2/3 1
frequency in π units
0 2/3 1
frequency in π units
Piecewise Linear Phase Response
Linear Phase Response
angle in π units
2/3
0
−2/3
angle in π units
0
−2/3
0 2/3 1
frequency in π units
0 2/3 1
frequency in π units
FIGURE 7.2 Frequency responses in Example 7.3
Type-1 linear-phase FIR filter: Symmetrical impulse response,
M odd In this case β =0,α =(M − 1)/2 isaninteger, and h(n) =
h(M − 1 − n), 0 ≤ n ≤ M − 1. Then we can show (see Problem P7.2)
that
⎡
⎤
(M−1)/2
∑
H(e jω )= ⎣ a(n) cos ωn⎦ e −jω(M−1)/2 (7.6)
n=0
where sequence a(n) isobtained from h(n) as
( ) M − 1
a(0) = h
: the middle sample
2
( )
M − 1
a(n) =2h − n , 1 ≤ n ≤ M − 3
2
2
Comparing (7.5) with (7.6), we have
H r (ω) =
(M−1)/2
∑
n=0
(7.7)
a(n) cos ωn (7.8)
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