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376 Chapter 7 FIR FILTER DESIGN
P7.2 The Type-1 linear-phase FIR filter is characterized by
h(n) =h(M − 1 − n)), 0 ≤ n ≤ M − 1, M odd
Show that its amplitude response H r(ω) isgiven by
H r(ω) =
L∑
n=0
a(n) cos(ωn), L = M − 1
2
where coefficients {a(n)} are obtained as defined in (7.6).
P7.3 The Type-2 linear-phase FIR filter is characterized by
h(n) =h(M − 1 − n), 0 ≤ n ≤ M − 1, M even
1. Show that its amplitude response H r(ω) isgiven by
M/2
∑
H r(ω) = b(n) cos { ω ( )}
n − 1 2
n=1
where coefficients {b(n)} are obtained as defined in (7.10).
2. Show that H r(ω) can be further expressed as
( ) ω ∑ L
H r(ω) =cos ˜b(n) cos(ωn),
2
n=0
L =
M
2 − 1
where coefficients ˜b(n) are given by
b(1) = ˜b(0) + 1 2˜b(1),
b(n) = 1 ] [˜b(n − 1) + ˜b(n) ,
2
b ( )
M 1
2 =
2˜b ( M
− 1) .
2
P7.4 The Type-3 linear-phase FIR filter is characterized by
2 ≤ n ≤
M
2 − 1,
h(n) =−h(M − 1 − n), 0 ≤ n ≤ M − 1, M odd
1. Show that its amplitude response H r(ω) isgiven by
H r(ω) =
(M−1)/2
∑
n=1
c(n) sin(ωn)
where coefficients {c(n)} are obtained as defined in (7.13).
2. Show that H r(ω) can be further expressed as
H r(ω) =sin(ω)
L∑
n=0
˜c(n) cos(ωn), L = M − 3
2
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