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FIR Filter Designs for Sampling Rate Conversion 513
Input Signal: x1(n) = cos(πn/8)
Input Signal: x2(n) = cos(πn/2)
1
1
Amplitude
0
Amplitude
0
−1
−1
0 8 16 24 32
n
0 8 16 24 32
n
Output signal: y1(n): D = 2
Output signal: y2(n): D = 2
1
1
Amplitude
0
Amplitude
0
−1
−1
0 4 8 12 16
m
FIGURE 9.26 Signal plots in Example 9.12
0 4 8 12 16
m
up to ω x,s . Let
ω p = ω x,p and ω s =
( 2π
D − ω x,s
)
(9.56)
be the passband and stopband edge frequencies, respectively, of the lowpass
linear-phase FIR filter given in (9.51). Then we have the following
filter design specifications:
H r (ω) ≤ 1 ± δ 1 for |ω| ∈[0,ω p ]
H r (ω) ≤±δ 2
for |ω| ∈[ω s ,π]
(9.57)
where ω p and ω s are as given in (9.56) and δ 1 and δ 2 are the passband and
stopband ripple parameters of the lowpass FIR filter, respectively. Note
that it does not matter what the spectrum X(ω) is. We simply require
that the product X(ω)H(ω) bevery small beginning at ω | =2π/D − ω x,s
so that k ≠0terms in (9.53) do not provide significant contribution in
the band [−ω x,s ,ω x,s ], which is required to be free of aliasing.
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