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506 Chapter 9 SAMPLING RATE CONVERSION
X(ω x )
We will allow filter to
substantially change
this band.
ω x
−π −ω −ωx,s x,p
0
ω x,p π
ω x,s
(a)
V(ω y )
−π
− 2π
I
− π I
0
(b)
π
I
ω x,p
I
ω x,s
I
2π
I
2π − ω x,s
I
π
ω y
FIGURE 9.22
Frequency parameters: (a) signal, (b) filter
be the passband and stopband edge frequencies, respectively, of the lowpass
linear-phase FIR filter given by
H(ω) =H r (ω)e jθ(ω) (9.51)
where H r (ω) isthe real-valued amplitude response and θ(ω) isthe unwrapped
phase response. Then we have the following filter design specifications:
1
I H r(ω) ≤ 1 ± δ 1 for |ω| ∈[0,ω p ]
1
I H r(ω) ≤±δ 2
for |ω| ∈[ω s ,π]
(9.52)
where ω p and ω s are as given in (9.50) and δ 1 and δ 2 are the passband
and stopband ripple parameters, respectively, of the lowpass FIR filter.
Comment: Instead of beginning the stopband at π/I, wewere able to
shift it to (2π − ω s ) /I. Ifω x,s ≪ π, then this will be an important consideration
to lower filter order. However, in the worst-case scenario of
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