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Preliminaries 305
1 + δ 1
1
1 − δ 1
Passband
Ripple
|H(e jω )|
Transition
Band
Stopband
Ripple
(a)
δ 2
0
0
0
ω p ω s π
ω
ω
R p
Decibels
A s
(b)
FIGURE 7.1
FIR filter specifications: (a) absolute (b) relative
• band [ω s ,π]iscalled the stopband, and δ 2 is the corresponding tolerance
(or ripple), and
• band [ω p ,ω s ]iscalled the transition band, and there are no restrictions
on the magnitude response in this band.
7.1.2 RELATIVE (DB) SPECIFICATIONS
Atypical absolute specification of a lowpass filter is shown in Figure 7.1b,
in which
• R p is the passband ripple in dB, and
• A s is the stopband attenuation in dB.
The parameters given in these two specifications are obviously related.
Since |H(e jω )| max in absolute specifications is equal to (1 + δ 1 ), we have
and
R p = −20 log 10
1 − δ 1
1+δ 1
> 0(≈ 0) (7.1)
A s = −20 log 10
δ 2
1+δ 1
> 0(≫ 1) (7.2)
□ EXAMPLE 7.1 In a certain filter’s specifications the passband ripple is 0.25 dB, and the stopband
attenuation is 50 dB. Determine δ 1 and δ 2.
Solution
Using (7.1), we obtain
R p =0.25 = −20 log 10
1 − δ 1
1+δ 1
⇒ δ 1 =0.0144
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