Analog and Digital Filters 565
R.6.16 The magnitude plot is usually given in decibels. Recall that a positive value in deci-
bels indicates amplifi cation and a negative value indicates attenuation.
Recall also that the dB is defi ned asdB = 20 log 10 [H(jw)]R.6.17 Real fi lters are specifi ed by means of tolerances (or fl uctuations) in the pass
and stop bands, referred to as ripples, which is also expressed in decibels, in
terms of its peak or the maximum value in the pass-band region denoted by αp,
and with the minimum attenuation in the stop band denoted by αs defi ned in
R.6.19.
R.6.18 For the case of the LPF illustrated in Figure 6.2, the equations below are the type
used to defi ne the tolerances in the pass, stop, and transition bands.
a. 1 − δp ≤ H(jw) ≤ 1 + δp for w ≤ wp (pass-band region)
b. H(jw) < δs for ws < w < ∞ (stop-band region)
c. δsH(jw)< 1 − δp for wp < w < ws (t ra n sit ion-ba nd reg ion)
where δp is called the pass-band ripple, δs the stop-band ripple, wp the fi lter’s cutoff
frequency, and ws the fi lter’s stop-band frequency.
R.6.19 The ripple tolerances in the pass and stop bands denoted by δp and δs are specifi ed
in decibels as
a. αp = − 20 log 10 (1 − δp) dB
b. αs = − 20 log 10 (δs) dB
R.6.20 The fi lter characteristic may be specifi ed in terms of the loss function L(jw) instead
of the gain function H(jw).
R.6.21 The loss function is defi ned as the inverse of the gain or transfer function. Then
L(jw) = ____H(jw)^1 , or
Vi(jw)
____V
o(jw)
, given in decibels as− 20 log 10 H(jw)Note that a 0 dB (gain or loss) implies no gain (or loss), meaning that the output
is equal to its input, or the fi lter (system) acts like a short circuit.
R.6.22 Filters are classifi ed according to the pass-band regions as
a. LPF
b. High-pass (HPF)
c. Band-reject (BRF) or band-stop (BSF)
d. Band-pass (BPF)
The magnitude and phase plots of these fi lters are shown in Figures 6.3 and 6.4.
R.6.23 Note that the concept of BW applies only to the LPF and BPF and is defi ned as
follows:BW (LPF) = wpBW (BPF) = wu − wl.R.6.24 More complicated fi lters with multiple pass- or stop-band regions can be created
using the basic four prototype structures defi ned in R.6.22 (LPF, HPF, BPF, BRF).