This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ndstop filter transfer function m ay also be obtained from t he corresponding bandpass filter transfer function. T he case of lowpass a nd high pass filters
is similar. I f H LP (s) a nd H H P ( s) a re t he t ransfer functions of a lowpass a nd a
highpass filter respectively (both with the same cutoff frequency), t hen Therefore, a high pass filter transfer function may also be obtained from t he corresponding lowpass filter transfer function. 504
6 7 F requency Response a nd A nalog Filters 7.5 E xercise E 7.2 B utterworth F ilters 505 t Using the qualitative method of sketching the frequency response, show that the system with
the pole-zero configuration in Fig. 7.18a is a highpass filter, and that with the configuration in
Fig. 7.18b is a bandpass filter. 'V t IH0(0)1 IH0°o)1 I
Gp G, Gs s plane
)E . ....•.. jooo
0 oop 00, 0 0-+ 0 (a) t t IH0°o)1 G- Re - + x
p (b) F ig. 7 .18 Pole-zero configuration of a highpass filter in Exercise E7.2. 7 .4-5 (b) IH(joo)1 I
Gp Re - + G, Practical Filters and Their Specifications F or ideal filters everything is black a nd white; t he gains are e ither zero o r u nity
over c ertain b ands. As we saw earlier, real life does n ot p ermit such a world view.
Things have t o b e g ray or shades of gray. I n p ractice, we c an realize a variety of
filter c haracteristics which c an only approach ideal characteristics.
A n i deal filter has a p assband ( unity gain) a nd a s topband (zero gain) w ith
a s udden t ransition from t he p assband t o t he s topband. T here is no t ransition
b and. F or p ractical (or realizable) filters, o n t he o ther h and, t he t ransition from
t he p assband t o t he s topband (or vice versa) is gradual, a nd t akes place over a finite
b and o f frequencies. Moreover, for realizable filters, t he gain c annot b e zero over
a finite b and ( Paley-Wiener condition). As a result, there c an n o t rue s topband
for practical filters. We therefore define a s topband t o b e a b and over which t he
g ain is below s ome s mall n umber G ., as illustrated in Fig. 7.19. Similarly, we define
a p assband t o b e a b and over which t he g ain is between 1 a nd s ome n umber
G p (G p < 1), as shown in Fig. 7.19. We have selected the p assband g ain of u nity
for convenience. I t c ould b e a ny c onstant. Usually t he gains are specified in t erms
o f decibels. T his is simply 20 t imes t he log (to base 10) of t he gain. T hus
G (dB) = 2 0log lO G
A g ain of unity i s 0 d B a nd a g ain of \1'2 is 3.01 d B, usually a pproximated by 3 dB.
Sometimes t he s pecification may be in t erms o f a ttenuation, which is t he n egative of
t he g ain in dB. T hus a g ain of 1/\1'2; t hat is, 0.707, is - 3 dB, b ut is a n a ttenuation
o f 3 dB.
In o ur d esign procedure we assume t hat G p ( minimum p assband g ain) a nd
G s ( maximum s topband g ain) a re specified. F igure 7.19 shows t he p assband,
t he s topband, a nd t he t ransition b and for t ypicallowpass, b andpass, highpass, a nd ( c) F ig. 7 .19 (d) Passband, stopband, and transitionband in vario...
View Full Document