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Unformatted text preview: osite effect of suppressing t he gain in t he vicinity ofwo, as shown
in Fig. 7.l4e). A zero on t he i maginary axis a t jwo will totally suppress t he gain
(zero gain) a t frequency woo R epeated zeros will further enhance t he effect. Also,
a closely-placed pair of a pole a nd a zero (dipole) t end t o cancel o ut each other's
influence on t he frequency response. Clearly, a proper placement of poles a nd zeros
can yield a variety of frequency-selective behavior. Using these observations, we
c an design lowpass, highpass, bandpass, a nd b andstop (or notch) filters.
P hase response can also be computed graphically. In Fig. 7.l4a, angles formed
by t he complex conjugate poles - 0 ± jwo a t w = 0 ( the origin) are equal a nd
opposite. As w increases from 0 up, t he angle /1 1 because of t he pole - 0 + jwo,
which has a negative value a t w = 0, is r educed in magnitude; t he angle /12 b ecause
of t he pole - 0 - jwo, which has a positive value a t w = 0, increases in magnitude.
As a r esult, /1 1+/12, t he s um of t he two angles, increases continuously, approaching a
value 7r as w - > 00. T he r esulting p hase response L H(jw) = - (/1 1 +/12) is i llustrated
in Fig. 7.l4c. Similar a rguments apply t o zeros a t - 0 ± jwo. T he r esulting phase
response L H(jw) = (<PI + <P2) is d epicted in Fig. 7.l4f. 7 F requency Response a nd A nalog Filters 500 We now focus o n simple filters, using t he i ntuitive insights gained in this discussion. T he d iscussion is essentially qualitative. 7 .4-2 l ow p ass Filters A t ypical lowpass filter has a m aximum g ain a t W = O. T herefore, we need ~o
p lace a pole (or poles) on t he r eal axis opposite t he origin (jw = 0 ), a s shown I n
F ig. 7.15a. T he t ransfer f unction of t his s ystem is
We H (s)=-s +w e
W e have chosen t he n umerator o f H (s) t o b e W e in order t o n ormalize t he dc gain
H(O) t o unity. I f d is t he d istance from t he pole - We t o a p oint j w (Fig. 7.15a),
t hen IH(jw)1 = We d w ith H(O) = 1. As W increases, d increases a nd IH(jw)1 d ecreases monotonically
w ith w , a s i llustrated i n Fig. 7.15d b y l abel n = 1. T his is c learly a lowpass filter
with gain e nhanced i n t he v icinity of W = O. n =l ,x:
c n =5 jOO o 7.4 F ilter Design b y P lacement o f Poles a nd Zeros Wall-to-wall Poles A n ideal lowpass filter characteristic (shaded) in Fig. 7.15d, has a const~nt
g ain of unity up t o f requency W e. T hen t he g ain drops s uddenly t o 0 for W > W e'
To achieve t he i deal lowpass characteristic, we need enhanced gain over t he e ntire
frequency b and from 0 t o W e. We know t hat t o e nhance a gain a t a ny frequency w ,
we need t o p lace a pole opposite w. T o achieve a n e nhanced gain for all frequencies
over t he b and (0 t o we), we n eed t o p lace a pole opposite every frequency in this
b and. I n o ther words, we need a c ontinuous w all o f p oles facing t he i maginary
axis opposite t he f requency b and 0 t o W e ( and from 0 t o -...
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