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Unformatted text preview: agnitude 0 T he log a mplitude is given by
2 Q. log amplitude =  20 log /1 + 2 j( 4
(0 O.Ola O.OSa O .la F ig. 7 .5 O.Sa Sa a SOa IOa For W «':: W n> For 6 W » Wn + (~:) 21 (7.20) t he log amplitude becomes IOOa log amplitude Errors in asymptotic approximation of a firstorder pole. L HUW)=L(l+j:) (:J =tanl(~) , ~  20 log 1 = 0 (7.21) t he log amplitude is
log amplitude ~ 201og 1(  :J 21 =  401og (:J (7.22a) and, for W » W =  40logw  40logw n «':: a, (7.22b) =  40u  40logw n Let us investigate the asymptotic behavior of this function. For (7.22c) T he two asymptotes are (i) zero for W < Wn , a nd (ii)  40u  40logw n for W >
· T he second asymptote is a s traight line with a slope of  40 d B/decade (or
 12dB/octave) when plotted against t he log W scale. I t begins a t W = Wn [see Eq.
(7.22b)]. T he a symptotes are depicted in Fig. 7.6a. T he e xact log amplitude is
given by [see Eq. (7.20)]
Wn a, tanl(~) ~  90 0 T he actual p lot along with the asymptotes is depicted in Fig. 7.4b. In this case, we
use a threeline segment asymptotic plot for greater accuracy. The asymptotes are
(i) a phase angle of 00 for W : :; a /lO, (ii) a phase angle of  90 0 for W ~ lOa, a nd
a s traight line with a slope  45°/decade connecting these two asymptote~ (from
W = a /IO t o lOa) crossing the W axis a t W = a /10. I t can be seen from FIg. 7.4b
t hat t he asymptotes are very close t o t he curve a nd t he maximum error is 5.7 0 • A
p lot of the e rror as a function of W is i llustrated in Fig. 7.5b. T he actual .plot can
be obtained b y adding the error to the asymptotic plot. T he phase functIOn for a
pole a t  a is shown in Fig. 7.4b. The phase for a zero at  a (shown d otted in Fig. log amplitude =  20 log 1{[ (:J 2]2 + 4(2 (:J 2 }t (7.23) Clearly, t he log amplitude in this case involves a parameter ( . For each value of
( , we have a different plot. For complex conjugate poles,t ( < 1. Hence, we must
sketch a family of curves for a n umber of values of ( in t he r ange 0 t o 1. This
is i llustrated in Fig. 7.6a. T he e rror between the actual plot and the asymptotes
t For ( ~ 1, t he t wo p oles in t he s econdorder factor a re no longer complex b ut r eal, a nd e ach o f
t hese two real poles c an b e d ealt w ith as s eparate f irstorder factors. 484 7 Frequency Response a nd Analog Filters 7.2 Bode P lots 485 20 10 i:I:I """ : ::r:: 0 b Il .9 5   ... ?:l
10
 20 10.5 I I q.707 ; 5:.t=lt··t_t_I+~H_T_t_,!___+1:rj rfI
.
I 6 O.loon 0.2con 0.500n oo=oon 200n 500n 100ln 90 , T,.4,,,.. .~_.rr_r_r<~ OF
~~~~~~§§~~~~rf~;:~~i~=liliri!Tlllr'rww~'As~ptotel
i I II
I' '" ~ ..g 1i i"f  30  60 i Ii +b....::.5~p....A~~'I~\I~t\~f.t:=_"_c::=_=_..L_=_=_;;t;~~;._=rl_+ll~~IDI' _
C=O.l t~ I I 90 I ! I : : i ii I 3 0 I !
30 I ·120 60  ++'t
 60 I
150 F ig. 7.7 Errors in the asymptotic approximation of a secondorder pole.  180
0.2oon F ig. 7 .6 0.500n OOn 2 con 500n Amplitude and phase response of a secondorder pole.
(7.24) is shown in Fig. 7.7. T he...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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