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.J:r 20
30 I i 0.05 0.01 ( u=2) ~ I't II . . _~rr . ,, ! " ............. I 0.1 I! 0.5 ~I I 1i I I! ! I ; I lea ! jI I ; , il 1 W l~ rJ 00=1
(u 0 0_ II
!i 100 50 10 (u = 0) ( u=I) .. , " 1I I I'
!I II
I I. , ' .., j = I) (u = 2) (7.13) T he t erm 2 0Iog(KaIa2/bIb3) is a c onstant. We observe t hat t he log amplitude is
a s um of four basic terms corresponding t o (i) a constant, (ii) a pole or zero a t
t he origin (201og Ijwl)' (iii) a firstorder pole or zero (20 log 11 + j w/all, a nd (iv)
complex c onjugate poles o r zeros (20 log 11 + jwb2/b3 + (jw)2 /b31l· We can sketch
these four basic t erms as functions of w and use t hem t o c onstruct t he logamplitude
plot of a ny d esired transfer function. Let us discuss each of t he terms. , 1 50"1==+=+=+++=I+I1C==+=+==l=rnr:m==+==+=+=mm==+=+=::r=m:m
II . .....
I:
II
II
90"b_ . . _~.~~;1;j;;m=±±hmfh:dddd~!db=b!:::k~:Hj
__
i 1I r
't ;
T 11"~ 5 0"·++IHH+i+  (b ~ ,  r    Il
I
!I
r+i~+~rH~+~+++H+~+~+++H~++4~++~0 0_
I
~  50"f·...........
 .. ~tL+ ~ 0• • "._ I + j I
f ~"~.~.~.~+=~+=66~~~~~~~~~~~~~~~~~db~~~~~
..
 . I
1"( iI
i!
.. _ISO" ! ..... , 0.05 0.01
( u=2) 0.1 0.5 ( u=I) 00=1
( u=O) + :ft·=
10 i ! 50 ( u= I ) 100
(u = 2) Fig. 7 .3 Amplitude and phase response of a pole or a zero a t the origin.
T he log a mplitude of this t erm is also a constant, 20 l og(K aIa2/bIb3). T he
p hase c ontribution from this t erm is zero. 2. P ole (or Z ero) a t t he origin
Log M agnitude t This p oint c an b e s hown as follows: Let Wj a nd W2 a long t he wscale correspond t o Uj a nd
along t he u ·scale. T hen l ogwj = Uj, a nd logw2 = U 2. T hen
U 2  Uj = loglO W 2  logJQwj = logJQ(w2/Wj)
T hus, if
(w2/wIJ = 10 (which is a decade)
t hen
U 2  Uj = loglO 10 = 1
a nd if
(W2/WIJ 2 (which is a n o ctave)
t hen
U 2  Uj = loglO 2 = 0.3010 = Such a pole gives rise t o t he t erm  20 log Ijwl, which can be expressed as
 20logljwl =  2010gw U2 7 Frequency Response and Analog Filters 4BO Note t hat equal increments in u are equivalent t o equal ratios on the wscale. Thus,
one unit along t he uscale is t he same as one decade along the wscale. This means
t hat t he a mplitude plot has a slope of  20 d B/decade or  20(0.3010) =  6.02
d B/octave ( commonly s tated as 6 dB/octave). Moreover, the amplitude plot crosses
t he waxis a t w = 1, since u = 10glQw = 0 when w = 1.
For t he c ase of a zero a t t he origin, t he logamplitude term is 20 log w. T his
is a straight line passing through w = 1 a nd having a slope of 20 d B/decade (or 6
dB/octave). T his plot is a mirror image about t he waxis of the plot for a pole a t
t he origin a nd is shown dotted in Fig. 7.3a.
Phase 7.2 Bode P lots 481 18
12
6 o  12 T he phase function corresponding to the pole a t t he origin is  Ljw [see Eq.
(7.12b)]. Thus 18 (7.15b) L H(jw) =  Ljw =  90° O .Ola T he phase is c onstant (  90°) for all values of w , as depicted in Fig....
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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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