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Unformatted text preview: o _ _ o o r o o ro _ _  7t/2 ( a) (b) (c) F ig. 7 .2 F requency response of an ideal (a) delay (b) differentiator (c) integrator.
Consequently IH(jw)1
;rhiS = w and L H(jw) =~ (7.7) amp~itude an~ p hase response is depicted in Fig. 7.2b. T he amplitUde response ;~:eases h nearly WIth frequency, a nd p hase response is constant (7r / 2) for all frequencies
h IS r esult c an be .explai~ed physically by recognizing t hat if a sinusoid cos w t is p assed t rough an Ideal dlfferentlator, t he o utput is  w sin w t = w cos (wt + !!:) T h C
h
o utp t '
'd
I' d .
2'
erelOre, t e
u .smusm aImp ItU e IS w t imes t he i nput a mplitude'' t hat is , t he a mpl't Ud e response
.
I
(gam) mcreases "mearly WIth frequency w Moreover t he 0 t t '
'd
d
h 'f'"
.,
U p u smusm un ergoes a
h
p( /ase) s .Iht 2' WIth r espect t o t he i nput cos wt. Therefore, t he p hase response is c onstant
7r 2 WIt frequency. e H
a~ideal d ifferentiator,.the a mplitude response (gain) is p roportional t o frequency
[I (J ) 1. wJ.' so t hat t he hIgherfrequency components are enhanced (see Fig. 7.2b). All ~ractJcal .slgnals are c ontaminated w ith noise, which, by its nature, is a broadband
(rap~dly varymg) signal.containing components of very high frequencies. A differentiator
ca~ I~crease t he nOIse ~Isprop?rtionately t o t he p oint of drowning o ut t he desired signal.
ThIS IS t he reason Why Ideal dlfferentiators are avoided in practice.
( c) A n i deal i ntegrator: T he t ransfer function of a n ideal integrator is [see Eq.
(6.56)J
1 H (s) = s
Therefore H (jw) = ~ =  j = ~ej"/2
w JW Consequently IH(jw)1 =; a nd w L H(jw) = % (7.8) ! his a mplitude ~nd p hase response is i llustrated in Fig. 7.2c. T he a mplitude response is
mversely proportIOnal t o frequency, a nd t he p hase shift is c onstant ( 7r / 2) w ith frequency. 476 7 Frequency Response a nd Analog Filters T his r esult c an b e e xplained physically by recognizing t hat if a sinusoid cos wt is passed
t hrough a n i deal i ntegrator, t he o utput is
s in wt =
cos (wt  ~). T herefore, t he
a mplitude r esponse is inversely p roportional t o w, a nd t he p hase response is c onstant
(  "/r / 2) w ith f requency.
Because i ts g ain is l /w, t he i deal i ntegrator s uppresses higherfrequency c omponents
b ut e nhances lowerfrequency c omponents w ith w < 1. C onsequently, noise signals ( if t hey
d o n ot c ontain a n a ppreciable a mount o f very low frequency components) a re s uppressed
( smoothed o ut) b y a n i ntegrator.
• t l:!. t 7.2 Bode P lots 7.2 4 77 Bode Plots Sketching frequency response plots is considerably facilitated by t he use of
logarithmic scales. T he a mplitude a nd p hase response plots as a function of w on a
logarithmic scale are known as t he B ode p lots. B y using t he a symptotic behavior
o f t he amplitude a nd t he p hase response, we can sketch these plots with remarkable
ease, even for higherorder transfer functions.
Let us consider a system with t he t ransfer function E xercise E...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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