Signal Processing and Linear Systems-B.P.Lathi copy

1 s etting s jw in this relationship yields eiwt h jwe

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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 hIgher-frequency components are enhanced (see Fig. 7.2b). All ~ractJcal .slgnals are c ontaminated w ith noise, which, by its nature, is a broad-band (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 = ~e-j"/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 higher-frequency c omponents b ut e nhances lower-frequency 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 higher-order transfer functions. Let us consider a system with t he t ransfer function E xercise E...
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