Unformatted text preview: put amplitude is 0 3;~s:~m IS ~.3:2, a nd the phase shift is 65.3°.
of the o utput is shifted with respect to th~t f t~m~ t e m put amplitude, and the phase
e m put by 65.3°. Therefore, the system
response to an input cos 2t is
y (t) = 0.372 cos (2t + 65.3°) The input cos 2t and the corres
t rated in Fig. 7.lb.
pon mg system response 0 .372cos(2t + 65.34°) are illus- 4 74 7 F requency R esponse a nd A nalog F ilters 7.1 F requency R esponse o f a n L TIC S ystem ( b) For t he i nput cos ( lOt-500), i nstead of computing t he values IH(jw)1 a nd L H(jw)
a s in p art ( a), w e s hall read t hem d irectly from t he frequency response plots in Fig. 7 .la
c orresponding t o w = 10. T hese are:
IH(jlO)1 = 0.894 a nd Ideal Delay 4 75 Ideal Differentiator Ideal Integrator L H(jlO) = 26° T herefore, for a sinusoidal i nput of frequency w = 10, t he o utput sinusoid amplitude is
0.894 times t he i nput a mplitude and t he o utput sinusoid is s hifted with respect t o t he
i nput sinusoid b y 26°. Therefore, yet), t he s ystem response to a n i nput cos (lOt - 50°), is o yet) = 0 .894cos(lOt - 50° + 26°) = 0 .894cos(lOt - 24°)
I f t he i nput w ere sin (lOt - 50°), t he response would b e 0.894 sin (lOt - 50° + 26°)
0.894 sin (lOt - 24°).
T he f requency response plots in Fig. 7 .la show t hat t he s ystem has high pass filtering
characteristics; i t r esponds well to sinusoids o f higher frequencies (w well above 5), a nd
suppresses sinusoids of lower frequencies (w well below 5). • o C omputer E xample C 7.1
P lot t he f requency response of t he t ransfer functions H (s) = s2~t;+2'
d en=[l 3 2 ];
a xis([loglO(.l) l og10(100) - 50 5 0])
s ubplot(211) , semilogx(w , 20*loglO( m ag))
s ubplot(212),semilogx(w,phase) 0 • E xample 7 .2
F ind a nd s ketch t he frequency response (amplitude a nd p hase response) for
( a) a n i deal delay of T seconds;
( b) a n i deal differentiator;
( c ) a n i deal i ntegrator. ( a) I deal d elay o f T s econds: T he t ransfer function of a n ideal delay is [see Eq.
H (s) = e - sT
H (jw) = e - jwT
L H(jw) = - wT
IH(jw)1 = 1
This a mplitude a nd p hase response is shown in Fig. 7.2a. T he a mplitude response is
c onstant ( unity) for all frequencies. T he p hase shift increases linearly w ith frequency with
a slope of - T. T his r esult can b e e xplained physically by recognizing t hat if a sinusoid
cos w t is passed t hrough a n ideal delay of T seconds, t he o utput is cos wet - T). T he
o utput sinusoid a mplitude is t he same as t hat o f t he i nput for all values of w. Therefore, t he
amplitUde r esponse (gain) is u nity for all frequencies. Moreover, t he o utput cos wet - T) =
cos (wt - wT) h as a p hase shift - wT w ith respect t o t he i nput cos wt. Therefore, t he p hase
response is linearly proportional t o t he frequency w w ith a slope - T.
( b) A n i deal d ifferentiator: T he t ransfer function of a n ideal differentiator is [see
H (s) = s
T herefore E==
L.H(jro) r o-L.H(j(JJ) 7t/2 r...
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