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

6 6 2 6s 2 h s 2s2 38 10 6s 5 r epeat p roblem

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Unformatted text preview: 9) is y(t) = IH(jw)1 cos [wt + 9 + LH(jw)] (7.5b) T his r esult, w here we have let s = jw, is valid only for asymptotically s table s ystems because t he r elationship (7.1) applies only for t he values of 8 lying in t he r egion of convergence f or H (s). For t he case of u nstable a nd m arginally s table s ystems, t his region does n ot i nclude t he i maginary axis s = jw (see also t he f ootnote o n p. 243). E quation ( 7.5) shows t hat for a sinusoidal i nput of r adian frequency w, t he s ystem r esponse is also a sinusoid o f t he s ame frequency w. T he a mplitude o f t he o utput s inusoid i s IH(jw)1 t imes t he i nput a mplitude, a nd t he p hase o f t he o utput s inusoid is s hifted b y L H(jw) w ith respect t o t he i nput p hase (see Fig. 7.1). For instance, if a c ertain s ystem has IH(j10)1 = 3 a nd LH(jIO) = - 30°, t hen t he s ystem amplifies a s inusoid o f frequency w = 10 b y a factor of 3 a nd delays i ts p hase by 30°. T he s ystem r esponse t o a n i nput 5 cos (lOt + 50°) is 3 x 5 cos (lOt + 50° - 30°) = 15 cos (lOt + 2 0°). C learly I H (jw) I is t he s ystem g ain, a nd a p lot of IH (jw) I versus w shows t he s ystem g ain as a function of frequency w. T his f unction is more commonly known as t he a mplitude r esponse. Similarly, L H(jw) is t he p hase r esponse a nd a p lot of LH (jw) v ersus w shows how t he s ystem modifies or changes t he p hase of t he i nput sinusoid. T hese two plots together, as functions of w, a re called t he f requency r esponse o f t he s ystem. O bserve t hat H (jw) h as t he i nformation of IH(jw)1 a nd L H(jw). For t his reason, H (jw) is also called t he f requency r esponse o f t he s ystem. T he f requency response plots show a t a glance how a system responds t o s inusoids of v arious frequencies. T hus, t he frequency response of a s ystem r epresents i ts filtering c haracteristic. E xample 7 .1 Find the frequency response (amplitude and phase response) of a system whose transfer function is H (s) = s + 0.1 s +5 Also, find the system response y(t) if the input f (t) is ( a) cos 2t ( b) cos (lOt - 50°). o 2 10 H ( 'w) J = j w + 0.1 jw + 5 Recall t hat t he magnitude of a complex number is equal to the square root of the sum of the squares of i ts real and imaginary parts. Hence and L H(jw) = t an-I (0~1) - tan-I (~) 2 10 (a) y ( t ) ; 0.372 cos (21 + 65.30) (b) F ig. 7 .1 Frequency response of an LTIC system in Example 7.1. Both the amplitUde a nd the phase response are de ic . . t· b p ted m Fig. 7.1a as functions of w These plots furnish the complete inf . . o rma Ion a o ut the freq t o smusoidal inputs. uency response of the system ( a) For the input f (t) = cos 2t, w = 2, a nd IH(j2)1 = J (2)2 + 0.01 _ J (2)2 + 25 - 0.372 • In this case o (0_ L H(j2) = t an - 1 (0~1) - tan - I G) = 87.1 ° - 21.8° = 65.30 . We also could have read these values directl fro Fig. 7 .la corresponding t o w = 2 T h' u l Y m t he frequency response plots in frequency w = 2, the amplitude gain of ~h:es t me.ans t hat for a sinusoidal input with In other words, the o ut...
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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.

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