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

Hence and l hjw t an i 01 tan i 2 10 a y

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Unformatted text preview: 7.1 Find the response of an LTJC system specified by d 2 + 3~ y dt dt if t he i nput is a sinusoid 20 sin (3t + 35°) 2 Answer: 10.23 sin (3t - 61.91°) 7 .1-1 H(8) = s(s + 2y(t) = 'dt + 51(t) !!.. K (s + a ll(s + a2) + bll(s2 + b2S + b3) (7.11a) where t he second-order factor (s2 + b2S + b3) is assumed t o have complex conjugate roots. We shall rearrange Eq. (7.11a) in the form \l S teady-State Response t o Causal Sinusoidal Inputs So far we discussed t he LTIC s ystem response t o everlasting sinusoidal i nputs ( starting a t t = - 00). I n practice, we are more interested in causal sinusoidal i nputs (sinusoids s tarting a t t = 0). Consider t he i nput e Jwt u(t), which s tarts a t t = 0 r ather t han a t t = - 00. I n this case F (s) = 1 /(s + jw). Moreover, according t o Eq. (6.51) H (s) = P (s)/Q(s), where Q(s) is t he c haracteristic polynomial given by Q(s) = (s - Al)(S - A2)'" (s - An). Hence, Y (s) = F (s)H(s) = P(S) (s _ AIl(8 _ A2)'" (8 - An)(8 - jw) I n t he p artial f raction expansion of t he r ight-hand side, let t he coefficients corresponding t o t he n t erms (8 - All, (8 - A2)'" (8 - An) b e kl, k2, . .. k n . T he coefficient corresponding t o t he l ast t erm (8 - jw) is P(8)/Q(8)ls=iw = H (jw). Hence, n k H (·) Y (8)= L:_i_+~ i =l 8 - Ai S - JW (7.11b) a nd -'('-1-+- ;-~_;:-'). . . ,(~1,. .+-:. : .L)_:_ :_; c:_ (1 + jW) [1 + j_b2_w + _(j_w_)2] b3 b H (jw) = _K_a_l_a_2 _ __ bl b3 jw bl 3 This equation shows t hat H (jw) is a complex function of w. T he a mplitude response IH(jw)1 a nd t he p hase response L H(jw) a re given by a nd a nd 1+jWI11+jWI IH(jw)1 = K ala2 1 ~ ~ l b b3 IjwI11+jWI11+jb2W + (jw)21 bl b3 b3 n y (t) = L : kieA't u(t) i =l '-.,--' t ransient c omponent Y tr ( t) + H (jw)eiwtu(t) (7.9) '-,,-" s teady-state c omponent Y S8 ( 7.l1c) L H(jw) = L (l (7.12a) + ~:)+L(l + ~:) - Ljw (t) For a n a symptotically stable system, t he c haracteristic mode terms eA,t decay with time, and, therefore, constitute t he so-called t ransient c omponent of t he response. T he l ast t erm H (jw )eiwt p ersists forever, a nd is t he s teady-state c omponent of t he response given by yss(t) = H (jw)eiwtu(t) F rom t he a rgument t hat led t o Eq. (7.5a), it follows t hat for a causal sinusoidal input cos wt, t he s teady-state response yss(t) is given by yss(t) = IH(jw)1 cos [wt + LH(jw)]u(t) (7.10) I n summary, IH ( j w)I cos [w t + LH ( j w)] is t he t otal response t o everlasting sinusoid cos wt, a nd is a lso t he s teady-state response t o t he same input applied a t t = O. _ L(I+ jw bl ) _ L[I+ jb2W b3 + (jw)2] b3 (7.12b) From Eq. (7.12b) we see t hat t he p hase function consists o f t he a ddition of only t hree kinds of terms: (i) t he phase of j w, which is 9 0° for all values of w, (ii) t he p hase for t he first-order t erm of t he form 1 + j w, a nd (iii) t he p hase of t he second-order term a 1 + j b2w + (jw)2] b3 b3 We can plot these three basic phase functions for w in the range 0 t o 0 0 a nd t hen, using these plots, we c an construct the phase fun...
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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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