Question

Problem 1: (15 points) Determine if the following statements are True or False and briefly describe the reasons for your answers. (2 points for the correct answer of True or False, and 3 points for the correct reasoning.) (a) The Laplace transform of an LTI system's zero-input response is always equal to the system's transfer function, H(s). (b) The transfer function of an LTI system is H(s) = e. Then, it can be concluded that the system is BIBO stable. (Hint: You won't be able to find the characteristic equation of this system. But, derive ho first and think about what it means to the relationship between the input signal and output signal.] (c) The Laplace transform of a signal is Then, according to the Final Value Theorem, it can be concluded that the final value (that is, when approaches infinity) of this signal must be -2.

Answer 1

always equal is always e @ Zero-input Response function H transfer - to system's act) [h - YCH - ole yp system as false Given statement zero llp Responce is nothing but who the Yp =0, onlly initial condition's are considere Cat But according to definiation of Transfe function, Transfer function is the to Up th Ratio of Laplace transform of olp where Initial conditions are zero ** ies, 2 Yls) H(s) xls) Tinitial conditions are Corsect state ment is esponse os Zero-state Response Response always equal to system's Transtus function s to system's Transfer

Let take a differential equation de de arty= det y hole atyp Zero state Response Yp = x=0 뽑 sy(s) -4LO)ty ts) zo y(s) (s ti] = 460) YGS) = 4(0) But Tip = Y() na stl XC57 H(s) = ē55 01 5 h (t) = $(t-5) y(t) = ) tehty x8 * YCH = xf *864-5) = *+-5) 96) = x(6-5) if x (t) is bounded the shifted ve e shifted version

"ob Evy F (x (4-5)] is also bounded it, Con for a Bounded Up, we getting a Bounded output. So, for system H(s) zés,is BiBo stable 12 xls) 3 S (5-3)(3+2) final value theorem ROC it 24) = H S x(s) på logot >00570 ; Re) 73 It act) = x(0) to con pe; accoje It S sto 12 s 5-3) 8+22017 I o. glad olie n the system contains - f because 1 B 83) right sided pole about l gw axis. taking Roc Re(s) < 3 singheal is stable to go - but en jed - of 13 so 200) = 483) 6+2 include jw axis = a 2-3 [acco=-2

in Time domain 12 XLS) = 56-336+2) * = 12 = -2 819 = 120.8 아녀들을02 349 lo2 X13= toodetest loob sta Roc selecting the given signal is stable ROE Re (8) <3 *) = -24649.-0.823+ 4C=) + lozētuch 2.cm = -2-0.8@#466) +1.2@ulo) = -2-0.8(60) Co) + 1-26) 1 Taco)=-2]