fa2005_exam3soln - / ECE 210/211 Analog Signal Processing...

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Unformatted text preview: / ECE 210/211 Analog Signal Processing Fall 2005 University of Illinois Allen, Basar, Kudeki Exam 3 Thursday, December 1, 2005 —— 7:00-8:15 PM Section: circle one Please clearly print your name and circle your section in the boxes above. This is a closed book and closed notes exam, but you are permitted both sides of one 8V2" X 11" crib sheet. Calculators are not allowed. Please show all of your work. Backs of pages may be used for scratch work if necessary. Good luck! Probleml Problem2 Problem3 Problem4 Problem 1 (25 points) a) The unit step response of an LTI system is g(t) = e—2t u(t — 3). What is the impulse response h(t) of this system? Simplify your answer. NH = 91.3 z» 264’: was) te’ztflera) -2? ”‘ h(t) _ 2x5 44 (/Jc '3) + b) If an LTI system has the input and output below, what is the system impulse response h(t)? y(t) f(t) E: t t _ l l 2 2 . “w m N ‘9: = '8 -— c) What is the impulse response h(t) of a LTI system if H((o) = 1 6—12“ ? "' \ l + 2 jco . 1/1 {I} 2 , lava) (W i f , l 1—: fi ) 1/4; +qu \_/ Ne‘a’bboz's“ _ ' a “(#1) MW» //-« h(t)— /‘"\/ -Lt $46 azure) g. d) What is the following integral? J2 rect 5(t — 2) dt = . e) Given an LTIC (linear, time invariant, causal) system with an impulse response h(t) = 5(t — 4) Is it BIBO stable? Explain. gte is. mg; M558 deal Problem 2 (25 points) You are given the following circuit IQ f(t) y(t) having an impulse response h(t) = e't u(t) with f(t) = t rect [t — a) Plot f(t) versus t. p H) b) What is the integral formula for y(t) in terms of h(t) and the input f(t)? 00 \GM) = Mk) i4 Cm) : f M1) gar: > A: 00 / ~00 : g 5 L44 (1:) (t’T)ALL/T(‘t”k';)d£ ’ 00 09 / r of e’LCchw/WU'L’flmfl- c) Sketch the functions f(t — T) for the following values of t. t=—1, t=0, t=2 / \ ((4) \ fl—I—T) , r—l} ,;-— “er-‘4» ——~—j~—"- “firm—*7 "C C Problem 2 (continued) d) Find the numerical value of y(t) at #._,____M> C y (0) = I ~ I ——_——— LGQ‘): f -& Opt>dt o ‘ f I ,, / ’ b ’ .— ’L ,é’ a OLL £9; {off ’ ya): ‘ L/‘PW’”"’ \ r ,{’%)’+ [41,51,011 7 1 \‘H' u r‘C RC} ((1) 2 a (Z L) a“: y(2)=' ‘ 7, ’2' —T Problem 3 (25 points) Consider the following system. 1 1 cos colt cos (ozt where F(co), H1(0)), H2(w) and Y(a)) are given by F(co) H1(co) 1 - l Mg) m on 3 5 7 —17 f—13’\—1o —8 8 101213 1517 —7 —5 _3 —15 —12 T H20») } Y(w) (D (D —2 2 —2 2 a) What are the frequencies co} and (02? (Hint: there are two distinct possibilities for (1)1.) (O is! 7/0 mat/5 § rod/5 :9 n b) The output y(t) goes through the following LTI system. Find the Fourier transform R((o) of the output. 1 M -3214; Mic) Amiga» : iath’rZHaHvQC—a \(MeJ +Yi0) e I = \((~>Zm(2w3 R(CO) = Problem 4 (25 points) IQ 3) + + f(t) 1 H y(t) For the linear circuit shown above, determine the frequency response H(03) = and the ' 1 ht. Hlm , :mpu se response ( ) T F E LE. > Y ' 1+3” H(°°)=————+j F 1‘” T w VOW? ‘34" a r t h(t): - 6 M H') H (W T *Jfi’” I J , _’ no . * v t / 2500’ 6 MR) b) A non-causal system shows an impulse response h(t) = e-at u(—t), where a is a real number. For which values of a is the system BIBO stable? Explain your answer in terms of BIBO stability criterion. 61 v o {A R > I ' 014 O > ‘ - "'“"""'*7 Jc BIBO stable if and only if: A < O Explanation: m L\ 0 l" (*5 “>0 5< 4< 0 CW» Mom/e m5 “hwjl Hi) {5 wasch Eu‘lférA/Lle M I Am), H441 Fargo Stun/Li glam wt. Med mo. Problem 4 (continued) c) One of the following systems, a or b, is LTIC (linear, time invariant, causal). Determine which one is LTIC and explain your answer. N) = t f(t) f0) LTIC system is: [3 Reason: gi'e’w‘ fl is 140+ ’rlhw‘fl {Wm pad and Magi-(e (Mt/kart be LTIC: a“) .7. t3.(+)_,t_p,(+) am.» eta—t») —‘> ['37, 6t) > tée) = k‘RCt't°) ‘ Hl(‘£"'tw) thaw-I: d) Given that h(t) = u(t) — t u(t), what is the Laplace transform I:I(s) ? J, .L 9 31' ...
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This note was uploaded on 06/15/2010 for the course ECE 313 taught by Professor Sarwate during the Spring '08 term at University of Illinois at Urbana–Champaign.

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fa2005_exam3soln - / ECE 210/211 Analog Signal Processing...

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