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Unformatted text preview: EE 350 EXAM II 3 March 2003 Last Name: Solui‘a on} First Name: ID number (Last 4 digits): Section: ‘DO‘ NOT TURN THIS PAGE UNTILYOU ARE TOLD TO DO SO Instructions. 1. You have two hours to complete this exam. 2. This is a closedbook exam. You are allowed to use both sides of a handwritten 8.5” by 11”
note sheet. ~ 3. Calculators are not allowed. 4. Solve each part of the problem inthe space following the question. If you need more space,
continue your solution on the reverse side labeling the page with the question number, for
example, “Problem 2.1) Continued.” No credit will be given to a solution that does not meet
this requirement. 5. Do not remove any pages from this exam. Loose papers will not be accepted and a grade of
zero will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey What you are doing.
To receive credit, you must show your work. Problem 1: (25 Points) 1. (5 points) The block diagram in Figure 1 contains three LTI systems whose impulse response functions
are h1(t), h; (t), and h3(t). The input to the overall system is f(t), and the output is y(t). Represent
the composite system using a single impulse response function h(t) so that 1105) = f (t) * h“):
and express h(t) in terms of h1(t), hg(t), and 11.30.). f(t) y(t) §z (t) Figure 1: System comprised of several subsystems speciﬁed by their impulse response function. allot) : 1am aunte) u) gait) = We) at big) (z)
#3 (+3 ; gitQ a< hJH'J. (,3) 44/50 mic. these, ace) Can be, expruyeﬁ and
We) = 0\4,H=3 +yzzt) egg—e) C1) Subséfévﬁl? QDVW£(Oﬂs (I) throw]? [3) ”1.1.0 (Y) ﬁl® ft) = — ﬂasks» + ‘Fl'Q’kMLH'J — (Heywoﬁ has) ’1 JFltHrEk‘c—D 1412,04.) 4:1“) New}
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hat) 2 2. (8 points) Consider a LTI system that has the impulse response h(t) = (2 — 25—5t) u(t). Is the system BIBO stable ? You must justify your answer to receive credit. jwlhwﬂoﬂﬁ = wlzzéﬁloQJc 6€c¢vse Z  2,2, 3. (12 points) It is known that a certain physical system with input f(t) and output y(t) can be repre sented by an ODE of the form
W) + 01W) = NW C 199 In order to determine the unknown parameters a and ,8, an engineer applies a unitstep input and
observes the zerostate unit~step response. Using this data, the engineer determines that the impulse response of the physical system is 4t er ‘H: T 5 w n “CEVPMQ mg
h(t)=3e "(t) X ff, tho, stEWb 0!an So Determine the value of the parameters a and ,3. a; ___ q bﬁ I n Sfec'EIu/i _l \ SOL/Q, (A) Law 406) a «66) 01mg. g(o) =0] tkw‘E [SJ 'pmﬂ
tl‘Q. taro—Ltw‘lz unit—5W feS/yonse. «acé: w Problem 2: (25 points)
1. (15 points) Use the graphical convolution method to determine an expression for the zerostate response
y(t) = f(t) * h(t) for
f(t) e—W2
h(t) —_ u(t + 2) — u.(t) In order to receive credit, clearly specify the regions of integration, and, for each region,
provide a sketch of f and h. Summarize your results by providing an analytical expression
for y(t) for the different regions of integration. Do not sketch y(t). .pcQ) th) kHz1') JOB $j £[t)k{‘I:"C)CQ‘C
+
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(asj 4/; _: —22. z —._—2<J_ L—ez')
# i e a i < 2. (10 points) Suppose that both f(t) and h(t) are causal signals and that W) = f(t) * Mt) Using the convolution integral, show that y(t) is also a causal signal. 3 0 Bacwvse, 9C?) I6 ox. Couvsvrg Sig/’V‘Q') ‘Céo. ' (SQC‘VR h(E 'T) 75 av coat/SQ Jain/Q; )7[[: ’C)
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O [Q ~Fw— C4?) £(CBL6 :30 a l'vw) Problem 3: (25 points) 1. (5 points) A LTI system with input f(t) and output y(t) is represented by the frequency response function
Jw2 Jw+1' Find the ODE representation of the system, and express your answer in standard form. HOW): (he the Lora: (uni 4;” WWW wywen—emémi £1 ] w"~.
014:" 5 (ﬂ ) Y N wZ.
.12.; ﬁll/w): ’96::— M ’2. doc/F: \\ dwy'i‘}; 2. (10 points) Once again consider the LTI system with input f(t) and output y(t) that has the frequency
response function _ _ jw2
Using phasor analysis techniques, determine the sinusoidal steady—state response of the system for the
input
f(t) = 5 + 3 cos(t + 20°).
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+149 = M w) a (t) =
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+ 21 99%)) a [Via .5 ago, rpos z’iSluA 6 live) 640:3 : g leaQ‘CaSGJtiOa‘ INN/0)) 3. (10 points) An engineer has generated the exact magnitude and phase response plots of a LTI system
using the MATLAB command script w = logspace(—3,3,1000);
bode([8,7,2,1,0], [4,0,7,2,0,1], w) Write a MATLAB script that calculates and plots the zerostate response of the same system to the input
f(t) : 28in(5t) over the interval 0 S t g 10 using a time vector containing 800 points. = Qmsyaue. Co) IOJ 800); t 4 = z =x .sm (acy '5)»
e 2' E89742) loo—1) Q : EﬁJOJ'Z/ZJO/ljJ j_ = QSM” (KQJ‘FJ‘EUJ
plat: (4:,&) xlahe/i CHE/mo. (see) ’) ﬂinchoi C ’ccn//.£uﬁ,’) 10 Problem 4: (25 points) 1, (6 points) Show that the set of three realvalued basis functions {451(t), ¢2(t), 45305)} deﬁned over the
interval 0 g t S 1 in Figure 2 are mutually orthogonal. ¢1m gm gm 11 2. (6 points) Once again consider the three basis functions shown in Figure 3 and determine if they are
orthonormal. #p) #1:) gm Figure 3: A set of there basis functions {¢1(t),¢2(t),¢3(t)}. 
“Mm = jmﬁwe =. 531+: ;/ /
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($2) 4’2.) :3‘4931ﬁ5ﬁ‘b ; InfI: ;) l/
\ '00 /
«53,459 ;§¢3chx£t 7! t e I Seem15¢ (l) tke. <25; (1:) ave, mu'me/(a of thcdoA‘Q DZ‘a
C2) <G5£JdJL> >1 ‘Fo‘t‘ E;g 2, 3 a, ScynaQS Ed), (65 ﬁfth 91+); ‘6 mvﬁug “tanner/ml, 12 3. (13 points) The basis functionsconsidered in parts 1 and 2 are shown below in Figure 4, and are now
used to represent the signal N) = St
over the interval 0 S t S 1 as
f(t) = c1‘131(t)+ CW2“) + 03¢3(t) + 63(15): Where e(t) is the approximation error. Determine the coefﬁcients c1, ca, and cs so that the energy of
the approximation error e(t) is minimized. gt) em +1 0.25 0.5 0.75 1.0 Figure 4: A set of there basis functions {¢1(t), ¢3(t), q53(t)} used to approximate f(t) = 8t over the
interval 0 S t _<_ 1. CL 7' <44“) 2. <‘E 95D (beca‘be’ <¢3~J¢5> 3 1/ OWL
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 SCHIANO,JEFFREYLDAS,ARNAB

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