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Unformatted text preview: EE 350 EXAM II 24 October 2002 Last Name: g (“)l GE 0 05 First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Weight Test Form A Instructions 1.
2.
3. You have two hours to complete this exam. This is a closedbook exam. You are allowed one 8.5” by 11” note sheet. Calculators are allowed. ' . Solve each part of the problem in the 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.” N 0 credit will be given to a solution that does not meet
this requirement. . Do not remove any pages from this exam. Loose papers will not be accepted and a grade of zero will be assigned. . 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. (15 points) A linear timeinvariant system with input f(t) and output y(t) is described by the ODE
17+2§I+y= 2f(t). Determine the impulse response h(t) of the system. FImQ— the zerosimie. raspoare. &(£) ‘50 0" W’éﬁﬁep
mput FL+,§ :Wl‘éﬂj then h(£\ = o’f/Jqéc 7“ 2 1:”,
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._ C  '2— yCO) :o C) + Z .3 cl = 2_ ' (07 :0 1‘— ‘C1 + CZ : L g —'t’ We)  [Z — 20"“99 let)
F——" O .— 2. (10 points) Another LTI system is described by the impulse response function
h(t) = eatu(t — ,6),
where a and ﬂ are realvalued constants. o (5 points) For what range of values of a and ﬂ is the system BIBO stable and causal ? In order
to receive credit, you must justify your answer. 0 For 620 ana oC<o the 984:9.“ b 5/50 84:“ng 045 thmlé = fwemtié = tEe’a—tl: o (5 points) For what range of values of a and ﬂ is the system not BIBO stable and noncausal ?
In order to receive credit, you must justify your answer. . Pg, g; ‘0 ”(+3 is as. nonchdeQ ScyﬂccQ 0L3 t: Z<). 77725 rhea/L5 ﬁle: Problem 2: (25 points) 1. (12 points) Use the graphical convolution method to determine an expression for the zerostate response
y(t) = f(t) a: h(t) for 4e2(t  1)u(t — 1)
2[u(t+ 1)  u(t + 3)] W)
N)
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 a sketch of y(t), as well
as an analytical expression for y(t). {(T) hit) b ("E't’ t ”l
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Summaw‘tmg 2 (7 points) Let 9(t) = f(t) * f(t).
and show that the energy E: of the signal f(t) is given by Er = 9(0) ;(t) 2 ‘PC'ﬂ‘kﬂﬂ zj‘ F( C) Hi: what 661(0) '—'— I 'Fét5F(‘C)£C E j“, 1, wet. (a) : E4: 3. (6 points) A system with input f(t) and output y(t) is comprised of three separate LTI systems whose
impulse response functions are denoted as h1(t), h; (t), and ha (t), respectively. The zerostate response
y(t) of the system to the input f(t) is given by W) = N) * [h1(t)  h2(t) * hs(t)]  Using separate blocks to represent the systems described by h1(t), h2(t), and h3(t), sketch a. block
diagram of the overall system clearly showing where the input f(t) enters and the output y(t) exits. If
a summer is needed, use + and — symbols to specify whether a given input is added to, or subtracted
from, the summer output. ‘F‘klh \L Ha Problem 3: (25 points) 1. (7 points) Consider a BIBO stable LTI system with input f(t) and output y(t) that is represented by
the ODE ﬂ+2y+2y=4f+2f+4ﬁ o (3 points) Determine the frequency response function, H (1w) = f’/ 17‘, and express your answer
in standard form How) = Z = W
I“ 0w)“ +“n—1(Jw)"_1+a1(1w)+ a0 Raphae ‘W n wad» (“an
(0'0“)? ‘Ar’ 4" 2%.») q + 2)? M
VEC WYW— ltd») +2] == [ng)1+ 20 7 + <4] f: : "(Cam/)1]; 4 0'9“)"; + ”E e (2 points) Using the frequency response function, determine the DC gain of the system. (DC. am} 5 ”(£035 L. e (2 points) Using the frequency response function, determine the high frequency AC gain of the
system. AC hfﬂk freauancd ﬂat}; i'S gmﬂ ampmﬂ “0  HC 3 — 2;,“ ”C >Z+im+~r
,gl/g—w d9“ wave {/W)Z " 1(zw) +13 9mm. can/MW“ *3 :Lm—sz (av/)7” 0° (007’
Hy». M5 6am =7 W V 2. (8 points) An engineer has written the following m—ﬁle to display the exact magnitude and phase plots
of the frequency response function H (1w) for the circuit shown in Figure 1. omega = logspace(1,4,500); bode([8e—1], [lo4,1], omega) Figure 1: Circuit with input voltage f(t) and output voltage y(t). o (2 points) By using only the arguments from the Bode command, determine the frequency
response function H (10)) and express it the standard form m "1—1 I O Q
H(Jw) Y _ 5111060) + b —1(Jw) +510“) +50~ ~ F _ (1w)" + an_1(Jw)"‘1+ 01(Jw) + ao o (2 points) State the range of frequencies for which the exact magnitude an phase plots are shown,
and specify how many points are contained in the vector omega. Omega, (“amid/rid 500 yolk}; r‘qnﬂlv from IO rag/[590
{:0 lo L! Faced/53¢“ o (4 points) Given that R1 = 200 D and R2 = 800 Q in the circuit shown in Figure 1, what must
be the value of L ? Us!!!” Jol'lilAijL J‘QISIcJﬂ ’ﬂ ﬁe P5050), £001th (Na/ft Cacti/02101:
(\a R2. F “a >’
Y = L "’ ~ = «+2.; .2. +
RN“). *ﬂw E I f“! “#97. ' 3. (10 points) An engineer located on a ship in the Paciﬁc has been sipping coconut milk from his EE
juice glass while monitoring a radio beacon. When he detects a weak oscillatory signal that cannot be
attributed to known radio sources, he increases the volume level of the receiver but the signal decays
away. In haste, to test his receiver, he tunes to a radio station located on a nearby island without
ﬁrst lowering the volume. He ends up spilling the coconut juice when the lyrics of Margaritaville
blast through his headphones. He throws a switch that places a circuit, with the frequency response function given below, in the signal path to modify the bass and treble response of the receiver. ~ 2
2: =0.1'7w/10 +1 H : _._—_
(1w) F Jul/103 + 1 o (8 points) In order to understand how the circuit affects the sound, ﬁnd the sinusoidal steady
state response y(t) of the circuit to the input f(t) = cos(10t) + cos(104t)
u,  "‘ W ﬁgs): 0,! 004,032 ‘H ‘ ZLHVUQ ’ T‘qﬂ'fg}; _ "”7 I000
V(W/Io3)" ‘H fig?” 4, 4;
’________._. A, _ z ”(0071:; = ‘Hgmﬂ‘a *l
{2411): (ogﬂo‘ta) =? F: I. Y! d .’ 0° ’ (Tam... °l— TW’.’ 0’0,
+ 9% ’X,0 2‘5 0 :2 W3; ‘(0 0 0L
reﬁne"L t} =O“”") N 0:] e 1] 7 01' you/Io“)
z wwme ,, (W = ~  _  449°
ﬁt ‘02 4(an' 107'"““?0’f) ._ 1,06
re '0 ﬂ a 44w" ’1’ ’
YL T. 0“ ~ ‘1 l: ; Cos (Io ‘1 t)
3;”) ; Ila. i V2. 6%") 3
7%
F (t) 2 (as 001:) + C05 (Io )
[5) = o.\ (as (Iot) 4 (o; (’0‘1 s; ) l
y h‘yk F/aythlﬂ 5'?"
ﬂow £ré'ﬁVeﬂg'bﬂ’ [791559) thrwyl') gym! a tum“ o (2 point) Will Jimmy Buffet be (1) upset because the engineer is attempting to drown out his
vocals by amplifying the bass response of the guitar and drums, or (2) be pleased that the
engineer is amplifying his highpitched vocals ? SW” my w.” 99 “engage the}: ﬁlo2, eny/néer )5 aWWVW‘E/y tk’ £9955 r95/0ﬁ)24 {a ilﬂw‘t lot Ca), bt‘éL 6% MIA" ﬁle byter («away/,9, UO(CVIIS.
10 04 (0u¢$Q) rc 3mm} PTaC'blCOJ “,va he 3M7;
“Q Wayloan it (“7Q {til/lef w a; J Problem 4: (25 points) 1. (10 points) Consider a set of complexvalued functions
45"“) = e] n we t deﬁned over the interval [0, To], where n is an integer and we 2 27r/To. Determine < d)”, ¢m > for the cases 72, = m and 71. 9e m. Case I 2 n=mT 1;,
«my» = f (eﬂwﬁxeimwtﬁlt = S 0012 5 To.
CM“ 7— n #m “a
T5 anut *‘ E __ j (IF/”Qua, £
<d2n a > = (evmmt) (3* > 60 Se 1
2 m a To 0
 Jn~m)w9t
— ,L..... [a
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BecOWSQ <2 ﬁlm/”5L” : (a5’(nm)Z)T tjgmm'm) 2” PI) 44;", d‘Jm5 >0 41” Wm Using your results, justify whether or not the signals (1),. (t) are 0 mutually orthogonal, or o mutually orthonormal. T/Re 51 y (1 all! an), 3W lyecause.
/ (Cpnj¢».>= "PW ”#07. The are or;
_  5T 3’.
moba {1/ wtlmnanmuﬂ, 11" 70/2 I} {o 1:17} <95”, ($7 0 2. (15 points) Two realvalued functions, f(t) and g(t), are deﬁned over an interval [—1, 1] and are exactly . represented by a set of three realvalued mutually orthonormal functions, ¢.(t), also deﬁned over the
interval [—1, 1], as H W)
W) ~2¢1(t) + 2¢a(t)
«mm + Mt) +ﬂ¢a(t) o (3 points) Determine the values of < f, 43, > for i: 1, 2 and 3. CZz<F,d>g>/<¢L,¢;) : “MD as <e>i,¢z>:l
I <‘FJ ¢z> 1C}, 20‘) (‘8 $37.:C3 = 2' t BQCWJSQ. Ee. =65 Parrot/a“: Theorem is a ”Malia
3 I E; z 3an <Mph) = CF+C§+ c3? = 9 o (8 points) Determine the value of positive, realvalued parameters a and ﬂ so that (1) the signals
f(t) and g(t) are orthogonal, and (2) that g(t) has energy E ‘= 9. UJInﬁ_ paYYchKNS Theorem) .3 1 z 7— 2. 2.
7. ~ 
:: icn<¢nn>ecl+‘1+‘3’0¢+’*é
h:) E
3 a
E; = “I => W C"
<4“ 5‘) :0 g) Flt) and) 9(6) ave, orﬂogoﬂi‘
a (pi?) ’ <'1¢5+Z¢3,00¢1+¢;+I$¢3> =0 0 O
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 SCHIANO,JEFFREYLDAS,ARNAB

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