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Unformatted text preview: NAME: g0 lk’ll\ b N Michigan State University
Department of Electrical and Computer Engineering ECE 366 INTRODUCTION TO SIGNAL PROCESSING
SPRING 2011 Practice Exam 2 April 14, 2011 Two sides of 8.5” x 11” paper consisting of handwritten notes are permitted. (Please
turn these notes in with your exam. And, don’t forget to write your name on your
“cheat” sheet!) Four tables are provided from Lathi (last four pages of this exam booklet). They are
Table 7.1 (Fourier Transform pairs), Table 7.2 (Fourier Transform properties), Table
5.1 (ZTransform pairs), and Table 5.2 (ZTransform properties). Calculators are NOT allowed.
You will have 60 minutes to complete this exam. For those problems that allow for partial credit, please show your work clearly to
maximize your score. Good luck! 1. (25 points) True/False. No partial credit will be given, so it is not necessary to show
your work. You will receive 2.5 points for each correct answer, —1 point for each
incorrect answer, and 0 points for no answer. — (a) The Fourier Transform of a rectangular pulse is a sine function, while the Fourier
Transform of a sine function is a rectangular pulse. l NLQ .
(b) In amplitude modulation (AM) radio, the signal to be transmitted is shifted in
frequency by multiplying it with a sinusoid prior to its transmission. True. (0) Consider a linear and timeinvariant (LTI) ﬁlter with a perfectly constant magni
tude response over its passband. This ﬁlter cannot distort an input signal whose
spectral energy falls totally within the ﬁlter passband. Fools e . The {X Hu‘ (:0le lkocUQ (k kokluk emf flan) l‘QS peruse lelgu do: 2 Olis‘lof'l' suck (UK IKPW+ Sl‘tkkl.
(d) Ideal interpolation is achieved by convolving a sampled signal with a continuous time (CT) sinc function. Trwe. (e) The discrete—time (DT) sinusoid cos(1.57rk +7r / 2) is equivalent to the DT sinusoid
cos(0.57rl~s + 1r/2). TT‘ False. cos Cl .YTF Krg) t (:05 (”0.517 b 4 $1.): Cos C0,:l’lTL: —'%")
(f) The DT sinusoid f [k] = Acos(2k) is a. periodic signal with po er equal to A72
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(Aeﬂ‘q is cellariooel. c. (g) Summing in DT is analogous to differentiating in CT, while differencing in DT is analogous to integrating in CT. False. SUMMIrch [IL wT (s analogoUJfo quoaroqlmam CTJ dLrlQ 0‘ (‘P‘PQPQAC («8 1“ ”T [S QILQLOBOLCJ 7!!) 0ft zfQA‘I‘HSAIkQD
(h) The sifting property is utilized to derive both the convolution integral and sum. wt CT: Truce. _ ‘ (i) The Z—Transform (ZT) exists for all DT signals that grow over time. lei‘e . The %T aloe: [UH/L UCLS’l ‘(od‘ J‘tﬁVUi/J‘ $445} are») ‘Fu‘l’er ‘Hram out expo/lawful] (“ir» on “’1!
(j) The Z‘Iransform is the Laplace Transform with a substitution of variables, w 'ch makes the unit circle important rather than the jaraxis. Tract I ____.__... .._.___._._._. mm——n.
___ W 2. (25 points) Show all your work to receive partial credit. Correct answers without
accompanying work will not receive full credit. Determine the Fourier Transform for the following signals: — (a) f1(t) = e'3‘U(t) (b) f2(t) = e‘3(“5)u(t — 5)
(C) fs(t) = 6‘3“"5)U(t) (d) f4(t) = e“mutt  5) (This problem can be solved by any means. But, as a hint, the entire problem can be
solved without integration!) CL) :FBM Tor—lb l1 ”9 l ) “or E a f >
e la) curate ) or O UDQ OLPNUQ Ok.+ F\ (“35 = Elam ll) WSW?) lull. “l “P‘Hw +«MQ~SLHA+FMPUM [kale ,7}— 2/1‘.Q\j
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tropeJ‘l'xG (A Teal) l‘L 'Df'kj LOQ <3erva cpl Extra sheet: 8P5 (“0) 2 ”i351 , _
0U £6): Q'HKC7L“I) ﬂag“) : e—tre—3L+—y)wé+vr] (Auk? POLF+ «)1 \HUA SCGLIN‘ MHIf/tcoalwq
Vfo?kf+a/V’ +km +‘M9 $LLTA‘F ff'OfLr+a./ [DQOJFIUQ ap‘l" 3. (25 points) Show all your work to receive partial credit. Correct answers without
accompanying work will not receive full credit. (3.) Consider the CT signal f (t) with spectrum F(w) shown below. The signal f (t) is
sampled at intervals of T = 1.0 sec to arrive at the sampled signal fs (t). Sketch
and carefully label the spectrum Fs(w) of the sampled signal f, (t) F(co)
1 —2n 27E (b) For the signal f (t) in part (a), what is the maximum sampling interval needed to prevent aliasing? FUD) r a. qltcw'i'tal i be K a, ICl'VL ml
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have,” _. . __.—...___.....=.=.— ,, 4. (20 points) Show all your work to receive partial credit. Correct answers without
accompanying work will not receive full credit. Determine the unilateral ZT for the following signals. (a) f1[lc] = 2"“2u[k — 2]
(b) f2[k] = k2’°‘2u[k — 2] CL) {(1:sz (5 “Heme/moi Swat/UH 213L823
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