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exam2_sol

# exam2_sol - ECE 301 Exam 2 Daniel Aguiar July 6 2011 This...

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Unformatted text preview: ECE 301 Exam 2 Daniel Aguiar July 6, 2011 This exam will be 60 minutes. It is closed notes, closed book and there are NO CALCULATORS to be used on the exam. There are FIVE problems. Problem 1 (10 points total) Find the impulse response for each system below. Part a (5 points) Part b (5 points) a: AH): 714% Mm g: 4Z~J= Wm) Problem 2 (20 points total) For each impulse response below, state if the system is causal, memoryless, and/or stable. N 0 work is necessary Part a (5 points) Part b (5 points) Part c (5 points) Part d (5 points) Pdrw ”d M“ Y‘“ P M) M" M 4/” (Jr P 4 L Ye; Y” r“ dr Problem 3 (20 points total) For each signal, ﬁnd and plot its Fourier Series coefﬁcients. Part a (10 points) 00 1:05): 2 [(t+1)(u(t+1+4k)—u(t+4k))+(1—t)(LI(t+4/<:)—u(t—1+4k))] k=—oo Part b (10 points) a .1“ I -174 .1 "JL 7L. J k) = 4 _. H1 1:: 7 (me/e J6 M6) 7' V k 7! ( N :21; Uté J": éJz £15 1 _. PTA/ﬁt Jack 41': 'fn‘icﬂu 0 / .— 0 “ . 0 2 ‘4 -IA; .é. U‘Z‘ké/ - ,L 2t - 7% -—— dz - ‘7 4*” ER 6); K, +07% - 3 J‘ 0'” ~’ I _ 1’ w~i ‘dW‘/I+ 2t V?“ i fledzue/zf] , ‘17? —{’/< = ##(I‘éw '6‘) W 2 _ F = W(’ “WV 9" '" /< 3,17% ‘Y'3"Z we 1 2 £7 <"' ' é Problem 4 (20 points total) If :1: (t) is real and even and has Fourier Series coefﬁeents ak, Part a (10 points) Show that ak = a_k. for k; > 0. Part b (10 points) Show that :1: (t) can be expressed as 00 2 x (t) 2 be + Z bkcos (LT/ct) 19:1 T and state the expression of bk in terms of mg. To receive full credit, Show all work. 'k 111% ' : .L .U T a. 4A TJT MUe .Jt . w a - [j «MW/”J7: UM 40:46 "/< - ‘7: T , I . qjkéryULJ _ 77 2((-U) e a 'r - if A ~ {66/ , .L U) M TUJU fwd MU— 17L?“ a 1 4k "" 'Kﬂ’é x: M): Z «K68 T =1” o0 - 1/956 -( L772 ~ Jl‘ T : Z akedkr f—do +12 4‘6 (t/ /<=-oa Laf [“4 00 °_‘: 2w .— 1H6 1 Z 4' U/a Ti 7L‘do + Z 4k€d ,- ﬂat 1 e F3! °° . ur , 2-1? _ uk’i 1)" Ti = 40 + 2— 4k {a T + e / km 5! ~ 40 Asd 90 HT ‘ ~ 24 k’d ”a k - do 7L Z 14K CIS{7— / 0 /<<d Problem 5 (30 points total) Using convolution, ﬁnd the output of the systems, speciﬁed by their impulse responses, and the inputs below. Part a (15 points) a: (t) = cos (2m) (u (t) — u (t — 3)) M25) 2 an) —6(t—- 1) Part b (15 points) \$[n] = UMI l—Z/l[n—5] h [n] = (g) ...
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