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sp2008_exam3soln - Problem 1(25 points a Given J(t = a...

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Problem 1 (25 points) a) Given J(t) = a e-atu(t) B F(OJ) = a. . Find the bandwidth W defined as IF(W)12 = _ 2 1 a + Jm (half power bandwidth) [your result will be in terms of a]. \ F~ )12- :: Ti IbJ-Y- !.- ~ J- en. Z;:' I +-L~'~; W:=0/ I + ~)' 2-1 Ct.) BW= ~ b) Given J(t) B ~(m) and J(t - to) B G(m), what is the relation between IF(OJ)I andIG(OJ)I? Explain. _ 'LJt f - 4 UJt~ I - G(0)~ F(w) e. J 0 ~ e. - I) 0--() I G-~)t ~ I F~)( I e--4!()~~ (I G ~)! = I f(<<»! h(t) 1 c) Given J(t) = 38(t) + 58(t -1) - 48(t - 2) , and o 1 2 t yet) = h(t) * fit). Sketchy(t). (Label axes carefully.) h(tJ= ~ ( t~) ~ [)::- 3~~)~ ~6(.G)+- z. z. -1 ~l~) S" 3 o 'f -t:- -1 d) Given below are the impulse responses of two different systems. In each case, state whether or not the system is BIBO stable and causal. Explain your answers. I k(t-) i) h(t) ~ reet( ~ ) Lt::=L t 2.,~ -2,'3 "J,..$"" L.J L,~}Jt :=: ~ ~ 11 (f) ~O /at tLD__ ~-~ ii) h(t) = cos mot u(t) 00 I l~wo tl ctt- --s:,> ~ o l1(t)::D jot t L D ~
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4k> Problem 2 (25 points) a) Given f(t) = 5rect ( 0~2 ), i) Sketchfit) andlabelaxes. ii) Find the frequency spectrum F(w), sketch and label axes. f€v)=- ~X"t24~) ~.z.C() - "2- r-~) ::: ~(OI{W) "t Iw iii) Use the above result to find the frequency spectrum of g(t).
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