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ece 153 final

# ece 153 final - ECE 153 FINAL EXAMINATION December 8 2003 l...

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Unformatted text preview: ECE 153 FINAL EXAMINATION December 8, 2003 ' l. The random variables X and Y have the joint density 4xy, Os'xsl,Osysl fx,Y(x’y)- , . Find the density of Name: O L- UT‘ 0 06 £2(}):_ ____ 74g?(}727)d7/Gewmi Befrwswt/ I W-e int/5+ 'Szmuﬁ‘amQO-Uél)’ 5a’hé{)’/ 05 51 05751 0557 s( 05 4—‘~ Name; 5 0 L. U TLC N 2. Let X1,X2,... be a sequence of independent identically-dis'tributed‘ random variables with density V ,2, 1 - 2 , 2 fx (x)= e 20 ,k=1,2,.... oo ~1— ELFCXu)]=ﬁ*;—r—'0‘~Sx€ d7: Consider the non—linear operation ‘_ 0—- 0 . “V2.7? 00 z. X, X20 '2. 1 2.3.2523. F(x)= 3 GEE-r X [email protected] Olf , 0, x<0 . 26 _ .0: Sb thtth I i ‘ ~ 1 ow a esum v 2 ' varLI—‘(mj = z: {mm—(scam) 21'! n 1 u Sn= F(X‘) ~ 0" n 131 k - — 70"}1F is a consistent, unbiased estimate of the standard deviation 0. That is, E[Sn] = o' and lim Var [Sn] =0 new]: @ZEiF—([email protected](% = o— Name; I 3. 1 X1 (t) and X2(t) are independent random telegraph signals (0 ) Nl N2(t X1(t)=X1(0)(-1) , X2(t)=X2(0) (-1) , Here N1 (t) and N2(t) are independent,'classical Poisson process with constant rates Al and 1.2 respectively. The random variables X1 (0) and X2(0) are independent of each other and independent of N1(t) and N2(t) with P<xl (0)=1)=p1 P(X2(0)=1>=p2 . Let Y(t) be the product Y(t) a. X1 (t) X2 (t) . Find the correlation function of Y(t) 5 é RY(t,s)»=E [Y(t) Y(s)1 EiW’WY‘Cw] = E[Xi(t221(t)z.(s)zl(5)] : E[Zt(W-§L(§)] E[Y1(f)X_2(\$/1 E {X} (timed : E [2:10) (~ Wily? — I 0 “N1, ‘5‘ ' [@Zwtt) ( )1 Because ECf)au0iX——2(f/ , are [MAQPQMAQM ll rt. w = € ‘ CerthdQ/V‘MV T149 Cage for win h “thtﬁgl ‘ EEC; 3 9_ 1):?{7-6! 9 tMtiQV‘(/ E Knit) 21(9)] 3 Q 90 Mad“ \anllx N —2(>L+,\2)(t~6l ElfttzfﬁM/J = Q 1745 wt hon/Le, Name: SOL-VTION 4. X(t) and Y(t) are independent, zero mean, wide sense stationary process with identical correlation functions Rx(1.')= Ry (1:) = R('c) and power spectral densities Sx (w)= Sy (w)=S(w) . Consider the new process Z(t) = X (t) cos (wot + 6) + Y(t) sin (wot + 9) The random variable 9 is independent of X(t) and Y(t) with density 2 E , OSBSJVZ 0 , otherwise . Show that Z(t) is wide sense stationary and express its power spectral density in terms of 8(a)) . ' E 2:55) 3 El t2l'c06(waf+9) +~EL ﬂ] SLMCwat¢6~):O [ 3 KO V KO 1 QZWS) = ElZCf) EMU : Elmﬂﬂfﬂ 005(wo‘tre)005(uw\$+6t) ' +E [YWYWJ wwo‘tw) MGWS-Ft?) ’17: tug '+ E (XE/2(a) W(wo‘t+ 6) MCwo 5+5) + £5555)?ch 004 COWS-+5) MWdfﬂa Name: gOLIZI (ON ' p 5. The input, X(t), and output, Y(t) , of a linear system are related via the differential equation 21m) + 3Y(t) = ix“) + X(t) . dt dt If the input is a wide sense stationary process with correlation function 1 41:] {w‘al’CLCu/f Rx<r)=§e , 6 6 0M: : 2a ‘0‘“ ‘ 2Hu 1 ﬁnd the output correlation function RY(1:) . . Consider m‘ath‘auclam 25+) and Yet) 16 OLAT'QV‘MIWQ Traushw fuue‘f‘ww. ELEM) é yum») Ctr-e Faun-2r Transforms 2(Iw2‘Zcrw2 +3>Ztcw= (cwﬂWHZlo‘w) Haw-YEW; z [+5.20 XU‘w) 3+2£w 9‘0 \ r~ -161..wa l 9z(w.7~~06:2L6 6 0m - MHw: _ 7_ \ ~ \ _ lH-Lu/I ;J—'—1 SiLW)—1H(LW)ISX(W)” [3+2L‘W'7'H‘W _ -l .. __I_ 4% ‘ C’sz ‘1 (3i)z+w7~ ...
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