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wss prob2

# wss prob2 - University of Illinois at Urbana—Champaign...

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Unformatted text preview: University of Illinois at Urbana—Champaign Department of Electrical and Computer Engineering ECE 434: RANDOM PROCESSES Spring 2004 Midsemester Exam 2 Wednesday, April 21, 520077:00pm, 165 Everitt Laboratory READ THESE COMMENTS BEFORE STARTING THE EXAM! This is a closed-book exam! You are allowed two sheets of handwritten notes (both sides). Calculators should not be necessary, but feel free to use one. Write your name on the answer booklet. There are ﬁve unequally weighted problems for a total of 50 points. A bonus problem worth 5 points is also included. Problems are not necessarily in order of difﬁculty. A correct answer does not guarantee credit; an incorrect answer does not guarantee loss of credit. Provide clear explanations, show all relevant work and justify your answers! If we cannot make sense of your writing or reasoning, you may loose points. Read each problem carefully and think before performing detailed calculations. Only the supplied answer booklet is to be handed in. No additional pages will be considered in the grading. You may want to work things through in the blank areas of the exam and then neatly transfer to the answer sheet the work you would like us to look at. Problem 1 (14/ 50, equally weighted parts) This problem has seven independent true /false questions. (a) A zero mean wide-sense stationary random process Xt with autocorrelation function RX(T) : 62—173 is mean square differentiable. 4m ‘ g 0Lng Q is hot oldemoo‘éhx at L: o_ (b) If X t is a wide-sense stationary random process, then Yt : X t + X P4 is also wide-sense .t t‘ . - _ v “may ELYtYsl- EL (wawnxw xwﬂ / we :ZQX (J99) + Qx (+~S+¢)+ Rx (ts-ct)» (c) Consider a discrete-time random process X k deﬁned so that the Xk’s are i.i.d. random variables with P[X;c : 1] : P[X;c = —1] = The continuous-time random process Y(t) : X ( M), where denotes the largest integer smaller or equal to t, is wide-sense stationary. : E (X0 I 1 :—_ O (Emfse , ' t l , a; ELYI'T/IQ]: 1“ I (d) The function RX(t, s) : cos(t + 8) cannot be the autocorrelation function of a random process X15. True. Qx¢ﬂ=£€><3120 But Ruff): Oosz‘t‘ A wide-sense stationary random process with autocorrelation function R X(T) : % is 1+TL m.s. differentiable. ,— €ﬁ<i§(\$ and i5 munmms _ (f) A wide-sense stationary random process with autocorrelation function R X(T) 2 H—lﬂ is not mean ergodic. / FOJSQ “TL 2: O :> XC'ﬁ) lS Mom €r6¢0iic l/CIQVO (g) The function Rxm :{ lsinmla ITI <1, 0, else is a valid autocorrelation function for a wide-sense stationary random process X15. Paige, Rx(o)7/l?xm)[ (+71?) ‘rs not \$933494} Problem 2 (12/ 50, equally weighted parts) Consider a discrete state discrete-time Markov chain X k with state set {1, 2,3} and one—step transition probability matrix P given by 1/8 7/8 0 P: 1/2 1/4 1/4 0 7/8 1/8 (Recall that P (2', j), the ith-row, jth—column entry of P, denotes the probability of transitioning from state 71 to state (a) What is the (ﬁrst order) stationary distribution 7r of the Markov chain? (b) Assuming that the Markov chain is initially in the stationary distribution that you calcu- lated in part (a) (i.e., assuming that P[X0 : 1'] : 7r(z') for 1' = 1, 2, 3), calculate the mean and autocorrelation function of the random process Y/:t-X0+t2-X1. (G) Let Z : Y; — My be a zero mean random process. Is Zf, wide-sense stationary? Justify your answer. [2f 7T: (Tr')nZ/TT5)' _..T( P: TVT E (PT.I)T[:O MemnwAICL 771+WL+715Z » , z ’ ., ~— — _ 4 , . (E) am]: it- Each t - EH1] (, tlxo]:él><,]— Eugzﬁig lLlL 7- : i: > Z V (’(i+f> [3 KY (ts): [iK’c ><o+tl><y)( SXD+ 85(1)} : 153 EzX31+fLSLEEXIL1+ (f31+’tZS)EZXoXt] stl=ir+¥e¢+iCF 13—0 l% I3 Xo I I 1 n9\ 1 2 3 3 3 X1 1 01 3 I a 3 I 1 3 P”°’X'“%é> We 0 cm 118;) raw-:1 o (7:72) (8-9 HX°X3l§éH1§¥12%-%W%%+é%é+é%%+9TE; ; QT H18); %+SC(+'CS)+ fStSth) QC) [5(th £0 (32 H18):— QY (‘6. 3) ~ ﬂy('f)'/I1Y(§> N001: Wluzn §:O , (2%,0) EO CV’C) (gut We Cah OJWIAgs 30m {>0, S>O ,taLS sac/1 f/Mt Q2 (f. S) i O z(f) is Hot Problem 3 (12/ 50, equally weighted parts) Let Ni, 0 S t < +00, be a Poisson random process with N0 = 0 and rate A > 0. (a) Find the joint probability that P(Nt1 2 n1, N,,2 : n2) for 152 > t1. (b) Let Xi = Nsds. Sketch a typical sample path of X, and ﬁnd E[Xt]. (C) Is Xi m.s. differentiable? ((1) Assume that A is a random variable that takes values 1, 27 and 3 with equal probability. Find the mean and variance of Nt. (0‘) [Cor “17/”,7/0 P (Na: VIM/V13: m): P(Nt,:m) P(N+L-/\/t,: hum) : EL WW“ “I! thym)! : 85% A“ t"’(+z~m”“n' V1,! (ﬁrm)! “7.): O for n). < I’ll t M1 E. Ext]: [0 kaOlS : Z 't do. msf ‘ InuarQ/(i of o\ Froqu; (Ant/i amt/mums IQNI’C) (cl) «if/VJ: HUMAN]: EfA/ellﬂjV;+EfA/{H22]% ~ + El/l/tl/KZSJ?’ (f+2t+3t)/3 :_ 2t E- 2. _ _‘: 2_ l {M}. [LU/Vt [H]: (t+t+ 4tz+zt+9t1+3t)/3 2 giwizt L l/wlA/J: £0th (EL—NJ) 4 1 %fl+ ZJC, Problem 4 (8/ 50, equally weighted parts) A Gaussian random process X1, is wide-sense stationary, has zero mean and autocorrelation function Rx(7') = 26‘”; (a) Is X L mean ergodic? Justify your answer. (b) Find the MMSE estimate of X3 given X1 : 1. Let another random process Yt be deﬁned as t . Y, 2/ ei‘sXsds. 0 (c) Compute the mean function of Yt. (d) Find P[Y3 Z 2] and express it in terms of the function on) = :00 x/1276ty’dt. (a) Yes th& RM?) —-> O (kl—>00) I (h) For TCTRVS’ MMSE: AMA/U71 A F 4 _ X3,MM\$E (Xl:|>: COX/(X5,X'> CUVfXIixl) < I“ EL><J> : QX(L)- R;((o).{ : .2Q—L: 01—! : 6:2 S 2.4ij ots : 0 {WV EEXWXV] duoiV f: f éwvz-ém'vl “WW ; u —u—v v—M c101 ﬂowfow a ‘e’ e 4. fju (Mo/(wk -é (— 7e ’ — Y 2 OZ P0931] : PL I~:e’67/JT'—7-€'°—] : (—72%) Problem 5 (4/50) . . A x > ‘i . . . Given a zero mean random process Xi With RX(t,s) : tzsze (Wm!) determine 1f1ts m.s. derivative K : X; exists. If so, determine Ry(t, s). (X 1 ~[tl *S__ L "5 JEQXmL/S): gfa (&§€ Se) SZO flail“ (2885+ SL€S> 3<O. d qt L - —5 L — CM gaggﬂm): (ﬁe—t6 )(Lse—ges) \$204420 (Zththlefw 2.561 Sled) 320 t<O —J( L “t (Me—ta (2363+SZQ5) 3<O,f;'0 S )‘x f ‘ L 1 (Me +ee*)(28e5+ stag) S<o ch {<0 I 87,0 {7/0 550 We C M an SQL “ (X . dsgx’agx/éggigx {<0 >1 exist O‘N‘X Cohuhumsy (In Parr/“VJ” 1w §<O {7/0 , f ? S<o O‘yq Cont; 'WW’V‘S CVC {3A1 éomhdmﬁes f: 0 0)» /§:O> -‘, Xt is M.S. diﬁerenuué®_ iQy(/t,s): wa'f.3); gt—gépﬂes) Bonus Problem (5/50) Let X, be a zero-mean, Wide—sense stationary random process with RX(T) 2 026(7) (Le.7 XL is white). Find hf such that the random process +00 Xshllisds ’00 14: has autocorrelation function K y(7‘) : 8’04“. EDGE]: fjemﬂtmds kyﬁ’) 0 EC YT «Y3? _ who HO ” V fwd“ 0W [:LXM‘Xv‘Xl ANNA) ﬁgmv) - (DO \‘rDO : [40 am [M 04v. (rim—v) AFC-M) WHO/V) “+90 4,00 CAB-n Ojc Vow} 2 I 7. Q 92 0‘ La OW {we 0W. V\$(V')I1(TIM)A*(VLM) v’::u«\/ _ 1 +120 ‘VIW AW—w- New dd 1 (TL <A(M)*A*<«M))(’C> [7‘ L } (kyt?))— (T Hue) Hm») V (Q'uU’CI) ’— Q\o§ 6:; 104 L 4‘ L L i M 1 , (w I, > 04+“) Dhjw d’jw V‘ ) jtw Om [905\$ng Soﬁa/1m IS HUN)” Griff/1W) (4%): ‘Véf Q:OUC {7,0 ) ...
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wss prob2 - University of Illinois at Urbana—Champaign...

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