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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 closedbook 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 widesense stationary random process Xt with autocorrelation function RX(T) : 62—173 is mean square differentiable.
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(b) If X t is a widesense stationary random process, then Yt : X t + X P4 is also widesense
.t t‘ .  _ v
“may ELYtYsl EL (wawnxw xwﬂ
/ we :ZQX (J99) + Qx (+~S+¢)+ Rx (tsct)» (c) Consider a discretetime 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 continuoustime random process
Y(t) : X ( M), where denotes the largest integer smaller or equal to t, is widesense 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 widesense stationary random process with autocorrelation function R X(T) : % is 1+TL
m.s. differentiable. ,— €ﬁ<i§($ and i5 munmms _ (f) A widesense 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 widesense 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 discretetime 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 ithrow, 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/:tX0+t2X1. (G) Let Z : Y; — My be a zero mean random process. Is Zf, widesense 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: >
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KY (ts): [iK’c ><o+tl><y)( SXD+ 85(1)}
: 153 EzX31+fLSLEEXIL1+ (f31+’tZS)EZXoXt] stl=ir+¥e¢+iCF 13—0 l% I3
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X1 1 01 3 I a 3 I 1 3
P”°’X'“%é> We 0 cm 118;) raw:1 o (7:72) (89
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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'
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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 widesense 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
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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
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/§:O> ‘, Xt is M.S. diﬁerenuué®_ iQy(/t,s): wa'f.3); gt—gépﬂes) Bonus Problem (5/50) Let X, be a zeromean, Wide—sense stationary random process with RX(T) 2 026(7) (Le.7 XL is
white). Find hf such that the random process +00
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This note was uploaded on 03/05/2012 for the course EE 7615 taught by Professor Naragipour during the Spring '11 term at LSU.
 Spring '11
 Naragipour

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