prelim08 - ECE 562 Spring 2008 Prelim Exam: Mar. 6, 2008...

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Unformatted text preview: ECE 562 Spring 2008 Prelim Exam: Mar. 6, 2008 Name: filu‘h 04: Rules: 0 Do not open this exam until you are instructed to do so. 0 This is exam is open-book and open-note. o No calculators are permitted. 0 No collaboration is permitted. 0 There is no penalty for guessing. 0 You must show your work to get full credit. 0 You have two hours to complete this exam. Problem Points Score Good Luck! 1. Consider the following discrete memoryless channel 1‘ Y I '1. .7 '1 3' I 1/2 1/6 1/3 0 0 1 1/6 1/2 1/3 0 0 7 0 0 1/3 1/2 1/6 a 0 0 1/3 1/6 1/2 (a) (2 points) Let X and Y denote the input and output, respectively, to the channel. Suppose'that X is uniform. Compute I (X ; Y). Clix-1y} : MY) — HUI!) IL ‘L i: ourlm. 1L Jean-mum a? ‘r 5! H(Y] —- dim-2— *;i-i é : ‘ql'lI’ 2 ’ Jc'lq ‘ ‘ J"}% .L .1 —' I‘Kl‘r) '3 ""lw-g— "I" (Iqé' ‘1' 11v) 2 Nee) *1 '03” ii; 0'4in hard ):,4,,'[u4'.°m 1‘{!;‘r) = HIV) — HUN) Hle) I", n“ hp”! 0“ P”), /0 Cluue f’lr/ “N Htifuut31t l :{ ‘r=3 E 2 Ldlwf 0 0+ (c 0m;- “,4. germ; 6‘ ¢ the) Ht‘f) = H(Y)+ HlEIY) : H(YIE) : MEN HUI“) =0. Am a; 2 C Cal} VIA-"L, Jthud] ‘ "er TLV; 'Hc TL] Lr [ML-en; if fl!) u- Wu: mam W Arm-um u new, M (b) (6 points) Determine the capacity of the channel. , 2. An Even Spread? Let X be a sequence of i.i.d. Bernoulli(1/2) random variables with length 872. We call the number of ones in X its weight. (a) (3 points) Ciiven that X has weight 4n, consider the event that the first 411. bits of X have weight n and the last 4n bits of X have weight 3n. - Compute or upper/lower bound this conditional probability as best you can. A 7. g 3? Lu WefJb'l ‘lVl-S 3:5 V! at m: .t 't w mu «1 - Jl _ n Min-E I I “EN-1L) L "i 1 TLuvrm “M “0,2 4 PC / _ 2 _ “ .1 1L 1 {wavK-fill-E) L F c) 2 u olm @un)‘ \ 2-En°(zll%): " 19 nos 1. (gm “)1 Lin-ll) New): “3"” : mm = rlwrtc) F4") P(A)_ \ .l | -?ub{'\l.ll{ L ~(:mp(;|\,,/ d- ‘_ full} '2 I L¥“_H)V 2 — L (b) (3 points) For large n, is this conditional probability large or small? Explain. TL; cippvup¥ coowleer 'H‘e. roL) Mum-,1 "F0 OH», /| K?.c‘A)-4>b d'J‘ vb...» ,5 (c) (3 points) Given that X has weight at least 5n, consider the event that X has weight at least 611. Compute or lower/upper bound this conditional probability as best you can. A - \xu Weak“ d’f leor‘l’ fl} 3-;{1 I“, Wow-'3‘“ t4 lufl‘ Ga} 7" K i 5" 0(L “.L ‘ 1) _FA f“ i t HM: i 2 e (r 5 Z '2 TL onw- L?“ “)1, kg“ P ) K=(‘ \_______._..—__._..._.J > 2- 9uD(% Mi] I new = '2’? + L3” = ,_ Mi); ,_L(_f.) a! kan Fu 9 {fad—EM) é (nitriharfll) n=tn g (vh‘up 7:?ng é m/ 5 (9 “*9 2‘ g“ DEM)- meut (hum —r.‘v({h:)é HA) C tam) 2-9» DEL-[Hi rlle) = 53%;) = rig) I 2-;l(v(%ntl- blflliUé POW é @A+I/2(2W“) -rn(b{{u (d) (3 points) For large n, is this conditional probability large or small? Explain. r 6* 9+ J1 HIM‘lf 0(3f’llé) “9(7Ilql_))o an e rem.» w J. FLVH -->I’ a: m—a—A. -n‘2 nowhl 'p'c‘hr’ r' I 3. Suppose that X1 is a Bernoulli(1 / 2) random variable. Let X2, X3, X4, ...be generated from X 1 via a cascade of binary symmetric channels " BSC X BSC X5 BSC "' The channels are independent, and each has crossover probability p 1-71 29 P 1—P' (a) (3 points) Compute H (X 1, X2) in terms of p. maxi) = Hlm+ Him») Him H H JP :5 x ,1 ,_.. ‘C V! o \u + r4— I L J L‘ I. \_/ Hi‘lwiyi) (b) (4 points) Compute Hm H(X1, . . . ,X"). 11—00 17, _ n H = ‘ 6-: Wax.) — g H H,...,x ) {mtg X‘ee )(K b9 9' Xv.) H U;\x.,..., L.-.) = uncut") = umlut) = Mr). A —| ) 1- __._“”"”°“ = ___—' 5‘“ “4” = um. I!“ n V. “'5” 4. Fix-free Codes A code is called fix-free if it is both prefix-free and suffix-free. (a) (3 points) A Harvard student claims that if there exists a fix-free code with codeword lengths £1, ..., 6,, then Is he correct? Why or why not? No (own ~11; to}: ctr) = 0 ,Q‘ =‘ C(2J=" ’{1=‘. “(‘- = u _. TL; (a It Ff a", {'69 L” {z 1 I 1 [- Tlg. Convcue I‘V'hv‘MJ’ "I a“ 0'“ {anew k“ 4h lull] 10 (b) (3 points) Is it possible to have a fix-free code with codeword lengths e,,...,en such that 3 ._[‘- = —? Z? 4- 1, If so. give an example. If not, explain why not. Ye}. (ouzhr +11 to): CU) = 00 cm :01 ct?) l o 7‘"! {1-1: -_ TLc C0 ck I'J’ 4}? ‘ 'r' ’9, M, I 11 5. A SecondAChance I Suppose that W is a random variable with alphabet W. Let W1 and Wg denote two guesses of W. Let Pc denote the probability that both guesses are wrong . A PC = Pr(W =,£ W1 and W # W2). Since there is no point in repeating a guess, we assume that W1 75 Wz as. (a) (6 points) Show that Hm) + Pelog(|W| — 2) + (1 — PC) 2 H(W|W1, W2). ‘ ;[ w ?al 5" E- : O °+LcrwJL 6 A A A « leElvzl'fiJ : le|w,/w.,) + l-WWzIW ~= HlElv‘vUvi/t) + lelvinkhs). =9 mwm. v1.) = HtElv'G‘lv'L) + t4lwlv'0.’fi"s) 12 (b) (3 points) Under what conditions does this inequality hold with equal- ity? U) E 1]- (Quafi) w :3 cynlhj Mill-y +0 ts»! II C (2) (was. E J W{ 04" Wt - ‘ ‘. {fuL‘ Cn‘via E :| W U tyudl’ ‘ +0 +Lg VAN: l'\ . W\fw',w73 13 6. A Second Chance 11 Consider a channel coding scenario in which the decoder gets two chances to correctly guess the transmitted message. We say that an error has oc- curred if both guesses are incorrect. Let 02 denote the capacity of the chan- nel under this setup, and let 0' denote the usual Shannon capacity. (a) (2 points) Explain why 02 2 C. (gunk, ‘~ fence“. 0! (Ignlw) Caroc!*\) - ackrzyrnj COJCJ‘ el'u’ r"‘. 40- EOCL C‘A Lg. +Vrnca Wk"! 9* +WV' jut/j (“’1 J.) WAI‘M) “Vi a,L:l,-”q fun} jinn. .fr'vwg WC Hill ‘Mvt l4 C énrfict W IND? (b) (6 points) Does Cg = C? Prove or disprove. of (7" K, a) x“ = 4‘ l w). a (Na K ‘ OVCI 2"Inu’2‘} fey» (Mg ‘I’es, w] P: -> or (WI Hg“) = 7( é; H(Pc) + Pctnfi +(l-Y:) waw => itw'.».,v>z)2 l :25 waul’e-VH ’—I‘+c => (ea 2’? C255 $ Cz—c. 15 LA I ). ru'ufi “- ...
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prelim08 - ECE 562 Spring 2008 Prelim Exam: Mar. 6, 2008...

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