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prelim07 - ECE 562 S pring 2007 Prelim Exam Mar 8 2007 Name...

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Unformatted text preview: ECE 562 S pring 2007 Prelim Exam: Mar. 8, 2007 Name: gllv‘l‘ufl 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 1 6 Good Luck! Score H 1. Let X1, X2, . . . ,Xn be independent. (a) Show that 'I(>1\,...,xn;\’) = %I(KL~;Y]X.,...,Y:-.I) (7| = :(Ettxttx....., m) — Hm 1 x,‘ x0” Y/] M , = E [ mg) — Htx: l xwx VJ] (L1 Evie/win”) 2 % CHUU— HUJYH \:l (mu: («vi-'+\‘ar.'u) "(IVT’J‘ ”JF’W‘I) 2 12:: I(>‘c’/Y) (b) Give a collection of Markov chain conditions that is necessary and sufficient for equality in (a). 2. Let X1, . . . ,Xn be i.i.d. Bemoulli(p) and let Y1, . . . ,Yn be i.i.d. Bemoulli(q). Show that 1 —_T—L-10ng(Y'1’ " ' aY’n) converges in probability and determine the limit in terms of p and q. a1 to; m (n... KY “1 We»; \«w of ”we. VW““", +L‘l’ (”V‘U" +0 E ~ Ia, mm] = ”W; F — (ewe; (l-r)‘ 3. Let C be a lossless source code for a binary source. C : {O,1}" I—> {0,1}*. Suppose that O is uniquely decodable, and that it makes a least one string shorter £0?) < n for some 50’. Show that it must then make at least one string longer fly“) > n for some fl. nrrm +Li+ (v.1 fiCfi-zj C A p“- JEN' 32; We w?“ )nw A (”+"l‘lfiu‘. fiw‘ 7‘ ' i 2’19)? f '2’“ : ‘1? 6‘00“ 3'8 éSPG‘G" {'Mtc, [0(2) 4 M [a] (ow; '78 ‘Hzr Iuefiwktfil Wurd- L9 f+r::‘l, l?w} ‘HH/ (0"4"J‘.‘+f kufl‘, lyr;u.l-‘iy ‘fvl‘c‘e Ht (0‘): if V“‘)‘vel\) /?laJa’rl/g This page intentionally left blank. 4. Compute the capacity of the following channel in terms of p and q. 0 l 2 3’ ”[1‘P—q 0 p q ‘ 0 1-10—61 q 27 Here 10 and q are nonnegative numbers satisfying p + q S 1. C = WW! 10190 = MN HM" HMS!) pm PM “(TH-i v‘r "W‘erwmt— ,I Ptk)‘ 10 CLMMQU U/ {VlfU/L Fouow Jfi/JVA’I‘OA \ ff ‘1’ '2— '2 Cf Y=j : O MCVWCJ-f m 0 HH 5/: Me) + Hms) : Hm ,, “(c r/ EMM- ‘por mm) (W) HCE) = MPH»), HWIE) = H(YIE=I) Nth) 1+ HWIE: o) rm: :0) g [. (NEH) + lv F/(Eze) =l w) quit“ if “‘0 fir V“:(‘QIM __g (Lug you UM3'A W“. 'I'lm c —. \M‘t’) — Mm) = — 6mm} lv (12/ ‘(W/l”) m 'L 7. WWW“) (WW 7 + rm + 7h; } 5. Consider two channels, p1(y|:r) and p2(y|x), in parallel. message channel X 17 (M53) \f p (ylx) 2 channel message W encoder 1 2 decoder W Suppose the first channel has capacity 01 and the second has capacity 02. Let C denote the capacity of the two channels in parallel. (a) Show carefully that C S min(Cl, 02). C 7- NM TU)?” (M For an) (9", ‘9 JA" 9”“ I’”"”""J "WW”, I(,(;'2) é IMHY)‘ (b) Suppose that p1 (ylx) is a binary symmetric channel (BSC) with crossover probability p and p2(y|x) is a BSC with crossover probability q. De- termine C in terms of p and q. Does equality hold for the bound in (a) in this case? 0 "—M O O o F ‘3 V ? 0 0 (J 0 l’f ”F TkC/ :f v1vi‘Va'mJ» +3 ‘Hag CL‘“"€' l' l" d f‘ Y‘ I- r W" r= M-w + IMO. E '4! H (a) J“ '1' ”'4‘ ”fftffln'l‘l Ln”: +gkc iVA- l M I 3 L 2 .t;: > _L t V = 77 '9 "’ ‘r w ' V '1 r=i‘ .1 Y. (c) Suppose we now allow coding between the two channels. The box marked “coder” can implement arbitrary transformations. message W message What do you think the capacity of this system is? Justify your answer as best you can. ‘ c = m C". (2/. V a TL; CQJCI c,“ Jere/a 'HL WU»): RV ‘ ‘ (0W W‘Vw'f.“ :4“ Q’f" Mli)fl«v 0’9 ((CM(‘JQ 1" RI I +La ffcuwo CLIVHL TLff W41” Clfllzli lO 6. Consider two realizations of an i.i.d. source with alphabet {a, b, c, d}. "'1 abdbcbba x a? baadcabc (a) Determine the probability of the second string under the maximum likelihood distribution of the first. VIN. Jv+a.‘Ltt.~M .r ‘“-< “the. III _ 31;; :[i_——] F32,_[8 7 5 5] t 2 9 ’ ll We. (b) Find the i.i.d. distribution p that maximizes 1 _’ 1 _. 5103P($1)+ ; 10gp(:y2). 7)? .4: +1»: {widen-Han a-P 1?: $5”,th .4 7: Wm} ‘Hne its Jvhttm-M Huh» wipwhf HEM'QJ =1 H?) .3 U. +L¢ +\)YG '£ 'x 12 (c) Express the distribution in (b) in terms of the types of (2'1 and :32. *‘iYfi (hf 72 U. fiwyir/ “HM. dVfivAJc 13 "P ...
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