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# fall07_midterm2 - University of Illinois l Fall 2007 ECE563...

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Unformatted text preview: University of Illinois l Fall 2007 ,/ ECE563 Moulin Midterm Exam Tuesday, November 13, 2007 30 1,0110 N 367' Name Score Please do not turn this page until requested to do so. «r -mn!<y=x«ﬁé\w mvsc ’ \\ was? A‘- eawmmmzmwmm ”A / Problem 1 [20 pts]. Consider the DMC with input and output alphabets X = 3) = {0, 1, - - - ,7} and transition probabilities Mylar) = { Give the capacity and the capacity—achieving distribution for this channel. = (93,34) = (0, 0) 01‘ (7, 7) .: (x,y) = (0,7) or (7,0) : else. OOH—I Onsh— o'LWuﬁ van tame m Mshwmic i .3? need h? 0604: we who”); 2;? Hum (54.17%: > ( TL: resﬂl‘ng Wham! PM; Ply/x4) U Wrm. 0 (LR)? CHM 96:0 (241,4 5529, ”56A 27M ﬂaky/[#5 ” t 1 kW) WOMM 11,69qu Ln 5: Wig”, W0C over \7/ 3—; MaXtmuS WWW? M) Arm) 1%?) . We hm lab/b9: W) W +W <3 4 3 (:63?me To whim/67L h(Y/7Q)/ 3145? SM?“ @2069 :O M “((0): ﬁd? 3J2: g; I awn/05 C, jYLWA/M‘f:o 9r ‘fv — 4;: EV%\f:1/Z,~;C Cam be {Edam WW9, Rﬂerasmjff (f @- 6 Problem 2 [20 pts]. Consider the channel with input and output alphabets X = y = {0,1} and law 39:32—163XieI/Vi, 2': 1,2,... where GB denotes modulo—2 addition, Y0 = 0, and W;- are independent Bernoulli random vari— ables With respective probabilities 2%., for i = 1,2, - - - . Give the capacity of this channel, and carefully justify your claim. 9,) 7h: WW oat/maﬁa Kongo/maﬁa}? Y; : Yo (49 Y6~I . ‘Eon “w 7:; 1053 K?) A N') V (Glam? )6»??? if Mewf7L\$) «timedk {f (mu/1;] 1’51 ’Wrm 57X) 3 9m owes: CoMMm'cJZ w M m assist? by am} '5» W i ‘ - . "6% We SIOVW Mali] 51-7 a) i . _ ‘ f3 4W g W60 W “PL 2 km . 7‘71“ b; >..‘__. “MW“? 1%”; Vﬁrt'wj 51/an 55m» 4/5” W/ 770 / 2?, Problem 3 [20 pts]. Consider the additive-noise channel Y = X + Z where Z is equal to 0. with probability 0.1, and equal to a normal random variable N(0, 1) with probability 0.9. Give. the capacity of this channel when each codeword as”(m) is subject to the unusual input power constraint 2” x2 (m) S W. Propose a method for achieving that capacity. i=1 i SW6; Z=0 wzéﬁ hat/129m ‘Vrrﬁz‘éx’h’fy (31230]: 0,i>/-- IMM/ Commie/r a» following WM 2 p 2mg M04530 5%563 Hl‘a’farm 2, 32mm 3 ﬁffﬁrnaﬁ‘wo Wt'vm'on ( Wilma; [) A-bo . 10w) \$- A? IM‘V“) = «ﬂ/‘m WW“) Ya): MW} H WW") = MYQV ”LZ‘) WM: 108) . A 0,! __. 3 29/3“) “2'2 763‘) ﬁg? #66 ‘i A fza/g‘j else. A go [A F’Jfagﬂﬁgoiﬁ) M3) Problem 4 [20 pts]. An iid N(0, 1) sequence X n is to be compressed and transmitted over a binary symmetric channel with crossover probability equal to 0.1. Give the minimum mean- squared error for reconstructing X n from the channel output. (71 “,5“ 9‘, at 6‘ e) a CWMW 0g ‘W; ”D§C : C = I» )72 (0.») I For gal) SWWQ/ Wﬂ .h/MC K<D): i203 ~L gar D6) Nasal , R (D) < C 7L0 Wall? lmmnw"; Wu VIM/mag 69 get < C ~zc ~20? halo-0) é) D>2 Problem 5 [20 pts]. Let X be uniformly distributed over the alphabet X = {(1, b, c}. The source X is to be compressed with reconstruction alphabet X = {a, b, c, d}. Consider the following three distortion matrices (1(33, 5:): 0 1 2 3 0 0 0 O 0 1 1 d1: ‘0 1 2 3 , d2: 1 1 1 1 , d3: 1 0 1 0 1 2 3 2 2 2 2 1 1 0 Derive the rate-distortion function for each of these three distortion assignments. /\ i @ Cain CLO/lien Zero Al‘Skol'ﬁbh £3 ‘dwoﬁm? X‘Iza 5le No Vlad h, h‘aydnw‘f; 59M) LL‘E :3; [R(D§:o gmu >>/o.! ' 9” riffs/dAAoi‘fon 57MB]; W 35%“de in ﬁrm; 0% push/Way»; I ﬂaw/2)] ; ﬂown) : I/ v;£;-/)2(D>:0 fwd/ab?) ‘ ‘ 7? @ ﬁfW’wf7mLe 0L Us quvwfem/‘fb b M ”a“ 95L 3 f L0 0 ‘ 005%;er and W Hal/co \$1 Hmo"w(' ...
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fall07_midterm2 - University of Illinois l Fall 2007 ECE563...

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