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# MTSoln - ECE 563 Midterm Exam Fall 2009 Name 1 A quaternary...

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Unformatted text preview: ECE 563 Midterm Exam Fall 2009 Name: 1. A quaternary channel has transition matrix Q 1—ﬁ ﬂ 0 0 _ 3 1—3 0 0 Q _ 0 0 1—7 7 0 0 7 1-7 a) Suppose that *y = 0, ﬂ = 1. What is the capacity? b) Suppose that *y = ﬂ. What is the capacity? c) Suppose that 7 7A ,8. What is the capacity? 05 wt“ Y=0.(5=\, HiX‘Y)=OJ yo C= WW I(x;v\=ww Wx) +4) X=‘IZ»3,H»=> '56,): ﬁ’yﬂlﬂlq w: Hum—mus», So C: I(X)Y)=\r\(Y)—H(Y\X\= 2-H!» c) “Lo,“ gap/wt CLMM is M L'wyu Sywwhéc 19m; .)~\/\L we Cameoruvﬁr iSSQ: GU‘L \$0 m exam-iv J’Lk ofbwq I i ,g, 1:71.?) Lie) £4" \$5M P' TVM .le». H (Y3=L\(¥2 %2 %»‘i')=-p\°s E— (We: 53 ‘ \* W?) W é. \4 (Y \x)=7iH(¥\¥=D¥§HW=i) ¥ WNW) ﬁr Uh he») - So C= "g,“ 1 " ”Hiﬂ-p*Mkm—h-rm(r) Wt. F’Ts oXFclmeé. gum 1-H.- -(be(”awaits-A-mmmmso (gs. pm mm =‘> [7* =1 ' 2. A binary source has distribution 10 = (0.89, 0.11). Determine if it is possible to encode (combined source code and channel code) so that the bit error rate is smaller than any given 6 > 0 for the ternary channel with transition matrix We COM \OvJ-UJaotmé 4L“ ‘9): Chara (N7! PG" I -.l. 3M, W CV\/¢m~t\ '3 gyMWm, m expat-L pod—5 A.» re\$w\-L “m MXDM Mal—wad ML!“ m-an, N’s-km M Foo: 1; —=-.) sup: so HM: wage»: \0332 m axwh HUDQ '3 HA)?- log 7-?- l 50 a C. a m- \ =0.\$!>Hls>) J: . So Sltvmmwx‘i Caévb TM :M‘Aws Ari-«3r '.J B 9055:“. 4—9 w:»&_ 9, vl’L-k'l' kid. Ei} gror Val-L Ts be7+‘VAV:\7 SMAH. 3. A Huffman code is used to encode a dictionary of 4,096 English words into sequences of bits. Prove that every codeword is at least two bits long if the most probable symbol has probability less than %. Does this conclusion depend on the size of the dictionary? PrOD‘C 57' CDAMAmi'iOAM I ASvaw. We 39 R Coéeydoré C(x\ Q” Mré X .gJNA :5 9w\y DAL b‘w’r ‘93 groan. rm axe, mt: Jim“ ’L w‘orvls, 74’ T: C‘LArJ’WA-i— AAA/e Cavx WV b-L DAL hunk MFA , 3,0 )c my)» 50.. «PL-L M499)" Fold», mg! or w Q0v~\A W543 W Cole «Loy-1» Wth New Consider 4/an Meade/:3 process. ID )( TS +0 radon. a Ceécwerl 0-9- kwaxk l, 713 probab‘lw/ mug, (1qu cc} he‘s-l as lava} OLS “It 3WM\$ 9” ”auburn” all J/LL aer-Mbg 91‘ caé¢wels OJ” Mela 96x59», LAM-m «kl-4. \oxslr shag, CowcféJr W cecal/é J-o his; SJ—a bl +1... antes)” eroa'”~ L3”; a‘n“‘a’vx.l°*——l’L-L wars [M w are»? by»; bum ,5“ he Are Won-Asih M 9% amp LL‘)’ VA: 9LC“)+‘H+P(QI\D I R5=P(L\)*“‘+P()At3 I Pf :P(1€)‘ [305mm LAOS) MAM— PAePgi-J’xé 5. M O\ CDAJFPA&EL+‘-\%Ai So Hr my)? LL HAL—l am?! COAX/War), 73 Cc!“ lens-l Z W's long, TVQ; LOAdel‘OA (\ﬁu W05: (lame w *L-L \$1?! 95 Mi Aid—Nash, .\ ’ (“ﬁll-x PMX <55 => \‘%\‘> ’5) 4. Use Stirling’s approximation (11! z 2mr (3)”) to show the following statements for large n. a) Show that (2 )~ NH“ ) b) Show that W ~ enH (p) where pk = nk/n. c) Consider a data compaction code that processes a source block of length 12 over a K — ary alphabet by first counting the frequency of the; kth source symbol (nk) for the actual block, then describing the sourceword in two parts, ﬁrst stating the composition (n0,n1,.. .,nK_1) of the block in a compact binary representation, then stating the speciﬁc sourceword by giving its position in an ordered list of all sourcewords with this composition. Is this a uniVersal code for large blocklength? ) l“ w. ’ m Ga“ 1V )‘h m 23”“ C‘ \0 (“40‘“«1— 1m QJ’“ mm m”?- “ “ (“4‘32“ \$540“ ‘2& spur YllW mj’zmw kbk— momma] k V_\_~_\& aim—Q-m—V bah-k) "RMA-l-‘QA‘QAA ,gwk-ﬁi‘¢x(w-k) = lam—“‘24- +t—‘9Mm= Hui :) g0 («3"9- QM?) ‘A N 1.1M ’1. J... b) §M\\°rly, 117%”: i'ﬁﬂﬂivw Tikm?‘ so twﬁxztlwmiqu ‘ZR‘Q’A‘A ’ikﬁhwwlh :Zkbﬁlgw’k‘: A“); ﬁ’“(\'”~ -,¥)=HLLP) 4. Use Stirling’s approximation (n! m \/ 2n7r (an) to show the following statements for large n. k a) Show that (Z) w eng(R)_ b) Show that 11% % enH(p) Where pk = nk/n. c) Consider a data compaction code that processes a source block of length n over a K - ary alphabet by ﬁrst counting the frequency of the kth source symbol (nk) for the actual block, then describing the sourceword in two parts, ﬁrst stating the composition (n0,n1, . . .,nK_1) of the block in a compact binary representation, then stating the speciﬁc sourceword by giving its position in an ordered list of all sourcewords with this composition. Is this a universal code for large blocklength? C3 E&L\ B\‘>!.-\\‘- Comm} mah‘m Mfg LN V\ o‘- vaf S\,M\30\3 \$0 WL m8 ”'0 MP2,. Ham 1 tars \a\~c.\( imbux ) w\/\~., odor/x5e. wwt 0 r 9 w~\03\ :5 PL Y V\‘ \ ngtp’: “(I ) JV: D‘\°5v‘+\°5ﬁx 1,; 7\\°'3 Q _' P So -¥\A‘:x £331; TS MVMUSQl «per large biotic \Mjlk ...
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MTSoln - ECE 563 Midterm Exam Fall 2009 Name 1 A quaternary...

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