Hw4SolnMain - ECE 5670 : Digital Communications ECE...

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Unformatted text preview: ECE 5670 : Digital Communications ECE department, Cornell University, Spring 2011 Homework 4 Solutions Instructor: Salman Avestimehr Office 325 Rhodes Hall «RT a v I _;' r .J 4- _, (.1 ) L L] -' ‘Pf( L \ I}: (133i) ‘RT g- L“ 21 Ta, 1911 "’ "’ - .. ('1’ XML ‘13.“! ‘4J)\1'&:) "Dz-5'0?) (i; 4r 4 . a 1v\1!'c I C10“ m l—ereuflj bed"! 1 _. “‘1‘ 4: h’c. ‘fiwfl " ii 2.7" 2 3' IA (‘ .» (g "' a! 4 _ 4 .3 X (t) f“ ! £_J 13‘L iafislisfij'fivmsv?) ' ! I'd-F a '1" (In) . , '9‘ -7 <'- __',‘-‘* "'7 - a “’ ')‘ ‘ ’ 44 PI _ 1.__ ll.). Ilia-(9r, p- bfi). . it? "I III C JEN-“Ir!fl JAJ [$3 ill/‘r M J, e1+i.f.§ “(a i-t'g “Pk-4349;] 1 .Lflfiun...'.1 l“ 51-. L- gl'menasu “w: arm-“mum i‘lika-M CJr‘J'fi‘W'é n. fllufi J “t I ' 2»? "LIT T‘m‘» lug! + WE‘LL J J-A'ML LC ‘J'tvi'r'n {a}; L. i {It I I r ‘J1 .3- Eds» ; \ J. \- ___H__ 31;!“ I fl-l’f‘C/l-d m a, . " I‘ ' {J 1 K :.433,|i=§%:c1(i¥#p=13) : Q i. l— J Arr I a» t J” ) t'm-Ifxr I At. 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(H._________,_ * ’- _ SNR h (M418 JAM—a)? firs R (R11 flit ijruk65JJ} of Imme) a geraniu‘) a“... t)”; +5 Ear; 6‘8 T4 cc: ) Tue Quinn!" +‘3 ?A'+ (C J v3 “Hy; n! M2 vat: c4 ram-luau jrqt-u;€x/ (Mi-l Jr'“ “"3 P'°l‘“”""110-*"7"/ *‘e MM“ 4 fie acme. WM, 4:: --' L'": ' ‘ I am ‘59 T—m, .1 R4. :6 11:: mm Hm! muc- MA “*W J " 1 . a . {)we 4" “HA? ’A “"4: afloa- 3 I. '3 4E {fanny-"((4% «GLGL’ () Ont I ‘3‘" <5) (e) The plot is shown below. R3,, saturates at log2 M. Because we use an M—point constel- lation every time slot, we cannot hope to achieve a greater rate than this. 7 —20 —1 0 0 1 O 20 30 40 If the distance between constellation points is much less than the standard deviation of the noise, many entries of the transmitted codeword will be flipped and a large amount of redundancy will be required in order to ameliorate this effect. This will result in a low rate of reliable communication. On the other hand, if the distance between constellation points is much greater than the noise standard deviation, few if any of the codewords entries will be flipped and we could easily improve the rate without significantly effecting the number of flipped codeWord entries, by increasing the constellation size. Thus in general we should space the constellation points at a distance roughly equal to the standard deviation of the noise. The conclusion is that in order to achieve higher rates, at low values of SNR we should use smaller constellation sizes, and at high values of SNR we should use larger constellation sizes. P' (H 99:1) ?'("(["3.811)12) 'Q‘ [((C‘).‘§:]).2 lg‘!%'):l)i)i(rFuIJ ,) », pr 0033.131 my» )i-uh rm-” z13'(C)2V3.:fi1f0|9,4°JPd‘Lfl‘)‘ 0 I I . . I - A ‘1 €"+r. a} E" é" 9'6 h J B 000.1". 1 {01’5" w- ‘HUHJ ; 4LE I)“ I tif ‘1..- r :4; 2 . 3 6‘”? 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H'IlTunLL'e) Lam HIE Liv! r1 (fl) (1"?! {id-J . ’11:: e“i"*”“" j“ 5 +3 2".” 0‘: 1—4 M II R (Lt ’ _ vac; :1.- E,“ J 0*, law-3 I3 (:35 Hum one "Mn-3 “1“ 1461...») be ‘cnj a; GIG-EVE": Am} 1r. 13w} firft-I . we a' ?It6/ “:6 [firm bu I: 1/ D‘O‘IE Mn ‘1 {gfic+;jfi o{ t? 6.115.465 C((‘L'fonj 9;“; 4" 1%: (I; Tar-Do. 1E5) if #IE (“46 ‘3 leg" ‘HM'I LI)? “Laue (ammu-HTI-J'ul" {JUN fine Ems“: Cltfimne‘ y, Pfihl'lff Km file ‘6“. Cl {:3le nor-Leo we how“ a?“ (*5 7—4,“), 7g? I“! (ML-r cf grog-15 “1’ 1x- verb} (4”? 1‘: {Dav/1‘ lir'jl' :""="“U‘/7' Clank?” Ian Z: I. [Mr/r AC4» E 4;; t" I'C‘*~ 3 1Ltli-r G.‘ L In": H ' Z '0‘_Luahfj} Elm/“fl HQM/firc N; 3%. m; W1 Ni KN“; 77:; {ELICI (cmmpnq'hflv-n ‘5 pc‘} Fwy“: 0+ (—a-(ps 9lw+fl 71th“ ‘1‘, 'PU‘H'I‘j HI”: 716641;?" W'M‘ flit GHQ-vi" 110 (E): we ram ("anti/("C 'tH'lk’l “ME Cuff-<91} ‘31 Em ivc {Lilith-‘5’ ,) ; L F. Q3) 3. (a) A simple and efficient way of storing the data is in an N X (D + 1) matrix. The first 03) (c) (61) entry of the nth row stores the number of data bits used in the generation of the nth coded symbol. The remaining D entries of the nth row store the indices of the data bits that are added together modulo two, to form the nth coded symbol. Typically no data bits can be decoded given only 100 coded symbols. In order to be able to decode at least one bit, we require a seed, that is a coded symbol formed from only one data bit. The expected number of such coded symbols is about 0.8(1001). The maximum possible number of bits that can be decoded from the 100 symbols is 100. This upper bound is very loose. Typically only a few data bits, if any, can be decoded given 200 coded symbols. Now the average number of seeds is 1.6, but the problem becomes one of sending another coded symbol that connects to the same seed symbol. The upper bound is now 200, but is still very loose. See the following graph (note the code may take many hours to run). There is a sharp transition at around N = 5000. The overhead needed to decode the 5000 data bits is very low when N is slightly larger than 5000. When N is much larger than 5000, this overhead becomes significant. As the graph below suggests, many additional symbols are typically required before the last few data bits can be decoded. Often in practice, an outer code is used (typically a simple block code) to circumvent this. number of bits decoded 4500 3500 -~-.n....m;_____;__ 3000 ._HH, 2500 -- 1500 ~-v~~~ 1000 -~---u 500 1000 2000 70600 4000 5000 6000 7000 3000 9000 10000 number of coded symbols received ...
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).

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Hw4SolnMain - ECE 5670 : Digital Communications ECE...

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