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Chapter 5 Low Density Parity Check Codes

(M ha kim tra chn l mt thp)

M ha kim tra chnlmtthp (LDPC), xutca Gallager trong 1962[19],v sau c pht hinbi MacKay [18] v Neal xut hinnh l mtlphcca m c th mang lihiusutrttt trn knh nhiutrng Gaussian (AWGN).Mha LDPC c thtcgiihnch tiu lign Shannon vigii m thct phctptrn mt knh AWGN, m turbo tthnvi cng mt kch thckhi v tl m.5.1 Giithiu

M ha LDPC l nhngvdcbitca m khituyn tnh.Cu trc camt m khituyntnh cm tbi ma trn G hoc ma trn kim trachnl H. Khnng sali k t trong mttm c xc nhbikhong cch tithiudmin. Dmin cnhngha l khilng t nhtca cchng trong ma trn G hocn c thcnhngha l slng t nht cc ct trong H c tng ti0.Tm camt m kim trachnlc hnh thnhbng cch kthpmtkhi cc chsthng tinnh phn vimtkhi ccchskim tra.Nhngchskim tra nycbiudindidng matrncgi l ma trnkim tra chnl.

Ma trn nyidin cho mttphp ccphng trnh tuyn tnh ngnht, v thitlpcc tm lb cc giiphp caccphng trnh

( )Trong ma trn nybnchsu tin l chs thng tin v bachscuicng l chskim

tra. Phng trnh kimtrachnl cho ma trn ny c thcara nh:x5= X1 x2 x3 (5.1)x6= X1 x2 x4 (5.2)x7= X1 x3 x4 (5.3)

Nhng ma trnkim tra chnl khngphi l rtngingii m khi thng tinbn trong l ln, do mtkthut m ha khc ca rabi Gallagercbitnnh l m ha LDPC. M ha LDPC l m xc nhbimt ma trn i.e, n chachyul 0s v ch c mtsnhca1s. M ha LDPC c thc chia thnh hai loi M ha LDPC c quy tc: Slngca 1 mi hng v ct l hngs.

M ho LDPCbt quy tc: Slngca 1 mi hng v ct khng cnh.Nhng mbt quy tc cho hiusut tthn so vi M c quy tc. Giskim tra tnh chnlmtthp ma trn H c N ct v M hng. Tl m c chobi R= 1-(M / N), th hai

pha tngng bao gm N bit node, cc node kim tra M v mtslngnht nhcacc cnh.Mi bit node, cgi l "left node", idin cho mt bit cat m. Mi ntkim tra, cgi l mt "right node" idin cho kim tra chnlca m. Mtcnhtntigiamt bit node v mt node kim tra khi v ch khi c mt 1 trong mctngng trong ma trnkim tra chnl. M ha LDPC c quy tc l nhng ma trn m ttccc node cng loi c cng mt mc,numccamt node l scnh cho mt

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node ln cn. M A(n, j, k) mtthp l mt m cachiu di khi n vis j ca 1strong mictca ma trn v s kca 1s trong mi hang ca ma trn. Ma trnkim trachnlca m ha LDPC c quy tc (3, 4) chinthdiy. Hnh 5,1 cho thyth hai pha lin quan ca n.

Hnh 5.1 thidin camt m LDPC c quy tc (3,6) chiu di 12. Cc leftnode idin cho cc variable node trong khi cc right node idin cho kim tracc nt. [20]

Ccphng trnh idinbinhng ma trn lun lun c thcgiiquytcung cp cho cc skim tra l tnghp r rng ca cc s thng tin. Phn tch mt mmtthp chiu di khiln l rt kh khn v slng bao la ca codewords thamgia. N l ngin phn tch mtqunth tonbt m nhvybi v sliuthngk camtqunth cho php mtmc trung bnh trn slng m khngphi l dxl trong m ringbit.T cc hnh vi qunth ta c thlp bo co thng k v cc thuctnh ca m ringbit. Cho phn tch mt ma trn ny c chia thnh submatrices j, misubmatric c cha duy nht 1 trong mict.

5.2 M LDPC trong MIMO

1 1 1 0 0 1 1 0 0 0 1 01 1 1 1 1 0 0 0 0 0 0 1

0 0 0 0 0 1 1 1 0 1 1 11 0 0 1 0 0 0 1 1 1 0 1

0 1 0 1 1 0 1 1 1 0 0 0

0 0 1 0 1 1 0 0 1 1 1 0

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Trong cc hthng MIMO, chuyntip m ha sali l iucnthit cho kt

nichtlng cao. Cc thut ton VBLAST csdng trong hthng MIMO chophp x l tuyn tnh thng tin. Cc vngpphivimtlpthut ton l shindincalitruyn v lpc pht hinu tin, m thng csingnal-to-noise ratio

(SNR) do mt tn hiuin nulling tuyn tnh, c nhiukhnngc gii m khngchnh xc. Tiu ha sng v tuyn l mtgii php ctholun trong chngtrc. nng cao hnna tin cyca lin ktv chngli lin k t l vnForward thchincchng trnh sali LDPC. Cc thtc m ha cbndliutruynvimts bitbov gip nhnbit xem c lixy ra trong qu trnh truyn. Mha LDPC l rthiuqu(ivi m ha v gii mphctp) . Sdng m ha LDPCgip thchintttimnng cahthng MIMO mt cch hiuqu.

5. 3 m haM LDPC l m tuyn tnh. Do ,n c thc thhinnh khng gian camtmatrnchnlkim tra , v d, l mtt m khi v ch khi

H xT= 0

T (5.4)

Sai "mt thp" p dng cho ma trn nn ngin. V d,nu c kch thc[(n / 2) x n], trong n l chn, sau n c thc yu cu cho H c 3 ct v 6

hng. Chng ta tham kho cc m lin ktnh l mt m LDPC c quy tc. Sn ginca H cho php hiu qu (ti u) gii m, trong khi tnh ngu nhinvm bo (theo nghaxc sut) mt m tt .[20]

Hnh 5.2 (a) Mt ma trn kim tra-chn ltngng trong hnh tam gic

thp hn. (b)ma trn kim tra chn l tnh gn ng hnh tam gic thp hn

5.3.1 M ha hiuquda trn khong tam gic thphn

Hiuquca ccb m ha pht sinh t cc ma trnkim tra chnl H v thutton c thc p dng chobtk(ngin)H. Cc hng ca H l clptuyntnh. Chng ta xem xt mt ma trnkim tra chnl H m x n trn mt lnhvc Galva F(GF (2)). M lin quan bao gm cc thitlpca n-tuplesxhn F nh

H xT= 0

T (4.5)

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ivi cc m tc cng sut knh, v c thi gian x l tuyn tnh chngta a ra ma trnkim tra chnlca ra trong hnh tam gic thphnbng cch sdngthut ton Greedy.iu ny c thcthchinnginvi hon v hng vct. Chng ta sdng thut ton Greedy A ,a ma trnbtumt hnh tam gic thphn.

Hnh 4.3 Sidincathut ton GreedyCt li cathut ton l bcngcho mrng.Thut ton Greedy Abtuvi matrn (1-r)lx l; Anh hnh 4.4 (a). Trongbckhito,dkinphnnh (1) cattc cc ctc phn loinhcbitn vphn cn lic phn loi l xa. Lnu tin cc thut ton thchinmtbc ny (1)lctcbitncspxpli hnh thnh cc ct uca ma trn A hinthnh trong hnh 4.4 (b). Gisrngcc ma trn cn li c hngbcmt,ctktnivi cc hngbng-mtc xc nhtrongbcth hai. Hy cc ct ny l c1..... ckv cho r1 .... rk cbcmt hng nhvy m ci cktnivi ri. Trong lnth hai p dngbcmt cc ctmi ny c

0) [Khito]Btuvimt matrn A, k khai mictclpcbitnvi sc xut1-. Hy A '= A

1) [Stop hoc mrng] Nu Acha khng phi l mt ct cbit n cng khng phi l hngca bc mt, sau u ra ma trntng t. mt khc thc hinbcmrng ng cho

2) [Khai bo bin nh bit] Btk ct A 'c kt ni vi bchang mt c khai bo c bitn. sau inbc 1.

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bitn v hng lin quan ca n cspxpdc theo mtng cho nh trongHnh 4.4 (c). Hnna, trong milnlpb sung ny ng cho cmrnghnna.Nu thtc ny khng chdnglitrc sau cc ng cho ktqudkinchiu di lv, do ,khong cch hng dkin c kch c(1-r-)lv khong cchct kch thc (1 -)lnhthhin trong hnh. 4.4 (d)

Hnh 5.4. (A) cho ma trn A. (b) Sau khi ngdngu tin cabcmt, (1 -)lctcbitncspxplito thnh u tin (1 - )lctca ma trn A. (c)Sau khi ngdngthhai cabcmt, kmicbitn ct v cc hng linquan ca hc sp xp li to thnh mt ng cho ca chiu di k. (D) Nu

th tc khngchm dt sm

sau

ng cho cmrngcchiu di lv, do ,khong cch hng l bng(1-r -)lv khong cch ctbng (1 -)l. [20]

Nu,mt khc, thtcchmdttrc khi ttc cc ctkt thc sau chng tinhncmt tam gic gnngbng cch spxpli cc ct cn li bn tri.Gisrng cc

phn cn licactbnglsau dkinktqukhong cch hngbng (1 - r -+)lv ktqukhong cch ctdkins lbng (1 - +)l.Sau khi sdngthut ton greedy v a ma trnca ra trong tam gic di hnhthc,btu m ha.Ma trnkim tra chnl l

H = Trong A l (mg) x (nm), B (mg) xg, T l (mg) x (mg), C l gx (nm), D c gxg, vcui cng E l gx (mg). Ttcnhng ma trnthatht v T l tam gic thphnvinhngngdc theo ng cho. Nhn ma trn ny t cnh tri

(5.7)

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Tip theo ma trn thu c

(5.8)

X = (s, p1, p2)Ni sbiuthphnhthng, p1 v p2 kthpbiuthphnchnl, p1 cchiu di g, v p2 c chiu di (m-g). Vic xc nhphng trnh H x

T= 0T chia tch

thnh haiphng trnh, cth lNh

AsT

+ Bp1T

+ Tp2T

= 0 (5.9)

(-ET-1A+C)sT + (-ET-1A+D)p1T = 0 (5.10)

Xc nh=(-ET-1A+D) v ginh cho thiim ny l nonsingular. Sau ,t(5.10)

p1

T= --1[(-ET-1A + C) sT] (5.11)

Sau xc nh AsT v Bp1T

v thm chng ti nhn ktqu T-1p2

T= -T-1[AsT+ Bp1T] (5.12)

Bctinx l v thct m ha

Bctinx l: u vo:Non-singularkim tra ma trnchnl H.

Output: Mt ma trntngngca hnh thc

nhvy

-ET-1B + D l non-singular

1.Tam gic: Thchin cc hng v ct hon v mang kim tra chnl ma trn H vo khonghnh tam gic thphn

2.

vi gbngkhong cch nhnht c th. Hnh thcthphn tam gic tcbng cch sdngthut ton Greedy v cng c khong cch l gim.2. Kim tra hng: Sdng Gaussian loibhiuquthchin php nhn

= trckim tra xem l non-singular , thchintiptcct hon v,mboc tnh ny.

Encoding: u vo: - chnlkim tra ma trnca cc hnh thc

nhvy-ET

-1B + D l non-singular v vector s Fn-m

.

Output: - vectorx= (S, p1, p2) , p1Fg, p1F

m-g nhvy HxT= 0T

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BNG II

Tm ttcathtcEncodingxut.Nihihaibc:MttinxlBcVBc mha thct

5.4 Giim:

Cc thut ton gii m csdng cho m ha LDPC cgi l thut ton quatin nhn, v cc thut ton clpilpli. L do cho tn ca chng mi vngca cc thut ton truyn thng ctruynt nt truyn thng kim tra cc nt, vkim tra cc nt trli cc nt truyn thng. Cc thong ipt cc nt truyn thng kim tra cc nt c tnh da trn gi tr quan st ca nt truyn thng v mts ccthng iptruynt cc nt ln cnkim tra nut tryn thng. Mt kha cnh quantrng m thng ipcginmt nt tryn thng v kim tra node c khngphia vo ti khon truyn thng c givng trct c n v. iu ny cngngivi thng iptruynt cc nt kim tra n cc nt truyn thng . Mtlp quantrngcathut ton qua truyn thng l thut ton truyns tin cy. Thut ton ny cmt trong nghin cuca Gallager [19], Cc thong ip thng qua cng cc cnh trongthut ton ny l xc sut, hay tin cy. Chnh xc hn, cc thng ip qua tmt nodethong ip v kim tra nt c l xc sut m v c mt gi trnhtnhca nt truynthng, v ttc cc gi trtruynt v trong vng trctkim tra sc cc nt v khchn c. Mt khc, cc thng iptruynt c n v l xc sut m v c mt gi trnht

nh cho ttc ccthng ip truyn cho c trong trc vng t cc nt thng bo khchn v[21].ivimtbinngu nhin nh phn x L (x) = Pr [x = 0] / Pr [x = 1] l khnngcax. Cho mtbin y ngu nhin, khnngiukinca x k hiu L (x| y) cnhnghal Pr [x = 0| y] / Pr [x = 1| y] Tngtnhvykhnngngnhpca x l ln L(x)v c iukinkhnngca x cho y l lnL (x| y).Nu x lbinngu nhin c xc sutngang nhau, sau L (x| y) = L (y| x) vi quy tc Bayes. V vynu y1 ......... yd l ngunhin clpbin, sau chng ta c

lnL(x |y1.yd)= (5.13)By gigisrng x1....... xi lbinnh phn ngu nhin v y1...... yi l ccbinngu

nhin. Ngoi ra k hiu F bi. Sau tnh ton ln L(x1... x`| y1; :::; Yl).Nu p= 2 Pr [x1= 0| y1] -1v q = 2Pr [x2= 0| y2] -1,sau 2Pr [x1 x2= 0| y1; Y2] -1 =pq.Do , 2PR [x1 ...x` = 0| y1...... y`] -1 = khi Pr [xi=0| yi] = L (xi|yi) = (1 +L(xi yi)), chng ta c 2Pr[xi= 0 yi] -1 = (L - 1)=(L + 1) =tanh(l/2), ti L = L(xi yi) v l= lnL.V vy, chng ta c c

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Ti li = lnL (xi j yi). Cc thut ton truyn tin cy cho M LDPC c thcbtngunt hai quan st. Trong vng 0, cc nt kim tra gi cng ttc cc ilog-khnngxy ra iukin trn gi tr quan st cah. V d,nu cc knh csdng l BSCvi xc sutli p, sau cc tin nhnu tin cginttc cc nt kim tra tipgip vimt nt thng ip l ln (1 - p) - ln p nu gi trca nt l s khng, v lkhng ng ca gi tr ny nu gi trca nt l mt. Trong ttc cc vng tip theoca cc thut ton mt nt kim tra c ginmt nt tin nhnlink thng qua khnngtheo n (5.14). Mt thng bo nt v gi n nt kim tra c ca n, iukin log-likelihood trn gi tr quan st ca n v trn nngnhpkhnngxy ra t cc ntkim tra link khc hn c sdng cc mi quan h (513).Cho m(l)vc l thng ip thng qua tin nhnt nt v kim tra nt c ti vng thlth ca

thut ton. Tngtnhvy, xc nh m(l)cv. Ti vng 0, m(0)vc l khnngngnhpnode v tin nhn c iukinv gi tr quan st ca mnh, l c lpca c. Chng ti

biuth gi tr nybi. Sau ccphng trnh cpnht c thc m tnh

trong Cv l tphpca cc nt kim tra scnhn nt v, v Vc l tphp cc nttin nhn,sckim tra nt c.Vic tnh ti cc nt kim tra c thcngin hahnnabng cch thchinchng trong t ng ngnhp min. V gi trca tanh (x) cth bcb, n l cnthittiptc theo di cc duhiuca n mt cch ring

bit. Hy l mtbnt cc sthc [-,] F x [0,] c xc nhbi(x): =(sgn (x), - tanh ( | x| / 2)) (set sgn (x) = 1 nu x1 v sgn (x) = 0 khc.) R rng l nhx, do , c tntimtchcnngnghcho-1.Hnna,(xy) =(x) +(y), trong addition l thnhphn chnh xc trong F v trong [0,]. Sau , n l rtd dng chothyrng(4)tngngvi

truyn tin tng c thc thc hin cho mt slng ti a vng.

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5.5 TlLDPC lin tcvbinLDPC:

Tl m hoctl thng tin camt m salichuyntip (FEC), gingnh m khituyn tnh, trng thi catnglng thng tin l hu ch (khng d phng).Nu mtl l k / n, cho mi k bit thng tin hu ch, cc coderto ra n bit dliu chung,trong

n-k l tha.Nu R l net bitrate(bitrate hu ch), gross bitrate ( bitrate raw) l R * n / k.Hngstlca m ny c ngha l ttc ccb m ha LDPC trong mng Laser my

pht c mttltngtcavicgidliu.Tl mbin thin l mt trong nhngtl m ti thng tin cgitiptc thayi hnh thcmtmng Laser my pht khc. thitktl mbin thin Puncturing[22] c thcsdng. Cc mbnhthngbng cch xa mt canhng k tchnl. Mt m (n, k) strthnh mt m (n-1, k). Trong on code tlcnhcthitktl m l R = 1/2, thng 1 bit ra ca 18 thay itl m R = 2/3.[22]

5.5.1Thitkca LDPC tl khc:

dl yutthitk v W e chiu di caphnchnlcat m. [23]ivibtk 0

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Fig5.6 So snh BLER v SNRng cong cho VBLAST uncoded v tc lintcLDPC m haVBLAST

V hnh 4.6 cho thyhiusutcathut ton pht hin VBLAST cithin khi m haLDPC csdng. Cho BLERca 10-11 c mtc m ha ca 2dB cthchin,docithin tin cyca thng tin lin lc, nng cao scmnhhiuqucahthng cho iukin sng mnh.Tl LDPC csdng l .

5.6.1 Simulation Kt qu cho t l khng i LDPCBiu m phng di y cho thy vic so snh cc VBLAST khng m ha viVBLAST Ti u ha v t l khng i LDPC, vi R =

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Fig5.7 So snh BLER v SNRng cong cho VBLAST uncoded, LDPC tc khng i mVBLAST v VBLAST uncoded sdng Ti u ha nt hng

Hnh 4.7, cho thy LDPC m ha VBLAST cho scithinca 1dB so vi tiu ha spxp VBLAST QR cho BLERf 10-11. C hai M ha v trtt phn loicsdngchngli truynli trong thut ton VBLAST

5.6.2 Mphngktqu cho LDPC tc khng i v tlbin LDPC

Cc LDPC tc khng i c tl , m l tl li sut thay itiptc thayibng cch sdng m ha tngt.Hiusut BLERcac hai cthhin tronghnh bn di,

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Fig5.8 So snh BLER v SNRng cong cho VBLAST uncoded, LDPC tl lintc m haVBLAST v tlbin LDP C m VBLAST.

Trong sung 4,8, c ththyrng m ha ca 3 dB tc so vi Uncoded thut tonVBLAST, v cithin 1 dB so vitl khng i M LDPC ctcbng cch sdng m LDPC tlbin cho BLERca 10-11

5.6.3 Mphngktqu cho LDPC tc khng i vbintl LDPC mnhshnlon vbtnyu.

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Hnh 5.9 So snh cc BLER v SNRngcong, tlkhngiLDPC VBLAST mha vbintlLDPC VBLAST m ha, cho yubtnviukinxo trnmnhm

Hnh 5.9 cho thy trong iukin sng yu, m ha c cung cpbi m LDPCl tthnnhiu so vi cc iukin sng mnh,tc khng i LDPC mt BLERca10

-11 ti SNR l 0,75 dB chdiiukinhnlonyu, m l iukin sng mnh mSNRca 1,75 dB l cnthit.