Transcription of 第八章 LDPC 码
1 LDPC LDPC RS BCH Turbo LDPC Mackay Turbo Turbo Turbo Mackay Gallager 1962 LDPC LDPC LDPC LDCP QC-LDPC LDPC LDPC S PA Turbo LDPC LDPC 1996 Mackay Spielman Wiberg Gallager 1962 LDPC Gallager Gallager LDPC 1981 Tanner LDPC Mackay LDPC LDPC Turbo Shannon LDPC 200 LDPC LDPC Turbo 1 Gallager Mackay LDPC LDPC LDPC LDPC LDPC Girth LDPC H H Mackay LDPC LDPC BIBD LDPC 2 Gallager LDPC MP Message Passing BP Belief Propagation MP
2 Tanner LDPC Tanner BP MP LDPC BP AWGN 1/2 107 LDPC 10-6 Shannon Shannon LDPC LDPC H H 1 H H 1 LDPC H regular irregular LDPC LDPC LDPC n, j, k LDPC H n m j k /m njk= j<k j<<m k<<n LDPC H (6,2,4) =101110110101011011H H 4 2 H R=(n-m)/n=(k-j)/k 4 1/2 LDPC Tanner n {xj, j=1, 2, .., n} m {ri, i=1, 2.}
3 , m} 1 Degree j j k k H Tanner 1x2x3x4x5x6x1 12 45rxxxx=+++2 135 6r xxx x=+++3 2346rxxxx=+++ 1 Tanner LDPC H H girth 4 girth=4 1 1x2x3x4x5x6x1 12 45rxxxx=+++2 135 6r xxx x=+++3 2346rxxxx=+++ 2 Girth 4 H 4 110110101011011101 = H 6 101110011 = 1H H1 H LDPC H 2L L LDPC 2 H 4 4 1 12 45rxxxx=+++ 2 135 6r xxx x=+++ 4 x1 x5 x1 x5 LDPC LDPC H G H LDPC H LDPC H Gallager H j 1 k
4 1 1 1 girth 4 j k H H H0 H0 H Mackay j k H 1 1 4 1A j k 1 1 3 33 3 j=3 k=6 R=1/2 H 1 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 1 2A 1A 2 H girth 2 m/2 m/2 m/2 3 4 H 1 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 1 1B 2B 1A 2A H girth Mackay 2 m/2 m/2 m/4 m/4 m 2 3 5 1 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 1 QC-LDPC LDPC LDPC QC-LDPC LDPC Quasi-Cyclic LDPC codes LDPC QC-LDPC QC-LDPC (G) ()( ),, ,ab ca bca b c G=.
5 G e ,eaaeaa G= = G a 1aG 11aaa ae = = (,,)A+ (,)A+ ( ,)A p ap-1=1 (mod p) ap=a (mod p) QC-LDPC H m n M(H) M(H) 0 1 L*L 0 L*L ijap H P 0100001000011000 = P 1 111111111111mn = M(H) 1 L L ijap mL nL H 1(n-1)11121n2(n-1)21222nm(n-1)m1m2mnaaaa aaaaaaaappp pppp pH=ppp p aij (i=1,2,..,m; j=1,2,..,n) aij ijap H aij 1 Finite Geometries LDPC a m ()ma GF q LDPC G n m n m H H j H k H 4 G H H hi,j=1 G i j girth 10 LDPC H LDPC H TH = [P , I] H G = [I, P] G 2()On Richardson LDPC 1) H ( -)(- )( -)( -)( -)(- )( - )mg nmmg gmg mggnmgggmgmn ABTH=CDE T H 6 -1I0-ETI H -1-1 ABT-ET A -ET B + D0 12sx= pp s p1 p2 Hx=0 12As + Bp + Tp = 0 -1-11(-ET A + C)
6 S + (-ET B + D)p = 0 -1 = (-ET B + D) -1-11p =- (-ET As +Cs) -121p = -T (As + Bp ) 2) 1 -1-11p =- (-ET As +Cs) As ()On -1T As -1T As = yAs = Ty ()On -1-ET As ()On Cs ()On -1-ET As + Cs ()On -1-11p =- (-ET As +Cs) 2()Og 2 -121p = -T (As + Bp ) As ()On 1Bp ()On 1As + Bp ()On -11-T (As + Bp ) -111-T (As + Bp ) = y-(As + Bp ) = Ty ()On 3) H ABTCDE -1-ET B + D 1 [ ] H ABTCDE g 2 [ ] = -1-1-1I 0 ABTABT-ETIC D E-ET A -ET B + D0 -1-ET B + D ABTCDE -1-ET B + D s 12sx= pp Hx=0 1 1 1p 2 2 2p 2()On g+ g H LDPC H G H G H 1 G G H Hsys H H1 H1 H 7 H 1 H H 1,11,21,(n-m)1,(n-m+1)2,12,22,(n-m)2,(n- m+1)2,(n-m+2)m,1m,2m,(n-m)m,(n-m+1)m,(n- m+2)m,(n-m+3)m,npppp000ppppp00H=ppppppp pi,j pi,jpi,jT=I pi,j-1=pi,jT H Hx = 0 1,(n-m+1) 11,1 11,221,(n-m) n-m2,(n-m+1) 12,(n-m+2) 22,1 12,222,(n-m) n-mm,(n-m+1) 1m,(n-m+2) 2m,n mm,1 1m,22m,(n-m) n-mpc =p x +p x ++pxpc +pc =p x +p x ++pxpc +pc ++p c =p x +p x ++px ()()()
7 T11,(n-m+1)1,1 11,221,(n-m) n-mT22,(n-m+2)2,1 12,222,(n-m) n-m2,(n-m+1) 1 Tmm,nm,1 1m,22m,(n-m) n-mm,(n-m+1) 1m,(n-m+2)2c =pp x +p x ++pxc =pp x +p x ++px+pcc =pp x +p x ++px+pc +pc + = 1n-m1mxxxcc ,, ,12mccc LDPC Bit-Flipping Weighted Majority-Logic Sum-Product
