Theory and Technology of Error Control Coding

Wireless Mobile Communication and Transmission Lab.

Theory and Technology ofError Control Coding

Chapter 7

Low Density Parity Check Codes

2/42Wireless Mobile Communication and Transmission Lab.

Outline

Introduction of LDPC codes

Encoding of LDPC codes

Construction of parity check matrix

Decoding of LDPC codes

Density evolution and EXIT



1960 1970 1980 1990 2000 2004

GallagerZyablov

Pinsker Tanner

MacKay

Neal

Wiberg

Davey

MacKay

Yu Kou

Shu Lin

Fossorier

SY Chung

Urbanke

Richardson

Burshtein

Miller

McEliece

Luby

Mitzenmacher

Spielman

......

Some important research

of LDPC codes since 1962



Regular LDPC code(6,4) parity check matrix H

Two classes of nodes in a Tanner graph (variable nodes and check nodes)

Check node j is connected to variable node i whenever element in H is 1

Bold line constructs a cycle of length 6 in a Tanner Graph

1 1 0 1 0 0

0 0 1 1 1 0

1 0 0 0 1 1

0 1 1 0 0 1

H

1v 2v 3v 4v 5v 6v

1c 2c 3c 4c

1 1v v

( , ) (3,2)v cd d

jiH



L D PC codes

R e g u la r G F(2 )

by th e we ig h t o fco lu m n (ro w)

by th e e le m e n t o fpa rity ch e ck m a trix

R eg u lar L D P C

Y

I r r eg u la r L D P C

N

G F ( q ) L D P C

N

Bin ar y L D P C

Y



rate=1/4, AWGN Channel, Thesis of M. C. Davey



Local girth distribution

histogram of variable nodes

Block length approaching infinity, the assumption of cycle freeness is asymptotically fulfilled

The relationship of girth, minimum distance and performance

2 4 6 8 10 12 14 16 18 20 220.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pe

rce

nt

of

vari

ab

le n

od

es

loop length (N=504)

2 4 6 8 10 12 14 16 18 20 220.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pe

rce

nt

of

vari

ab

le n

od

es

loop length (N=1008)

2 4 6 8 10 12 14 16 18 20 220.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pe

rce

nt

of

vari

ab

le n

od

es

loop length (N=10000)2 4 6 8 10 12 14 16 18 20 22

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pe

rce

nt

of

vari

ab

le n

od

es

loop length (N=4000)


Outline








H=[P|I]

G＝ [I|P’]

C=M*G



1 1 1

0

0

I A B T A B T

ET I C D E ET A C ET B D

1 21 1

1

0

( ) ( ) 0

T T T

T T

AS BP TP

ET A C S ET B D P




Outline








Random construction methods

Structured construction methods



Gallager method

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

0 0 0

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

0 0 0

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

0 0 0

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

0 0 0

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

1 1 1

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

1 0 0

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

0 0 1

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

0 1 0

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

0 0 0

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

0 0 0

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

1 0 0

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

0 1 0

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

0 0 0

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

0 0 1

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

0 0 0

111

1

11

1 1 1

1

1

111

1

11

1



Mackay methods



Bit-filling



Extended Bit-filling



Hesuristic girth distribution

max,6,4, lllg

2/

1

max 22l

kkkg



Progressive edge growth (PEG)



Random construction methods

Structured construction methods



FG-LDPC:EG-LDPC and PG-LDPC

n points and J lines : n*J incidense matrix H

Each line is composed of p points There is one and only one line between two points Each point lies on q lines Any pare of lines has only one common point or no common point



Partial geometry LDPC

Steiner 2-design;

Net or transversal design (TD);

Generalized quadrangle (GQ);

Proper PG



BIBD-LDPC

),,,,( brkv ),( AX

vrbk 11 krv)9,8,7,6,5,4,3,2,1(X

7,5,3,9,4,2,8,6,1,8,4,3,7,6,2,9,5,1

,9,6,3,8,5,2,7,4,1,9,8,7,6,5,4,3,2,1A

bvjihH



Block-LDPC

0 1 0 1

2 1 2 1


Outline








Bit flipping method

Belief propagation and related methods

Weighted bit flipping methods



Bit flipping method =0 =1

v a ria b le no d e s toc he c k no d e s

1

1

1

2

1

0

c he c k no d e s tov a ria b le no d e s

v a ria b le no d e s toc he c k no d e s

Connected to two

unsatisfied check nodes: flipped



Bit flipping method





Belief propagation method

All the effective decoding strategies for LDPC codes are message passing algorithms

The best algorithm known is the Belief Propagation algorithm

(1) Complicated calculations are distributed among simple node processors

(2) After several iterations, the solution of the global problem is available

(3) BP algorithm is the optimal if there are no cycles or ignore cycles



Belief propagation method (log domain) Probability information transmitting among connected codes through the edge Two types of message: The probability that one bit is 1 or 0, obtained via the connected checks nodes other than

the check node that received the probability. The conditional probability of that one check node is satisfied if one connected bit is 1 or

0

)0(

)1(ln

mn

mnmn q

qz

)0(

)1(ln

mn

mnmn r

rL

nn

nn y

P

PF

2

2

)0(

)1(ln

mn

mnmn

nmNn mn

mnmn T

TL

z

zTstepHorizontal

1

1ln

exp1

exp11

\' '

'

mnMmmnnn

mnMmnmnmn

LFz

LFzstepVertical

\

\'

'2



Belief propagation method: message passing in two steps

mc

1v

v

nv

1(

)k

mL

q

()k m

Lq

( )knmL q

( )kmnL r

()k

m

Lr

1(

)km

Lr

nv

1c

c

mc

1(

)k

nL

r

()kn

Lr

( )kmnL r

1( )knmL q

1(

)k

n

Lq

11(

)k

nL

q



UMP-BP based (min sum)

mNnmnm

mn

mnmn

mnnmNnmn

zif

zif

zL mnm

2mod

0,0

0,1

min1 '' \



Normalized UMP-BP based Reduce the complexity of horizontal step: The function value is greatly

decided by the variable with minimum absolute value, L2 is greater than L1, Normalized factor is used to compensate the performance loss

2

1

21

1

2

\2

\

\1

''

' '

'

' '

'

min1

exp1

exp11

exp1

exp11

ln1

1ln

LE

LLE

LE

LE

zLL

z

z

z

z

T

TLL

mmse

mnnmNnUMP

nmNn mn

mn

nmNn mn

mn

mn

mnBP

mnm



Bit flipping method




BPSK Modulation: The smaller the absolute value, the fewer the reliability

Output of the check node

Flipping the variable node n with largest weight


1mv 2mv 3mv

mcmlmmm vvv 21

nmNn

m y)(

min12

mlv

nmNn

mm yr)(

min12

nMmmmn yE

min12




nnMm

mmn yyE

min12

Some improvements of WBF algorithm

Consider the reliability of the bit (MWBF):

Modified check node output (IMWBF):

nnMm

mnmnIMWBF ywe

',, 12

2',

2

\)(2 /2/2min mni

nmNiwyL

22, /212

1 n

nMmmnIMWBF yLe

2

1

21

LE

LLE

d

d

c

v

'

' \

', min

nnmNnmn yw




( )

( )

n nm mj A n

m ii B m

q r w

w K num y T

Some improvements of WBF algorithm

Consider both of the maximum and minimum symbols (LP):

Add a check weight factor (MLP):

Consider the ratio (RRWBF):

( ) ( )

/ 2 , 0

( ) / 2 , 1

min , max

n m mnm

n m m m

m i m ii B m i B m

y l cr

y u l c

l y u y

nMm

mnmn RE /12

maxm

nmn

y

yR

nMm

mmn

n Ty

E 121

)(mNn

nm yT




kk sssL 22113

2

13 LLE

kkkkk

k

k

sLE

sLE

sLE

sEssEssE

ssEsEssE

ssEssEsE

1

21

11

2

1

221

22212

12121

nmNnyNsssSn

\,/4,,, '0121 '

Developed from IMWBF which is a counterpart to Normalized BP Based algorithm

Consider all the symbol in each check with the constraint of extrinsic information:

Linear combination



Outline







Density Evolution

Messages passed in the factor graph are random variables. The calculations performed under the SPA are functions of random variables.

Messages passed through the graph are conditionally independent Symmetry Condition

1)()0(1)( ][][ vdll QFPFFP

dvvPP lle

0 )()( )(


EXIT

VND CND

AWGN channel output

Iterative Decoding of LDPC

Decision

1 2 2,VND ,

0

( , , ) ( 1)[ ( )]bE A s s A ch

EI I d R J d J I

N

1

,

(1 )( , )

1E

A CND E c

c

J II I d J

d


EXIT

Documents

Theory and Technology of Error Control Coding