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

3/42Wireless Mobile Communication and Transmission Lab.

Introduction of LDPC codes

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

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Introduction of LDPC codes

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

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Introduction of LDPC codes

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

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Introduction of LDPC codes

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

7/42Wireless Mobile Communication and Transmission Lab.

Introduction of LDPC codes

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)

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Outline

Introduction of LDPC codes

Encoding of LDPC codes

Construction of parity check matrix

Decoding of LDPC codes

Density evolution and EXIT

9/42Wireless Mobile Communication and Transmission Lab.

Encoding of LDPC codes

H=[P|I]

G＝ [I|P’]

C=M*G

10/42Wireless Mobile Communication and Transmission Lab.

Encoding of LDPC codes

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

11/42Wireless Mobile Communication and Transmission Lab.

Encoding of LDPC codes

12/42Wireless 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

13/42Wireless Mobile Communication and Transmission Lab.

Construction of parity check matrix

Random construction methods

Structured construction methods

14/42Wireless Mobile Communication and Transmission Lab.

Construction of parity check matrix

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

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Construction of parity check matrix

Mackay methods

16/42Wireless Mobile Communication and Transmission Lab.

Construction of parity check matrix

Bit-filling

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Construction of parity check matrix

Extended Bit-filling

18/42Wireless Mobile Communication and Transmission Lab.

Construction of parity check matrix

Hesuristic girth distribution

max,6,4, lllg

2/

1

max 22l

kkkg

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Construction of parity check matrix

Progressive edge growth (PEG)

20/42Wireless Mobile Communication and Transmission Lab.

Construction of parity check matrix

Random construction methods

Structured construction methods

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Construction of parity check matrix

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

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Construction of parity check matrix

Partial geometry LDPC

Steiner 2-design;

Net or transversal design (TD);

Generalized quadrangle (GQ);

Proper PG

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Construction of parity check matrix

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

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Construction of parity check matrix

Block-LDPC

0 1 0 1

2 1 2 1

25/42Wireless 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

26/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

Bit flipping method

Belief propagation and related methods

Weighted bit flipping methods

27/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

28/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

Bit flipping method

Belief propagation and related methods

Weighted bit flipping methods

29/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

30/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

31/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

32/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

UMP-BP based (min sum)

mNnmnm

mn

mnmn

mnnmNnmn

zif

zif

zL mnm

2mod

0,0

0,1

min1 '' \

33/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

34/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

Bit flipping method

Belief propagation and related methods

Weighted bit flipping methods

35/42Wireless Mobile Communication and Transmission Lab.

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

Output of the check node

Flipping the variable node n with largest weight

Decoding of LDPC codes

1mv 2mv 3mv

mcmlmmm vvv 21

nmNn

m y)(

min12

mlv

nmNn

mm yr)(

min12

nMmmmn yE

min12

Weighted bit flipping methods

36/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

Weighted bit flipping methods

37/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

( )

( )

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

Weighted bit flipping methods

38/42Wireless Mobile Communication and Transmission Lab.

Decoding of LDPC codes

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

Weighted bit flipping methods

39/42Wireless 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

40/42Wireless Mobile Communication and Transmission Lab.

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 )()( )(

41/42Wireless Mobile Communication and Transmission Lab.

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

42/42Wireless Mobile Communication and Transmission Lab.

EXIT