Post on 15-Jan-2016
description
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
4/42Wireless Mobile Communication and Transmission Lab.
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
5/42Wireless Mobile Communication and Transmission Lab.
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
6/42Wireless Mobile Communication and Transmission Lab.
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)
8/42Wireless 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
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
15/42Wireless Mobile Communication and Transmission Lab.
Construction of parity check matrix
Mackay methods
16/42Wireless Mobile Communication and Transmission Lab.
Construction of parity check matrix
Bit-filling
17/42Wireless Mobile Communication and Transmission Lab.
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
19/42Wireless Mobile Communication and Transmission Lab.
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
21/42Wireless Mobile Communication and Transmission Lab.
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
22/42Wireless Mobile Communication and Transmission Lab.
Construction of parity check matrix
Partial geometry LDPC
Steiner 2-design;
Net or transversal design (TD);
Generalized quadrangle (GQ);
Proper PG
23/42Wireless Mobile Communication and Transmission Lab.
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
24/42Wireless Mobile Communication and Transmission Lab.
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