Decoding and Performance of Nonbinary LDPC codes

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Decoding and Performance of Nonbinary LDPC codes

Decoding and Performance of

Nonbinary LDPC codes

Discussion and Perspectives

David Declercq

ETIS ENSEA/UCP/CNRS UMR 8051

NewCom SWP1 meeting - 28th april 2005


Ususal Definition

C H =

LDPC

n

c ∈ GF (q) ×N : H.c

o

GF (q)

= 0

C G =

LDPC

nc = G.u, ∀u ∈ GF (q) ×Ko

H(M × N) : parity check matrix with entries from GF(q),

G(N × K) : codeword generating matrix obtained from H by Gauss

elimination on GF(q).

Length

8

><

>:

of the codeword N R = K/N

of the information symbols K (if H is full rank)

of the redundant symbols M

Density of H :

d H =

nb. nonzeros elements in H

MN

LDPC : d H

N→+∞

−→ 0

2 NewCom SWP1 meeting - 28th april 2005


Claim

NonBinary LDPC codes decoded under iterative Belief Propagation are

good candidates for :

1. medium code lengths (500 ≤N≤ 3000 coded bits),

2. high spectral efficiency communications (M-QAM with M≥ 16),

As for now, I will consider only :

regular (d v , d c ) nonbinary LDPC codes,

AWGN channels either with BPSK inputs or M-QAM inputs.

3 NewCom SWP1 meeting - 28th april 2005


Factor Graph of a regular

(d v , d c ) = (2, 4) code in GF (8) .

H =

3 0 7 1 0 0 0 2

4 3 0 5 0 0 1 0

0

0

0 5 0 3 6 0 7

6

0

0 1 2 1 0

c c c c c c c c

0 1

2 3 4

5 6 7

c c c c c c c c

0 1

2 3 4

5 6 7

Interleaver

.

.

Codeword

: message

Parity Checks

Π

Codewords

Parity Checks

4 NewCom SWP1 meeting - 28th april 2005


Why choosing d v = 2

(3, 6) code, N=5000 bits (2, 4) code, N=5000 bits

10 −1 Code Regulier de Gallager (3,6)

10 −1 Code Regulier de Gallager (2,4)

10 −2

GF[64]

GF[256]

10 −2

10 −3

10 −3

BER

BER

GF[2]

10 −4

GF[2]

10 −4

GF[16]

10 −5

GF[4]

GF[16]

10 −5

GF[64]

GF[4]

seuil binaire

GF[256]

1 1.2 1.4 1.6 1.8 2 2.2 2.4

E b

/N 0

(in dB)

10 −6

0.5 1 1.5 2 2.5 3 3.5 4

E /N (in dB)

b 0

consequence : non-binary LDPC codes should be ultra-sparse [McKay’98]

if q ≤ 32, it is possible to consider irregularity in the graph,

if q ≥ 64, almost none irregularity can be considered and d v = 2.

5 NewCom SWP1 meeting - 28th april 2005


BP decoding of LDPC codes over

GF(q)

[ITW03] L. Barnault and D. Declercq, Fast Decoding Algorithm dor LDPC over GF (2 q )

1 st step Data pass (term by term product) :

U tp = L ×.

dvY

v=1,v≠t

. V pv t = 1 . . . d v

2 nd step Permutation step (nonzeros values in H) :

U pc = P h(x) U vp

where P h(x) is a q × q permutation matrix corresponding to h(x) ∈ GF (q)

3 rd step Check Pass with F(.) the Fourier transform in GF (q) :

0

V tp = F @

dcY

c=1,c≠t

1

. F(U pc ) A

t = 1 . . . d c

6 NewCom SWP1 meeting - 28th april 2005


Simplification of BP decoding, the

LogEMS algo.

[IEEE Trans. Com. 05] D. Declercq and M.P. Fossorier Decoding Algorithms for LDPC codes over GF(q), submitted

[Gretsi 05] A. Voicila, D. Declercq, M.P. Fossorier and F. Verdier, Algorithme simplifié de décodage des codes LDPC sur

GF(q), submitted (in french)

Goal

Solution

: decode nonbinary LDPC codes with “reasonable” complexity compared to existing

competing codes (Turbo-Codes, binary LDPC codes, RS codes, etc).

: by replacing the check node update with a generalization of the Min-Sum algorithm in

GF(2) : LogEMS = “Extended Min-Sum algorithm in the log. domain”.

complexity per Check Node :

BP BP/FFT LogEMS


O d c .q 2” O (d c .q. log 2 (q)) O

“d ”

2

c .n m

with n m


Performance of LDPC codes for

N=848 bits (ATM size)

(2, 4) code over GF(256) (2, 8) code over GF(128)

10 −1

10 −1

10 −2

BP

10 −2

SP59

BP

Frame Error Rate

10 −3

10 0 E b

/N 0

(in dB)

SP59

Frame Error Rate

10 0 E b

/N 0

(in dB)

10 −3

10 −4

10 −4

(n m

,n c

)=(13,3) factor correction

(n

10 −5 m

,n c

)=(13,3) offset correction

(n m

,n c

)=(13,3) without correction

(n m

,n c

)=(7,2) factor correction

(n ,n )=(7,2) offset correction

m c

(n ,n )=(7,2) without correction

10 −6 m c

0 0.5 1 1.5 2 2.5 3 3.5

10 −5

(n ,n )=(13,3) factor correction

m c

(n ,n )=(13,3) offset correction

m c

(n ,n )=(13,3) without correction

m c

(n ,n )=(7,2) factor correction

m c

(n ,n )=(7,2) offset correction

m c

(n ,n )=(7,2) without correction

m c

10 −6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

8 NewCom SWP1 meeting - 28th april 2005


NonBinary LDPC codes for high

order QAM signals

[ISITA 04] D. Declercq, M. Colas and G. Gelle, Regular GF (2 q )-LDPC coded Modulations for higher order QAM-AWGN

Channels

LDPC

GF(Q)

Demapping

BP

M−QAM

BPSK

if the channel is memoryless and Q > M

demapping.

AWGN

⇒ no loss of performance due to the

9 NewCom SWP1 meeting - 28th april 2005


Performance results on

QAM-AWGN channels

GF (256) (2, 4) codes on 16-QAM and 256-QAM, Gray labelling

16-QAM

256-QAM

4

Performance of regular (2,x)−GF(256) LDPC codes at BER=10e−5

8

Performance of regular (2,x)−GF(256) LDPC codes at BER=10e−5

7.5

3.5

3

Capacity of the

16−QAM channel

7

6.5

Capacity of the

256−QAM channel

Capacity (bit/Hz/s)

2.5

Capacity (bit/Hz/s)

6

5.5

5

2

4.5

1.5

N=2000 bits

N=8000 bits

N=40000 bits

4

3.5

N=2000 bits

N=8000 bits

N=40000 bits

1

1 2 3 4 5 6 7 8

E b

/N 0

3

6 7 8 9 10 11 12 13 14 15 16

E /N b 0

10 NewCom SWP1 meeting - 28th april 2005


Key Issues in Nonbinary LDPC

codes - open to discussion

tomorrow

Under BP decoding, the class of powerfull nonbinary LDPC codes is rather small

(d v = 2, concentrated checks).

change the decoder (but obviously more complex),

focus on the carefull design of the parity matrix (H with long cycles, choice of the

matrix values, etc).

For M-QAM constellations, take advantage of the properties that if q ≥ M :

the LDPC decoder is insensitive to the labelling choice,

the messages at the decoder input are independant (no loss of information between

demapper and decoder).

maybe allow some change at the constellation/mapping level

Finally, when q is relatively small q ≤ 32, optimize the code irregularity by means of

Density Evolution.

11 NewCom SWP1 meeting - 28th april 2005

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