Design of Short Irregular LDPC Codes for a Markov-modulated

Gaussian Channel

W. Proß

1,2

, M. Otesteanu

and F. Quint

Faculty of Electronics and Telecommunications, Politehnica University of Timisoara, Timisoara, Romania

Faculty of Electrical Engineering and Information Technology, University of Applied Sciences, Karlsruhe, Germany

Keywords:

Irregular LDPC Codes, Density-evolution, Downhill-simplex, Markov-modulated Gaussian Channel.

Abstract:

This paper deals with the design of short irregular Low-Density Parity-Check (LDPC) codes. An optimization

method for the underlying symbol-node degree-distribution (SNDD) of an irregular LDPC code is introduced,

which is based on the Downhill-Simplex (DHS) algorithm. In order to compare our method with the opti-

mization described in (Hu et al., 2005), which is based on a simpliﬁed version of the DHS algorithm, we

ﬁrst designed a rate 0.5 irregular LDPC code of length n = 504 for an Additive White Gaussian Noise Chan-

nel (AWGNC). The proposed optimization method was then used to design an irregular LDPC code for a

Markov-modulated Gaussian Channel (MMGC). The decoding performance of the resulting LDPC code is

then compared to the design based on the Density-Evolution (DE) method.

1 INTRODUCTION

The importance of channel coding has increased

rapidly together with the still vast growing market in

the ﬁeld of digital signal processing. One channel

code that is more and more signiﬁcant is the Low-

Density Parity-Check (LDPC) code. The principle

of this linear block code has already been published

in 1962 by Robert Gallager (Gallager, 1962). After

LDPC codes had been forgotten for decades, mainly

because of their computational burden, they were re-

discovered by MacKay and Neal in 1995 (MacKay

and Neal, 1995). Since then lots of design techniques

have been developed, yielding in LDPC codes opti-

mized with respect to different design criteria (e.g.

low error-ﬂoor, performance close to capacity, hard-

ware implementation). A commonly used tool for

the design of a class of LDPC codes called irregular

LDPC codes is Density-Evolution (DE) (Richardson

et al., 2001)(Luby et al., 2001). In (Eckford, 2004)

the author derived Density-Evolution for Markov-

modulated channels and based on that designed an

irregular LDPC code for a Markov-modulated Gaus-

sian Channel (MMGC).

In section 2 we brieﬂy explain irregular LDPC

codes and their design. In the next section we intro-

duce the DHS-based design that we propose and then

show results in section 4.

2 IRREGULAR LDPC CODES

Low-Density Parity-Check (LDPC) codes are based

on a sparse Parity-Check Matrix (PCM). The n

columns of a PCM stand for the n symbols of a LDPC

codeword and each row represents one of m = n − k

unique parity-check equations with k being the num-

ber of information symbols. The code rate is then

r =

. An alternative representation is obtained by

use of a Tanner-graph (Tanner, 1981). Such a bi-

partite graph consists of n symbol-nodes (SN) and

m check-nodes (CN) corresponding to the n columns

and m rows of the PCM respectively. The SNs and

CNs are connected dependent on the nonzero entries

in the PCM. Considering an AWGNC the decoding

of LDPC codes is done using the Belief-Propagation

(BP) algorithm (Gallager, 1962) or an approximation

of it (e.g. the Min-Sum (MS) decoder) (Hu et al.,

2002). The LDPC code of interest in this work is

the irregular LDPC code that, in contrast to regular

LDPC codes, exhibit several row weights and col-

umn weights. They are described by use of polyno-

mials. The following polynomial is used to specify

the symbol-node degree-distribution (SNDD).

λ(x) =

∑

j=1

(1)

is the fraction of SNs that have j+1 connected

Proß W., Otesteanu M. and Quint F..

Design of Short Irregular LDPC Codes for a Markov-modulated Gaussian Channel.

DOI: 10.5220/0004017000310034

In Proceedings of the International Conference on Signal Processing and Multimedia Applications and Wireless Information Networks and Systems

(SIGMAP-2012), pages 31-34

ISBN: 978-989-8565-25-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

edges and d

is the maximum number of adjacent

edges. The description of the check-node degree-

distribution is likewise.

Density-Evolution (DE) is a powerful tool to ana-

lyze the asymptotic performance of a LDPC code en-

semble described by a pair of degree-distributions (for

the SNs and CNs respectively). In (Richardson et al.,

2001) and (Luby et al., 2001) the authors showed the

possibility of designing good irregular LDPC codes

based on DE. In (Luby et al., 1998) and (Richard-

son and Urbanke, 2001) a concentration theorem is

proved that states, that the performance of an ensem-

ble of LDPC codes decoded with a BP-decoder is con-

centrated around the average performance of the en-

semble. The analysis of LDPC codes using DE is

based on the concentration theorem and on the as-

sumption of a cycle-free code. It is well known that

the shorter the LDPC code the more cycles occur.

Furthermore for short LDPC codes the length of the

cycles is short with respect to the decoding iterations

required in average, which leads to an harmful im-

pact on the decoding performance. In (Amin et al.,

1999) it can be seen that the gap between the pre-

dicted performance based on DE and the real perfor-

mance increases inversely proportional to the block-

length. Furthermore the concentration theorem does

not hold for short LDPC codes. This can be seen in

(MacKay et al., 1999) where a signiﬁcant variation of

the decoding performance over an ensemble of LDPC

codes is shown. Thus DE is not an appropriate tool

for the design of short LDPC codes. That is the rea-

son for Hu et al. to consider the Downhill-Simplex

(DHS) optimization (Nelder and Mead, 1965) for the

design of short LDPC codes in (Hu et al., 2005).

3 DHS-BASED DESIGN

We adopt the concept of designing short LDPC

codes by means of the DHS algorithm and the

Progressive-Edge-Growth (PEG) algorithm, but in

contrast to (Hu et al., 2005) we establish the whole

DHS-algorithm instead of using a simpliﬁed version.

The DHS optimization is based on a simplex S =

{λ

,λ

,··· ,λ

,λ

N+1

} with each vertex λ

represent-

ing a unique SNDD and thus consists of N values

{λ

i, j

}

j=1

. During the optimization process the ver-

tices are constantly sorted according to their function

evaluations so that f(λ

) ≤ f(λ

) ≤ ·· · ≤ f(λ

) ≤

f(λ

N+1

). In the context of SNDD-optimization the

function evaluation is represented by the computation

of the Word-Error-Ratio (WER). For the AWGNC we

used the Min-Sum-decoder (Hu et al., 2002) and for

the MMGC we decoded by means of the Estimation-

Decoder (ED) as described in (Proß et al., 2010).

The DHS algorithm always tries to replace λ

N+1

(the

worst vertex) by a better one. This is done based on

an operation called Reﬂection which is computed by

′

+ α(

′

−λ

N+1

) (2)

with α = 1 and

′

being the centroid of the simplex

(computed without considering λ

N+1

) on which the

worst vertex is reﬂected. It is calculated according to

′

∑

i=1

. (3)

Depending on the WER of the reﬂected vertex λ

one of the following four operations is processed:

Inward Contraction:

= λ

N+1

+ β(

′

−λ

N+1

); (β = 0.5) (4)

Outward Contraction:

′

+ β(

′

−λ

N+1

); (β = 0.5) (5)

Reduction:

new

= λ

+ σ(λ

−λ

) ∀i\ 1;(σ = 0.5) (6)

The computation of the Expansion operation is

based on equation 5 with β = 2. The whole algorithm

can be seen in Algorithm 1.

is the average distance of the vertices to the

centroid

λ of the simplex and is computed by

N + 1

N+1

∑

i=1

∑

j=1

(λ

i, j

−

)

. (7)

λ is calculated as in equation 3 except that the sum

includes λ

N+1

. Two constraints have to be considered

when optimizing a SNDD:

0 < λ

< 1 ∀ j \ N (8)

0 <

N−1

∑

j=1

< 1 (9)

We respect the constraints in the same way as in

(Hu et al., 2005). In order to reduce the probability of

converging to a local minimum, we repeat the DHS

optimization ten times. The threshold r

thres

in Algo-

rithm 1 is set to 1e

−4

for the ﬁrst nine rounds. In

the last round we then include the resulting SNDDs

of the nine previous rounds when creating the initial

simplex and set r

thres

= 1e

−10

. For the ﬁrst round

the i

vertex λ

= {λ

i,1

,...,λ

i,N

} of the simplex S =

{λ

,λ

,...,λ

,λ

N+1

} is initialized as follows:

i, j











0.5−

N−1

,∀i\ N,∀ j \ i

0.5+

, j = i

random[0,r

max

] ,i = N + 1

(10)

SIGMAP2012-InternationalConferenceonSignalProcessingandMultimediaApplications

with

max

= 1 −

j−1

∑

l=1

i,l

(11)

The initializations of the next eight start-simplezes are

done randomly.

4 RESULTS

We designed two short irregular LDPC codes using

the optimization method described in section 3. In

order to compare our design with the one described

in (Hu et al., 2005) we designed one irregular LDPC

code with the same code rate r = 0.5 and blocklength

n = 504. We thereby used the only channel-model

that the authors in (Hu et al., 2005) designed for,

Algorithm 1: DHS optimization of the SNDD.

1: S

initial

= {λ

,λ

,...,λ

,λ

N+1

} ⊲ create initial

simplex

2: while (r

> r

thres

) do

3: SORT VERTICES;

4: COMPUTE REFLECTION;

⊲ f(λ

) in between worst and 2.worst

5: if f(λ

) < f(λ

N+1

) then

6: COMPUTE OUTWARDCONTRACTION;

7: if f(λ

) < f(λ

) then

8: λ

N+1

← λ

9: else

10: PERFORM REDUCTION;

11: end if

⊲ f(λ

) worse than worst or equal

12: else if f (λ

N+1

) ≤ f(λ

) then

13: COMPUTE INWARDCONTRACTION;

14: if f(λ

) < f(λ

N+1

) then

15: λ

N+1

← λ

16: else

17: PERFORM REDUCTION;

18: end if

⊲ f(λ

) better than best or equal

19: else if f (λ

≤ f(λ

) then

20: COMPUTE EXPANSION;

21: if f(λ

) < f(λ

) then

22: λ

N+1

← λ

23: else

24: λ

N+1

← λ

25: end if

⊲ f(λ

) in between best and 2.worst

26: else if f (λ

) < f(λ

) ≤ f(λ

) then

27: λ

N+1

← λ

28: end if

29: end while

which is the AWGNC. The second design was done

for comparison purposes with the DE-based design

in (Eckford, 2004), where the author designed a rate

r = 0.608 irregular LDPC code for a MMGC. Thus

we designed a LDPC code with the same code rate for

the MMGC as well and chose the blocklength to be

n = 576. For both designs the maximum symbol-node

degree was set to d

= 15. Based on the designed

SNDD a PCM was constructed using the PEG algo-

rithm and based on a following simulation the Bit-

Error-Ratio (BER) and the Word-Error-Ratio (WER)

were computed for different values of

The resulting SNDD for the AWGNC was

λ(x) = 0.42958x

+ 0.40154x

+ 0.00017x

0.07714x

+ 0.0001x

+ 0.00362x

+ 0.00085x

0.06449x

+ 0.00028x

+ 0.00029x

0.00347x

+ 0.01379x

+ 0.00031x

+ 0.00438x

The simulation-results can be seen in Figure 1. For

comparison purposes the results based on the SNDD

of (Hu et al., 2005) are depicted as well. All curves

in Figure 1 are based on 100 decoding iterations with

the Min-Sum decoder (Hu et al., 2002). It is well

seen that the performance of our LDPC code beats

the one from (Hu et al., 2005) with up to 0.25dB for

the BER and up to 0.35dB for the WER.

The design results for the MMGC can be seen in

Figure 2. For each of the three simulations a MMGC

with P

good

= 0.6 and P

bad

= 0.3 and an ED-decoder

as described in (Proß et al., 2010) was used. The

ﬁrst simulation is based on the design results in (Eck-

ford, 2004) that are obtained by using DE. The second

simulation evaluates the decoding performance by us-

ing our design method described in section 3. The re-

sulting SNDD was λ(x) = 0.35489x

+ 0.24392x

0.18349x

+ 0.12821x

+ 0.04756x

+ 0.00259x

0.00221x

+ 0.00227x

+ 0.00231x

+ 0.00239x

0.00661x

+ 0.00279x

+ 0.01083x

+ 0.00995x

We additionally added the results of a simulation with

a regular LDPC code having a SNDD of λ(x) = x

The results in Figure 2 show that the irregular LDPC

code designed with our DHS-method beats the reg-

ular LDPC code as well as the irregular LDPC code

designed with DE in terms of BER and WER.

5 CONCLUSIONS

In this paper an optimization method for the symbol-

node degree-distribution of irregular LDPC codes is

introduced, that is based on the Downhill-Simplex al-

gorithm. We proved that the decoding performance

increases when designing short irregular LDPC codes

with our design method instead of the simpliﬁed DHS

version in (Hu et al., 2005). Furthermore an irregular

DesignofShortIrregularLDPCCodesforaMarkov-modulatedGaussianChannel

0 1 2 3

−6

−5

−4

−3

−2

−1

(dB)

BER - Hu

BER - DHS

WER - Hu

WER - DHS

Figure 1: Two r = 0.5 LDPC codes with n = 504 and an

AWGNC. DHS refers to the Downhill-Simplex based de-

sign proposed in this paper and Hu refers to the design re-

sults obtained in (Hu et al., 2005).

0 1 2 3 4

5 6

−7

−6

−5

−4

−3

−2

−1

(dB)

BER - DE

BER - DHS

BER - reg

WER - DE

WER - DHS

WER - reg

Figure 2: Three rate 0.608 LDPC codes of length n = 576

with a MMGC and Estimation-Decoding. DE stands for the

Density-Evolution design, DHS for the Downhill-Simplex

based design described in section 3 and reg for a regular

LDPC code.

LDPC code of length n = 576 is designed with the

proposed algorithm for a Markov-modulated Gaus-

sian Channel. The results of a following simulation

reveal a superior decoding performance of the LDPC

code designed with our method compared to the de-

sign by means of Density-Evolution.

ACKNOWLEDGEMENTS

This work is part of the project MERSES and has

been supported by the European Union and the Ger-

man state Baden-W¨urttemberg.

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