design of ldpc-coded bicm using a semi-gaussian approximation1

8/7/2019 Design of LDPC-Coded BICM Using a Semi-Gaussian Approximation1

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JOURNAL OF ELECTRONICS(CHINA)

Design of LDPC-Coded BICM Using a Semi-Gaussian Approximation 1

Huang Jie Zhang Fan Zhu Jinkang

(PCN&SS Lab, University of Science and Technology of China, Hefei 230027 )

Abstract This paper investigates analysis and design of Low-Density Parity-Check (LDPC) coded Bit InterleavedCoded Modulation (BICM) over additive white Gaussian noise (AWGN) channel. It focuses on Gray-labeled 8-aryPhase-Shift-Keying (8PSK) modulation and employs a Maximum A Posteriori (MAP) symbol-to-bit metric calculator at thereceiver. An equivalent model of a BICM communication channel with ideal interleaving is presented. The probabilitydistribution function of log-likelihood ratio messages from the MAP receiver can be approximated by a mixture of symmetric Gaussian densities. As a result semi-Gaussian approximation can be used to analyze the decoder. Extrinsicinformation transfer charts are employed to describe the convergence behavior of LDPC decoder. The design of irregularLDPC codes reduces to a linear programming problem on two-dimensional variable edge-degree distribution. This methodallows irregular code design in a wider range of rates without any limit on the maximum node degree and can be used todesign irregular codes having rates varying from 0.5275 to 0.9099. The designed convergence thresholds are only a fewtenths, even a few hundredths of a decibel from the capacity limits. It is shown by Monte Carlo simulations that, when theblock length is 30,000, these codes operate about 0.62-0.75 dB from the capacity limit at a bit error rate of . 610

Key words I rregular Low-Density Parity-Check (LDPC) codes; EXtrinsic Information Transfer (EXIT) charts;Semi-Gaussian assumption

I. Introduction

Trellis-coded modulation [1], MultiLevel Coding [2] (MLC) and Bit-Interleaved Coded-Modulation [3] (BICM) providebandwidth and power-efficiency with pragmatic architectures. Recently, turbo-like codes, including Low-DensityParity-Check [4] (LDPC) codes have been under intense study for their near-capacity performance on several binary-inputchannels [5]. In this area, the most recent research has surrounded tools for the design of LDPC codes for higher-ordersignaling. The tools require the understanding of the convergence behavior of the iterative decoding algorithm [6-9] .

LDPCencoder

Mapper andmodulator

AWGNchannel

MAP symbol-to-bit metric

calculator

LDPCdecoder

Fig.1 Block diagram of LDPC-coded BICM

Among the tools, Density Evolution [6] (DE) has been used to design LDPC coded MLC [10,11] ; DE with All GaussianApproximation [8] (AGA) and EXtrinsic Information Transfer [12] (EXIT) charts has been used to design LDPC codedBICM [13,14] . Recently a more accurate one-dimensional algorithm called Semi-Gaussian Approximation (SGA) has beenintroduced in Ref.[15] to design LDPC codes for BPSK signaling over AWGN channel. Designing codes using DErequires intensive computation and a long search [5,6] . Designing codes using EXIT charts is simple, but it ignoresinformation about the probability distribution function (pdf) of messages and results in a loss in accuracy [16] . Comparedwith the AGA algorithm, which has serious limitations on the maximum node degree and also on the rate of the code [8],SGA algorithm has no limitation on the maximum node degree and works for a wide range of code rates [15] . In Ref.[15],the author has designed a code of rate 0.9497 with check nodes of degree 120, which has a gap of only 0.0340 decibelfrom the Shannon limit.

In this paper, we examine the design of LPDC-coded BICM over AWGN using SGA algorithm, without loss of generality, focusing on Gray-labeled 8PSK. We approximate the pdf of Log-Likelihood Ratio (LLR) messages fromdemodulator by a mixture of symmetric Gaussian densities. EXIT charts are used to describe the convergence behavior of the decoder and the design of irregular LDPC codes reduces to a linear program.

This paper is presented as follows. In Section II, we describe the system model under study and analyze its capacity.In Section III, we analyze the demodulator. We show that the LLR messages from the demodulator can be wellapproximated by a mixture of symmetric Gaussian distributions. In Section IV, we use SGA algorithm in the analysis of LDPC-Coded BICM. In Sections V, we address the problem of design irregular LDPC codes and present a group of codes

to verify the capabilities of our method. We conclude this paper in Section VI.II. System Model and Capacity Analysis

1Manuscript received date: September 28, 2005; revised date: January 1, 2006.Communication author: Huang Jie, born in 1981, male, Ph.D. candidate. PCN&SS Lab, University of Science and Technology of China, Hefei 230027, China. [email protected]


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M The system model is shown in Fig.1 . Since an LDPC encoder is employed, a bit interleaver is not necessary here [14] .

The M -ary modulator takes code bits at a time and maps them to a complex symbol. The AWGN channel adds

noise sample,2logm=

I Qn n jn= + , to each such symbol, where I n and are independent Gaussian random variables with zeromean and variance

Qn2n . The demodulator is a Maximum A Posteriori (MAP) symbol-to-bit metric calculator

[3] analyzedbelow. Its output is passed to decoder, which employs the sum-product decoding [17] algorithm.

Assume that the M -ary modulation is modeled by a two dimensional equiprobable signal set and each signal

corresponds to a binary label of length m. We can calculate the capacity of M -ary BICM via

10{ }

M i iA a

==

[3]

1

1 10

20 0

( | )1

log2 (

k j b

M

imi

k b ja A

p y aC m E

p y a

=

= =

| )

=

(1)

where y represents the received signal and k represents the subset of signals inbA A whose labels equal b in the k- th bit,(bit 0 is the rightmost bit). For M -ary modulation with unit average symbol energy, it is conventional to

describe the channel Signal-to-Noise Rate (SNR) by

{0,1,..., 1}k m

)20/ 1/(2s nE N = and to report this number in decibels as

10 010log / sE N . Fig.4 of Ref.[3] shows the BICM capacity versus channel SNR for 4PSK, 8PSK and 16QAM with Gray

labeling and natural labeling over AWGN. The natural labeling provides a lower capacity than the Gray labeling. When

the capacity is not more than 1.5, Gray labeled-4PSK provides the same capacity as Gray-labeled 8PSK does. When the

capacity is between 0.7 and 2.7, the curve of Gray-labeled 8PSK is approximately linear.

III. Receiver Analysis

Throughout this paper, we assume 8PSK modulation. Since Gray labeling provides a higher capacity than the naturallabeling, we focus on Gray labeling for the modulator.

Let k x denote the k-th code bit ( k = 0, 1, 2) of 8PSK label 2 1 0( , , )x x x x= ( 0x is the rightmost bit). The LLR of the code bit

k x given received sample y is written as

0

1

0 1

22

22

* 2 *2 2

1exp( || || )

2( | ) ln

1exp( || || )

2

1 1max ( || || ) max ( || || )

2 2

k

k

k k

w A nk

v A n

w A v An n

y w

L x yy v

2y w y

=

v

= (2)

where *max ( , ) max( , ) ln(1 exp(| |))x y x y x y= + + and the term ln(1 exp(| |))x y+ can be implemented by a look-up table or apiece-wise linear function approximation [18] .

Consider the constellation of Gray-labeled 8PSK in Fig.2 , with labels for the signalpoints , respectively. As in Ref.[14], we use notations

1,

1to denote the sets of signals that are

far and near from the decision boundary for bit

(000,001,011,010,110,111,101,100)

0 1 2 3 4 5 6 7( , , , , , , , )v v v v v v v vf

xSnxS

1x . The definition of notations 2 , for bitf xS 2nxS 2x is the same.

1 0 3 4 7 1 1 2 5 6{ , , , }, { , , , };f n

x xS v v v v S v v v v= =

2 1 2 5 6 2 0 3 4 7{ , , , }, { , , , };f n

x xS v v v v S v v v v= =


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Channel

Input t o t he next i ter ati on

Output f r om t he previ ous i t erat i on

Fig.4 Depth-one tree for a (3, 6) regular LDPC code

Given the mean value of LLR messages from check nodes , ,and the mean value of LLR messages from

channel , , the mean value of LLR messages from variable nodes with degree , ,ischk m

im*m i

* ( 1)im m i m= + chk

hk

(3)

For Gray-labeled 8PSK modulation, in iterative decoding process we have

,0 0 ( 1)i cm m i m= +

,1 1 ( 1)i f f chk m m i m= +

,1 1 ( 1)i n n chk m m i m= +

,2 2 ( 1)i f f chk m m i m= +

,2 2 ( 1)i n n chk m m i m= +

where , ( , ) and ( , ) are mean values of LLR messages from variable nodes of degree which is

transmitted through the 0- th,1- th and 2- th levels, respectively.

i,2,i nm,2,i f m,0im ,1,i nm,1,i f m

If we have -PSK, then we have levels of bits, hence if v is the maximum left degree, there are sets of m m2m d


s ( ) satisfying the following conditions:, ,2 ,0 1i j vi d j m

1

,2 0

1vd m

i ji j

= =

= (4)

,0 ,1 , 1

2 2 2

...v v vd d d

i i i

i i ii i i

m= = =

= = = (5)

where,i j means the fraction of edges connected to variable nodes of degree which are transmitted through thei

-th level.j

[6]Obviously the equivalent channel model shown in Fig.3 satisfies the channel symmetry condition . the SPD decodersatisfies the check node symmetry condition and the variable node symmetry condition [6] . Thus we just need to analyze thedecoder as if the all-zero codeword was transmitted. Thus for Gray-labeled 8psk modulation, the pdf of messages fromvariable nodes can be described by a mixture of symmetric Gaussian densities:

,1,0 ,0 ,0 ,1 ,1

2

,1 ,2 ,2,1 ,1 ,2 ,2 ,2 ,2

( ) ( ( ,2 ) ( ,2 )2

( ,2 ) ( ,2 ) ( ,2 )2 2 2

vd i

chk i i i i f i f i

i i ii n i n i f i f i n i n

G m G m m G m m

G m m G m m G m m

=

= +

+ +

)

+ (6)

At the beginning of iteration, , it becomes0chk m =


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,1 ,1,0 0 0 1 1 1 1

2

,2 ,22 2 2 2

( ( , 2 ) ( , 2 ) ( , 2 )2 2

( , 2 ) ( , 2 ))2 2

vd i i

i f f i

i if f n n

G m m G m m G m m

G m m G m m

=

+ +

+

n n + (7)

We use mutual information based EXIT chart to describe the behavior of SPD decoder. An EXIT chart is a curve

0with two axes: and , and a parameter

0inI out I I ( , )out inI f I I = , which is the information from the channel. Any point of

this curve shows that if the knowledge from the previous iteration is , using this together with the channel informationinI


0I , the information at the output of this iteration would be . It has been shown in Ref.[9] that when the pdf of messages is a mixture of symmetric Gaussian densities, mutual information combines linearly to form the overall mutualinformation. That is, for pdf described by Eq.

out I

(6) , its corresponding mutual information is

,1,0 ,0 ,1

2

,1 ,2 ,2,1 ,2 ,2

( ) ( ( 2 ) ( 2 )2

( 2 ) ( 2 ) ( 2 ))2 2 2

vd i

chk i i i f i

i i ii n i f i n

I m J m J m

J m J m J m

=

= +

+ +

+ (8)

where function is defined in Ref.[12].J

Fixing the check edge-degree distributionk

s, we refer to the EXIT chart of a code whose variable nodes are all of degree and transmitted through thei -th level byj

, ( )i jf x , and we call it an elementary EXIT chart. The EXIT chart

of an irregular code with variable edge-degree distribution 1, ,

2 0

( ) ( )vd m

i j i ji j

f x f

= == ,i j s can be written as x .

V. Design of LDPC-Coded BICM System

We use linear program to design irregular LDPC codes. Our goal is to maximize the design rate of the code whilehaving the EXIT chart of the irregular code satisfy the condition that ( )f x x> , for all 0 , where 0[ ,1)x x x is the initialmutual information from the channel. Being above x for all x guarantees convergence to zero message-error rate for aninfinite block-length code.

The design rate of the code is,1 ( / / / )k i

k i j

jR k = i , and hence, for fixed parameters of channel SNR and check

edge-degree distribution s, the design problem can be formulated as the following linear program:k

vd 1,

i=2 0

1

, ,2 0

,0 ,1 , 1

2 2 2

1

0 , ,2 0

maximize

subject to 0, 1,

... , and

[ ,1), ( ) .

v

v v v

v

mi j

j

d m

i j i ji j

d d d i i i m

i i i

d m

i j i ji j

i

i i i

x x f x x

=

= =

= = =

= =

=

= = =

>

(9)

In the above formulation, we have assumed that the elementary EXIT charts are given. In practice, to find these

curves, we need to know the variable edge-degree distribution ,i j s. We need ,i j s to associate every input inI to itsequivalent input pdf. In other words, prior to the design,

,i j s are not known, and as a result, we can not find , ( )i jf x s to

solve the linear program above.

We use the two-step algorithm presented in Ref.[15] to solve this problem. At first we assume that the inputmessage to the iteration, Fig.4 , has a single symmetric Gaussian density instead of a Gaussian mixture. Using thisassumption, we can get those elementary EXIT charts

, ( )i jf x s through simulations and get the distribution ,i j s through

solving the linear program above.

After finding the appropriate distribution,i j s based on a single Gaussian assumption, we use this degree distribution

for finding the correct elementary EXIT charts based on a Gaussian mixture assumption described by Eq. (6) . Then we usethese corrected

, ( )i jf x s to do linear program above again to get the final degree distribution ,i j s. The algorithm can be

summarized as follows.


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(1). Map every to the associated single symmetric Gaussian density.I in(2). Get the charts

, ( )i j x s through simulations.

(3). Get the temporary distribution,i j s through solving the linear program.

(4). Use this temporary distribution for mapping every to the Gaussian mixture distribution described by Eq. (6) .inI

(5). Get the charts, ( )i j x s through simulations.

(6). Get the final distribution,i j s through solving the linear program.

1Manuscript received date: Septem 2005; revised date: January 1, 2006.Communication author: Huang Jie, born in 1981, male, Ph.D. candidate. PCN&SS Lab, University of Science and Technology of China, Hefei 230027, China. [email protected]

ber 28,

Let ,i j be the fraction of variable nodes having degree and transmitted throughi j -th level. Thus we have

v

,, d 1

,i=2 0

= i ji j mi j

j

i

i

=

Then the design problem of Eq. (9) can be formulated as

vd 1

,i=2 0

,

,0 , 12 2

1 1

0 , ,2 0 2 0

minimize

subject to 0,

1... ,and

[ ,1), ( ) ( ) .

v v

v v

m

i jj

i j

d d

i i mi i

d d m m

i j i j i ji j i j

i

m

,x x f x i

=

= =

= = = =

= = =

>

(10)

x

,

For the convenience of designing finite block length irregular LDPC codes, the problem of Eq. (10) can be formulatedas a linear integer program problem. For example, if we want to design a code of block length . The problem of Eq.

0m N

(10) can be formulated as

vd 1

,

i=2 0

, ,

,0 , 1 02 2

1 1

0 , ,2 0 2 0

minimize

subject to 0, is an integer,

... , and

[ ,1), ( ) ( ) .

v v

v v

m

i j

j

i j i j

d d

i i mi i

d d m m

i j i j i ji j i j

ia

a a

a a N

x x a f x ia

=

= =

= = = =

= = =

>

(11)

x

where is the number of variable nodes having degree and transmitted through,i ja i j -th level.

We have used our design method to design a number of irregular codes with various rates and a block length 30,000.With the analysis in Section II, we conclude that when the capacity is less than 1.5, QPSK would be a better choice than

8PSK because of its lower complexity. Therefore we have not designed any code with rate less than 0.5. In the design of the codes, we have avoided any variable node with degrees higher than 30 and we have limited the number of useddegrees to 10. These constraints make the designed codes more practical for implementation. For check nodes, we haveused a regular structure as recommended in Ref.[6,8]. In the design of each code, first, by the BICM capacity versuschannel SNR curves we choose an appropriate channel SNR for a given aim of rate. Then, we choose an appropriate check node degree by studying the elementary EXIT charts for different values of check-node degree. Finally, we maximize thecode rate for the given channel SNR. The design results are presented in Tab.1.

In Ref.[14], the authors have used EXIT charts to design a code of rate (the capacity is 2) for Gray-labeled8PSK over AWGN, which has a gap of 0.35 dB to the BICM capacity. In Ref.[13], the authors have used DE with AGA todesign a code of rate (the capacity is 2) for Gray-labeled 16QAM over AWGN, which has a gap of 0.25 dB to theBICM capacity. Compared with these methods, our method shows significant accuracy.

2 /3

1/ 2

For all the codes presented in Tab.1, we have performed Monte Carlo simulations. The simulation results, togetherwith their respective capacity limits, are shown in Fig.5 . We note that these codes operate about 0.62-0.75 dB from thecapacity limit at a bit error rate of . 610


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Tab.1

List of irregular codes of block length 30,000 designed by semi-Gaussian methodDegree Code1 Code2 Code3 Code4 Code5

sequence

2,5,9890,6132

3,4947,106,1745

5,2335,1,0

6,1739,3,0

30,974,0,2123

_

_

_

_

_

2,0,9657,9656

3,1556,343,344

4,6048,0,0

5,5,0,0

12,1616,0,0

13,57,0,0

25,1,0,0

26,531,0,0

27,186,0,0

_

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

6 6 6

7 7 7

8 8 8

9

1 ,0 ,1 ,2

2 ,0 ,1 ,2

3 ,0 ,1 ,2

4 ,0 ,1 ,2

5 ,0 ,1 ,2

6 ,0 ,1 ,2

7 ,0 ,1 ,2

8 ,0 ,1 ,2

9

, , ,

, , ,

, , ,

, , ,

, , ,

, , ,

, , ,

, , ,

,

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v

v d d d

v d d d

v d d d

v d d d

v d d d

v d d d

v d d d

v d d d

v d

d a a a

d a a a

d a a a

d a a a

d a a a

d a a a

d a a a

d a a a

d a9 9

10 10 10

,0 ,1 ,2

10 ,0 ,1 ,2

, ,

, , ,v v

v v v

d d

v d d d

a a

d a a a

2,0,10000,3220

3,3031,0,6778

4,3874,0,2

11,428,0,0

12,2667,0,0

_

_

_

_

_

2,0,5622,9998

3,5509,3445,1

4,1,2,1

6,1448,0,0

7,1794,8,0

30,1248,923,0

_

_

_

_

2,3669,8021,9986

3,3,0,2

5,5826,0,0

6,230,0,0

28,1,0,12

29,271,1979,0

_

_

_

_

40 30 20 12 10cd

Convergence

threshold(Channels

0.2108 0.2695 0.3030 0.3400 0.4430

)n

Rate 0.9099 0.8139 0.7591 0.6985 0.5275

Gap to capacity

limit (dB)

0.0622 0.0129 0.0167 0.0216 0.3892

VI. Conclusions

We present an equivalent model of a BICM communication channel with ideal interleaving and we show that the pdf of LLR messages from the MAP demodulator can be described by a mixture of symmetric Gaussian densities. We use thesemi-Gaussian approximation method, instead of the all-Gaussian approximation method, for code analysis and design viaEXIT charts. The design of irregular LDPC codes reduces to a linear program of two-dimension variable edge-degreedistribution

,i j s. Codes with various rates have been designed. They have thresholds only a few tenths, even a few

hundredths of a decibel from their respective capacity limits. It is shown by Monte Carlo simulations that these codesoperate only 0.62-0.75 dB from their capacity limits at a bit error rate of . 610



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Fig.5 Performances of designed LDPC codes

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design of ldpc-coded bicm using a semi-gaussian approximation1

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