2013 TCOM Ira Plnc

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2013 1

Design of Irregular Repeat-Accumulate CodedPhysical-Layer Network Coding

for Gaussian Two-way Relay ChannelsTao Huang, Student Member, IEEE, Tao Yang, Member, IEEE,

Jinhong Yuan, Senior Member, IEEE, and Ingmar Land, Senior Member, IEEE

Abstract—This paper addresses the design of irregular repeataccumulate (IRA) codes for coded physical-layer network coding(PNC) for the binary-input Gaussian two-way relay channel,assuming perfect synchronization and equal received power atthe relay. The design is based on a nontrivial extension of EXIT-chart based design. Specifically, we analyze the components ofthe IRA-PNC scheme and propose an approach to model thesoft information exchanged between these components. Then, wedevelop upper and lower bounds on the extrinsic informationtransfer functions to characterize the iterative process of comput-ing the network-coded information. Based on that, we constructoptimized IRA codes to minimize the computation error at therelay. The optimized IRA-PNC has considerable performanceimprovement over the existing regular RA coded PNC. For arate 3/4 code, as an example, we observed improvements of 2.6dB, and the optimized IRA-PNC scheme is only about 1.7 dBaway from the capacity upper bound of the Gaussian two-wayrelay channel.

Index Terms—Two-way relay channel, physical-layer networkcoding, compute-and-forward, Repeat-Accumulate codes, conver-gence analysis, EXIT chart.

I. INTRODUCTION

EFFICIENT communication over two-way relay channels(TWRCs) has attracted intensive research efforts, in light

of the discovery of network coding [1]. From the literature,several classical relaying mechanisms have been extended toTWRCs. Among those, an amplify-and-forward based scheme,namely analog network coding [3–5], is simple to implement,but suffers from noise amplification and unnecessary powerconsumption [6]. A complete decoding-based scheme, wherethe relay fully decodes both users’ messages and then formsthe network-coded message, can reduce the effect of noiseamplification but suffers from multiplexing loss [2, 7, 8].

The physical-layer network coding (PNC) technique pro-posed in [6, 9, 10] can further improve the performance of

Paper approved by K. Narayanan, the Editor for Coding Theory andApplications of the IEEE Communications Society. Manuscript receivedSeptember 11, 2011; revised May 24, 2012.

This work has been presented in part at the 8th International Symposiumon Wireless Communication Systems (ISWCS), Aachen, Germany, November2011 [44].

T. Huang and J. Yuan are with School of Electrical Engineering andTelecommunications, University of New South Wales, Sydney, Australia (e-mail: [email protected], [email protected]).

T. Yang is with Wireless and Networking Technologies Laboratory(WNTL), Commonwealth Scientific and Industrial Research Organization(CSIRO), Sydney, Australia (e-mail: [email protected]).

I. Land is with Institute of Telecommunication Research, University ofSouth Australia, Adelaide, Australia (e-mail: [email protected]).

a two-way relay system. Rather than decoding both users’messages individually, the relay in a PNC scheme only com-putes and forwards [11] sufficient information to the two users,which can avoid the drawbacks of the analog network codingscheme and the complete decoding-based scheme. It has beenshown in [7] from the information theoretic perspective that,the PNC scheme can achieve within 1/2 bit of the capacity ofthe Gaussian TWRCs and is asymptotically optimal at a highsignal-to-noise ratio (SNR).

The benefits of PNC have been recently demonstrated invarious work. In [12], the authors showed that the PNC schemeachieves a higher maximum sum-rate, and a lower sum-biterror rate than the conventional transmission scheme for anumber of practical scenarios. The work in [13] investigatedthe optimization of the PNC constellation with asynchronoustransmission using QPSK and 16-QAM modulation, anddemonstrated significantly improved end-to-end throughput.A modified higher order phase amplitude modulation wasproposed in [14] to resolve the ambiguity mapping at the relayin a binary PNC scheme. Although most work assume perfectsynchronization, [15] showed that even with synchronizationerrors, the PNC scheme can still provide a higher capacitythan the complete decoding-based scheme. The work in [16–18] proposed several approaches to mitigate the effects broughtby the asynchronous transmission in PNC schemes. A numberof works also extended the PNC scheme to multiple-inputmultiple-output systems [19–25] or networks with multiplerelays [26].

While most works considered uncoded PNC, [2] investi-gated a regular repeat-accumulated (RA) coded PNC schemeand proposed iterative message passing decoding algorithmsfor the channel coded PNC. Later, a convolutional coded PNCscheme with modified Viterbi and BCJR algorithms was stud-ied in [27]. The work in [28] further extended the work in [13]to a convolutional-coded system. In [29], an asymptoticallytight bound was derived for the error probability of a channel-coded PNC (CPNC) scheme, which showed that its error-rateperformance is degraded by at most ln 2 (in linear SNR scale)relative to the single-user case.

In this paper, we consider binary-input Gaussian TWRCswith perfect synchronization and equal power allocation forthe two users, as in the pioneering work [2]. Applicationsof such a scenario can be found in satellite communicationsand deep-space communications [2, 6]. We investigate the PNCscheme from a binary channel code design’s perspective. The


goal is to design a practical binary code for the CPNC schemesuch that it performs close to the capacity limit of a binary-input Gaussian TWRC. In particular, we investigate irregularRA (IRA) coded PNC schemes for binary-input GaussianTWRC. In the IRA-PNC scheme under consideration, therelay computes a binary network codeword, from its receivednoisy ternary superimposed signal sequence, which is thenforwarded to the users. To carry out this computation, itis required to extend the conventional Tanner graph [30] toan equivalent Tanner graph (ETG), defined over a ternarysuperimposed signal domain [2]. The presence of the ternarysignal leads to the challenges in the convergence analysis anddesign of a capacity-approaching IRA-PNC scheme.

In this paper, we address the challenges in the convergenceanalysis and design of IRA coded PNC schemes. Our maincontributions are:

1. We analyze the component decoders of the IRA-PNCscheme and derive the generalized update rules for thesecomponents in terms of log-likelihood ratios (LLRs).

2. Then we propose two models for the soft information ex-changed among the components decoders and develop boundson the approximation of the extrinsic information transfer(EXIT) functions of the IRA-PNC scheme. Based on that, wecarry out an EXIT chart curve-fitting technique to constructoptimized IRA codes. Our developed IRA-PNC schemes havesignificantly improved performance compared to the existingregular RA coded PNC schemes. We show an example wherethe designed IRA-PNC scheme performs only 1.7 dB awayfrom the capacity upper bound of the Gaussian TWRC, andoutperforms the existing RA coded PNC scheme by 2.6 dB.

3. We compare our developed IRA-PNC scheme with acomplete decoding-based network coding scheme, in whichthe relay completely decodes both users’ messages, usingiterative multi-user detection and decoding, and then formthe network-coded message. For fairness, both the CPNCscheme and the complete decoding-based scheme utilize theircorresponding optimal codes and power allocations. Numericalresults show that the CPNC scheme can significantly outper-form the complete decoding-based scheme if the code rate issufficiently high. This agrees with the information theoreticresult in [7, 31].

The paper is organized as follows: Section II introduces thesystem model and the CPNC scheme. Section III analyzesthe components of the IRA-PNC scheme and presents thecomputation operation. In Section IV, we perform an EXITchart analysis and use curve-fitting techniques to optimizethe degree distributions of the IRA-PNC scheme. In SectionV, we present numerical results. Possible generalizations arepresented in Section VI and Section VII concludes the paper.

II. SYSTEM MODEL

We consider a Gaussian TWRC where two single-antennausers, denoted by A and B, exchange information via an inter-mediate single-antenna relay. The users and the relay operatein half-duplex mode and there is no direct link between theusers. The transmission protocol under consideration employstwo time-slots for each round of information exchange. In the

Uplink

Downlink

Nb Encoder R BPSK

An

Ay Decoder

A

By Decoder

B

Bn

Encoder A

sE

Ry Nb

BPSKAc Ax

Encoder B

BPSKBc

RelayComputation

Bb

Ab

Bb(A)

Nb

Ab

Bb

Ab(B)

Nb

Rc

Bx

Rx

Rn

sE

RE

Fig. 1. Architecture of a two-way relay system operated with CPNC. Therelay computes the network-coded message bN = bA⊕bB without explicitdecoding of both users’ individual messages. Here, “+” denotes in the linearaddition in real values and “⊕” denotes the modulo-2 addition.

first time-slot (uplink phase), the users transmit their signals tothe relay. In the second time-slot (downlink phase), the relaybroadcasts to the two users. At each node, the received signalis corrupted by additive white Gaussian noise (AWGN).

Now we illustrate the CPNC scheme for the binary-inputGaussian TWRC. The block diagrams of the CPNC schemeis depicted in Fig. 1, which follows [2]. We first considerthe uplink phase. Let bA = [bA (1) , · · · , bA (k)] ∈ 0, 1kdenote the length-k binary message sequences of user A.This message sequence is encoded with a binary linear chan-nel code and the resulting codeword is denoted by cA =[cA (1) , · · · , cA (n)] ∈ 0, 1n, where n denotes the lengthof the codeword. The code rate per user is given by R = k/n.The users’ codewords are modulated via binary phase shiftkeying (BPSK) (0 7→ −1, 1 7→ +1), resulting in the signalsequences xA = 2cA − 1 ∈ −1,+1n. The encoder andmodulation for user B are the same as user A, with similarnotations. The signal sequences of user A and user B aretransmitted simultaneously.

It is noteworthy that, in general, the two users in a CPNCscheme of a TWRC may have different data rates and differentsignal power. In this paper, however, we will follow thepioneering work [2] by limiting our discussion to the caseswhere the two users have identical data rates and the samereceived symbol energies. Assuming perfect synchronization[2], the signal received by the relay is

yR =√EsxA+

√EsxB+nR =

√Es(xA+xB)+nR, (1)

where Es is the received symbol energy per-user and nR isthe AWGN vector at the relay. The variance of the noise is σ2

and the per-user SNR in the uplink phase is given by Es/σ2.

In light of the notion of network coding [1], the relay needsto deliver a network-coded message to the two users. In thispaper, we define the network-coded (NC) message sequence asbN , bA⊕bB , where “⊕” denotes the element-wise modulo-2 addition operation. Upon receiving yR, the task of the relay


is to compute an estimate of the NC message sequence bN ,given as

bN = FR (yR) . (2)

A computation error at the relay is declared if bN = bN .In the downlink phase, as shown in Fig. 1, the relay

re-encodes the computed NC message sequence bN into acodeword cR, which is then BPSK-modulated, resulting signalsequence xR. This signal is broadcast to the two users. Then,user A first decodes the NC message bN . If the NC messageis correctly recovered by both the relay and user A, user Acan correctly recover user B’s message by performing

bB = bA ⊕ bN , (3)

with the help of its own knowledge of bA. In contrast, if acomputation error happens at the relay, a decoding error willhappen1. The operation at user B is similar to that of userA. This completes one round of information exchange. Moredetails about the downlink phase operation can be found in[2].

In the above CPNC scheme, the operation of decodingthe NC message bN at each user in the downlink phase isa standard single-user decoding. Thus, the key issue in thedecoding of the CPNC scheme is to efficiently compute theNC message bN at the relay in the uplink phase, i.e., Eq. (2).In this paper, we will only investigate the computation of theNC message bN at the relay (as in [2]).

III. IRREGULAR REPEAT-ACCUMULATE CODEDPHYSICAL-LAYER NETWORK CODING

In general, any binary linear channel code, such as aconvolutional code [27], a turbo code and a low-density parity-check (LDPC) code [32], can be employed in the CPNCscheme. In this paper, we consider IRA codes, since theirencoding is simpler than that of LDPC codes, and theirstructure allows a more flexible design than Turbo-codes. Inparticular, IRA codes have a flexible code structure, definedby the degree distribution of the variable nodes and checknodes, which allows for convenient design by curve-fittingin EXIT charts [33]. Here we consider non-systematic IRAcodes. Our analysis and design also apply to a CPNC schemewith systematic IRA codes or other types of codes.

A. Encoding with an IRA Code

Consider the system in Section II. In the uplink phase,user A’s message bits bA(t), t = 1, · · · , k, are repeated dvtimes, where dv ∈ 2, 3, · · · specifies the length of repetition.The repetition, or variable node (VN), degree distributionis given by λ(dv), dv ∈ 2, 3, · · · where λ(dv) is theportion of message bits with repetition length dv . Notice thatλ(dv) ≥ 0,

∑∞dv=2 λ(dv) = 1. The repeated bit sequence

is permuted by a random interleaver, denoted by π(·). Theinterleaved sequence is encoded by a series of parity-check

1There could be a minor case where the NC message bN is wronglycomputed by the relay but the final decoding result at a user is correct.However, the probability of such a case vanishes as k increases. We willnot consider this trivial case in this paper.

INTERLEAVER

Variable node

Check node

Accumulator

(1)yR

(2)yR

(n)yR

NCbits

(1)bN

Equivalent Encoding ProcessComputation Process

(2)bN

(k)bN

(1)cs

(2)cs

(n)cs

(2)bs

(k)bs

2CNf

MAX

CNf cd

1CNf

2CNf

2CNf

1CNf(1)bs

Channel observation

2

VNf

3VNf

MAX

VNf vd

3VNf

3

VNf

2VNf

Inner DecoderOuter Decoder

Fig. 2. Equivalent Tanner graph of an IRA-PNC scheme.

codes of degrees dc, where dc ∈ 1, 2, · · · . The check node(CN) degree distribution is given by ρ(dc), dc ∈ 1, 2, · · · where ρ(dc) is the portion of CNs whose number of connectededges is dc+1. We denote the average CN and VN degrees bydc and dv , respectively. The parity-check coded bits are thenpassed through an accumulator (ACC), generating the codedsequence cA. The same operation is performed at user B. Theirregularity of this code resides in various repetition lengths(VN degrees) and various CN degrees.

It is noteworthy that when the two users transmit with thesame rate, the same code is employed [2]. Then, the modulo-sum of the two users’ codewords is still a codeword of thecode. This is to ensure that the relay is able to compute thelinear network coded codeword without decoding individualuser’s codeword, as we will see later.

B. Iterative Computation of the NC message bN at the Relay

The algorithm in [2] only applies to VNs of degree 3 andCNs of degree 2. Here, we develop a computation algorithmthat applies to VNs and CNs of any degree. Then, we representthis algorithm in a log-likelihood ratio (LLR) format whichwill be required in the subsequent EXIT chart analysis.

Let us define bs , bA + bB ∈ 0, 1, 2k and cs , cA +cB ∈ 0, 1, 2n as a superimposed message sequence and asuperimposed codeword, respectively. Consider the followinglinear processing

y′R = yR + 2

√Es = 2

√Es(cA + cB) + nR, (4)

where y′R is equivalent to yR for the purpose of computing

bN . From (4) we see that the signal y′R is a noisy copy of

the superimposed codeword cs. To compute the desired NCmessage, a “virtual encoding” process [2], which maps eachsuperimposed message bs to a superimposed codeword cs,is required. For an IRA coded PNC scheme, specifically, this“virtual encoding” process can be described via an equivalentTanner graph (ETG), formed by superimposing two conven-tional Tanner graphs [2] of the same single-user IRA code,as shown in Fig. 2. The structure of the ETG resembles thatof the single-user IRA code. However, there are two majordifferences:

1) The inputs and outputs have ternary symbols, i.e.,bs ∈ 0, 1, 2k and cs ∈ 0, 1, 2n. The message


exchanged between the component nodes consists of theprobabilities for the ternary symbols.

2) The ETG features an equivalent CN function and anequivalent VN function, denoted by fCN (·) and fVN (·),respectively, which are different from those of the single-user case.

Given y′R, the relay first exploits the ETG to compute

an estimate of the ternary superimposed message sequence,denoted by bs. Next, given bs, the estimated NC messagesequence bN is obtained by calculating the modulo-2 of bs,i.e.,

bN (t) =

0 if bs(t) = 0 or 2,1 if bs(t) = 1,

(5)

for t = 1, · · · , k.Now we briefly illustrate how to iteratively compute the

superimposed message sequence bs, based on the ETG givenabove. Consider a node in the ETG which has L edges. Theternary a priori message to the lth edge of this node, l =

1, · · · , L, is denoted by P (l) = [p(l)0 , p

(l)1 , p

(l)2 ], in which

p(l)θ is the probability that the lth edge takes on the value of θ,

θ ∈ 0, 1, 2. The collection of P (l) of all edges is denoted byP , P (1), · · · , P (L). In the iterative computation process, anode takes the a priori probabilities P to calculate the extrinsicprobabilities, according to its update rule. For the lth edge,l = 1, · · · , L, the ternary extrinsic probabilities are denotedby Q(l) = [q

(l)0 , q

(l)1 , q

(l)2 ], and the collection of them for all

edges is denoted by Q , Q(1), · · · , Q(L). The update rulecan then be generally written as

Q = f(P ). (6)

Here, we use “P ” and “Q” to distinguish the a priori proba-bilities from the extrinsic probabilities.

Initially, the relay calculates the ternary intrinsic probabil-ities based on the channel observation y′

R:

pCHθ = p(cs = θ|y′R),

=

γ exp(− (y′

R−θ·(2√Es))

2

2σ2

), θ = 0, 2,

2γ exp(− (y′

R−θ·(2√Es))

2

2σ2

), θ = 1,

(7)

where we have omitted the time-index, and γ is a normaliza-tion factor to ensure that pCH

0 +pCH1 +pCH

2 = 1. These intrinsicprobabilities are collected as PCH = [pCH

0 , pCH1 , pCH

2 ], andthey are only available to the accumulator. For the componentdecoders, the initialized a priori probabilities of each edge areP (l) = [1/4, 1/2, 1/4], l = 1, · · · , L [2].

In the process of computing bs, the ternary messages areiteratively exchanged among the component nodes in the ETG,in a similar fashion as the conventional iterative decodingof the single-user IRA code. As the receiver iterates, theternary messages are refined using the update rules (6) ofthe component nodes, which will be detailed next. After anumber of iterations, the computation process converges and adecision is made towards the estimated superimposed messagesequence bs. Then the estimated NC message bN can beobtained according to (5).

C. Update Rules

1) Update Rule with Probabilities: Let us first consider aCN with degree dc = 2. There are L = 3 edges2 connectedto this CN. Recall that the a priori messages available to thefirst and second edge are given by P (1) =

[p(1)0 , p

(1)1 , p

(1)2

]and

P (2) =[p(2)0 , p

(2)1 , p

(2)2

], respectively. The extrinsic message

of the third edge, denoted by Q(3) =[q(3)0 , q

(3)1 , q

(3)2

], can be

obtained as Q(3) = f2CN

(P (1), P (2)

), where the update rule

f2CN(·) for a CN of degree 2 is given by [2]

q(3)0 = γ

(p(1)0 p

(2)0 +

p(1)1 p

(2)1

2+ p

(1)2 p

(2)2

), (8)

q(3)1 = γ

(p(1)1 p

(2)2 + p

(1)2 p

(2)1 + p

(1)1 p

(2)0 + p

(1)0 p

(2)1

), (9)

q(3)2 = γ

(p(1)0 p

(2)2 +

p(1)1 p

(2)1

2+ p

(1)2 p

(2)0

). (10)

Here, γ is a normalization factor to ensure that q(3)0 + q(3)1 +

q(3)2 = 1.

In general, for a CN with a degree dc > 2, the updatefunction fdc

CN (·) can be obtained by successively utilizing thedegree-2 CN update rule, given by

Γ(2) = f2CN(P

(1), P (2)),

...

Γ(l) = f2CN(Γ

(l−1), P (l)),

...

Q(dc+1) = Γ(dc) = f2CN(Γ

(dc−1), P (dc)).

We refer to the above approach as a successive update.2) Update Rule with LLRs: The ternary probabilities ex-

changed in the CPNC decoders put challenges on the analysisand design of the scheme. We next represent the update rulein terms of LLRs, which will be required in our subsequentEXIT chart analysis. For the lth edge of a component node inthe ETG, the LLR couple associated with the a priori (ternary)probabilities are defined as

Λ(l)P , log

(p(l)0 + p

(l)2

p(l)1

)and Ω

(l)P , log

(p(l)0

p(l)2

), (14)

which are sufficient statistics of p0, p1, p2.From (5), we see that values bs(t) = 0 and bs(t) = 2 of the

superimposed message are both mapped to the NC messagebit bN (t) = 0. Therefore, Λ(l)

P is related to the LLR of thebinary NC message bit, and it has a pivotal role in the iterativecomputation process. To distinguish the two LLR values in(14), we refer to Λ

(l)P as the primary LLR and Ω

(l)P as the

2Following the common approach in literature for IRA codes [33], a CNwith degree dc has dc edges connected to the interleaver and one additionaledge connected to the ACC. A VN with degree dv has dv edges connectedto the interleaver.


Λ(3)Q = log

(q(3)0 + q

(3)2

q(3)1

)(a)= log

p(1)0 p

(2)0 +

p(1)1 p

(2)1

2 + p(1)2 p

(2)2 + p

(1)0 p

(2)2 +

p(1)1 p

(2)1

2 + p(1)2 p

(2)0

p(1)1 p

(2)2 + p

(1)2 p

(2)1 + p

(1)1 p

(2)0 + p

(1)0 p

(2)1

,

= log

1 + exp(Λ(1)P

)exp

(Λ(2)P

)exp

(Λ(1)P

)+ exp

(Λ(2)P

) . (11)

[Λ(3)Q ,Ω

(3)Q

]= f2

CN

([Λ(1)P ,Ω

(1)P

],[Λ(2)P ,Ω

(2)P

]),

=

log1 + exp

(Λ(1)P

)exp

(Λ(2)P

)exp

(Λ(1)P

)+ exp

(Λ(2)P

) , log

1 + exp(Ω

(1)P

)exp

(Ω

(2)P

)+KCN

exp(Ω

(2)P

)+ exp

(Ω

(1)P

)+KCN

. (12)

[Λ(l)Q , Ω(l)

Q

]= fdv

VN

([Λ(1)P ,Ω

(1)P

], · · · ,

[Λ(l−1)P ,Ω

(l−1)P

],[Λ(l+1)P ,Ω

(l+1)P

], · · · ,

[Λ(dv+1)P ,Ω

(dv+1)P

]),

=

(dv − 2) log 2 +

dv∑l′=1,l′ =l

Λ(l′)P +KVN,

dv∑l′=1,l′ =l

Ω(l′)P

. (13)

secondary LLR. Similarly, the primary and secondary LLRsassociated with the extrinsic probabilities are defined as

Λ(l)Q , log

(q(l)0 + q

(l)2

q(l)1

)and Ω

(l)Q , log

(q(l)0

q(l)2

). (15)

Consider a CN with dc = 2. The primary extrinsic LLR ofthe third edge is calculated by (11) where

(a)= follows from

(8)-(10). The secondary extrinsic LLR is calculated as

Ω(3)Q = log

(q(3)0

q(3)2

),

= log

1 + exp(Ω

(1)P

)exp

(Ω

(2)P

)+KCN

exp(Ω

(2)P

)+ exp

(Ω

(1)P

)+KCN

, (16)

where

KCN =

[1 + exp

(Ω

(1)P

)] [1 + exp

(Ω

(2)P

)]2 exp

(Λ(1)P

)exp

(Λ(2)P

) . (17)

Now, the update rule in terms of LLRs of a CN of dc = 2is given by (12). The update rule of a CN with dc > 2 canbe calculated using the successive update approach describedpreviously. In the sequel, we will use fdc

CN (·) to denote theupdate rule of a CN of degree dc in LLRs. A property of theupdate rule of a CN is presented next, which will be used laterin the next section.

Property 1: For a CN with degree dc, we have the outputsecondary LLR Ω

(l)Q = 0 as long as there exists an edge l′,

l′ = l, such that the input secondary LLR Ω(l′)P = 0.

Explanation: In (16), if any of Ω(1)P or Ω(2)

P equals to zero,the term Ω

(3)Q will be zero. Consider the successive update

rule, we obtain Property 1.

The derivation of the VN update rule fVN (·) in LLRs issimilar and it is given by (13) where

KVN = log

1 +

dv∏l′=1,l′ =l

exp(Ω

(l′)P

)dv∏

l′=1,l′ =l

(1 + exp

(Ω

(l′)P

)) . (18)

The derivation is given in Appendix A.

IV. CONVERGENCE BEHAVIOR ANALYSIS ANDOPTIMIZATION OF IRA-PNC SCHEMES

It is well-known that in the conventional single-user AWGNchannel, the performance of an iteratively decoded IRA code islargely affected by its VN degree distribution λ(dv), and theCN degree distribution ρ(dc). The optimal performance canbe achieved using the EXIT chart curve-fitting technique [33].Now, we adopt this methodology in designing the IRA-PNCscheme, so as to approach the capacity limit of the binary-inputGaussian TWRC. However, for the two-user CPNC scheme,there lacks a method to characterize the EXIT behaviors w.r.t.the ternary probabilities, that are exchanged in the iterativecomputation process.

In this section, we first propose a method to model the softinformation exchanged among the components of the IRA-PNC scheme. This will enable us to obtain upper and lowerbounds on the approximation of the EXIT functions. Based onthat, we design optimal component codes via curve-fitting.

A. Modeling of EXIT Functions

To carry out convergence behavior analysis, we partition theETG of the IRA-PNC scheme into two parts: an inner compo-nent decoder consisting of the combined CN and ACC (CN-ACC) decoder, and an outer component decoder consisting ofthe VN decoder. The idea of the EXIT chart technique is to


Λ

Fig. 3. Histogram of the one-side extrinsic primary LLR output andHistogram of the extrinsic secondary LLR output from the VN decoder duringiterative computation process.

predict the behavior of the iterative process by solely lookingat the input/output mutual information of the two individualcomponent decoders of the CPNC scheme.

Unlike the decoding of a conventional binary IRA codewhere binary probabilities are exchanged between the com-ponent decoders, the soft information exchanged between theCPNC component decoders have a ternary form. The ternaryprobabilities (or soft information) of the CPNC scheme canalso be written in terms of the primary LLR Λ and thesecondary LLR Ω, as in the previous section. In particular,the primary LLR Λ is related to the NC message bN to becomputed. For simplicity, we omit the time index here. Wedenote bN the random variable w.r.t. the NC message bit anddenote Λ the random variable w.r.t. the primiary LLR. Thus,the mutual information between bN and the input primary LLRΛP , IA = I (bN ; ΛP ), will be used for tracking the a prioriinformation of a component decoder of the CPNC scheme.Similarly, the mutual information between bN and the outputprimary extrinsic LLR ΛQ, IE = I (bN ; ΛQ), will be used fortracking the corresponding extrinsic information. An outputmutual information of IE = 1 means that all NC message bitsbN can be decoded error free.

The relationship of the input-output mutual information, i.e.,the EXIT function, of the inner component decoder (CN-ACCdecoder) with CN degree distribution ρ(dc) can be written as

IE = TInner(IA,P(ΩP ), ρ(dc), Es/σ

2), (19)

where P(ΩP ) denotes the probability density function (PDF)of the secondary LLR ΩP . We remark that, unlike the con-ventional single-user case, the EXIT function of the CPNCscheme is also affected by the PDF of the secondary LLRΩP . Similarly, the EXIT function of the outer componentdecoder (VN decoder) with VN degree distribution λ(dv) canbe written as

IE = TOuter (IA,P(ΩP ), λ(dv)) . (20)

Notice that the EXIT function of the VN decoder is notaffected by the SNR, since it is not directly connected to thechannel observation.

Numerical results show that the PDF of the primary LLRapproaches a consistent Gaussian-like distribution [34] with itsmean equal to half of its variance, with the increasing numberof iterations, as shown in Fig. 3. Thus, similar to [34], we canapproximate the primary a priori LLR as3

ΛP =σ2Λ

2(1− 2bN ) + nΛ, (21)

where nΛ is a Gaussian random variable with variance σ2Λ.

However, the PDF of the secondary LLR, as shown in Fig. 3,is not a Gaussian-like distribution. This makes the analyticaltreatment of the EXIT functions difficult. In order to tackle thisproblem, we propose two models for the secondary a prioriinformation.

Model I: We assume that perfect secondary LLR is availablein this model, that is,

ΩP =

+Ψ if bs = 0,

0 if bs = 1,

−Ψ if bs = 2,

(22)

where Ψ denotes a large positive value, e.g., 30,used in our simulation. Since the actual decodingprocess does not have perfect a priori information onthe secondary LLR ΩP for the component decoders,we have IE = TInner

(IA,P(ΩP ), ρ(dc), Es/σ

2)

≤TInner


2)

for the inner componentdecoder (CN-ACC decoder). Thus, we can obtain an upperbound for the approximation of the EXIT function of theinner component decoder by using Model I. Similarly, wecan obtain an upper bound on the approximation of the EXITfunction of the outer component decoder using Model I.

Model II: We assume the a priori secondary LLR ΩP iscompletely absent, i.e., ΩP = 0.

As the actual decoding retains certain a priori in-formation on the secondary LLR ΩP for the compo-nent decoders, setting ΩP to zero will result in an in-formation loss. Following the data processing inequality[35], we have IE = TInner


2)

≥TInner


2)

for the inner component de-coder and this also applies to the outer component decoder.Thus, a lower bound of the approximation of the EXITfunction can be obtained using Model II.

B. EXIT Charts

We next show the EXIT functions of the component de-coders of the IRA-PNC scheme using the two a prior informa-tion models developed earlier. The EXIT functions of the innerCN-ACC decoder with CN degrees dc = 1, · · · , 5 are shownin Fig. 4. These EXIT functions are obtained via simulationswhere Model I and Model II are used to construct the a prioriinformation. Clearly, the EXIT function obtained with ModelI is always higher than with Model II. This suggests thatthe availability of the secondary LLR ΩP can contribute toa higher output extrinsic information. From Fig. 4, we alsoobserve that the gap between the EXIT functions with Model

3Here we have omitted the time-index for simplicity.


Fig. 4. Comparison between Model I and Model II for inner CN-ACCdecoder with various CN degrees. The code rate is 1/3, and the per-user SNRis Eb/N0 = 2.2 dB.

I and with Model II diminishes as the CN degree dc increases.When dc ≥ 2, the gap is almost unnoticeable. Here we givean intuitive explanation for this behavior. Let us consider thea priori information model I, the probability of ΩP = 0 is 0.5since 50% of message bits have bN = 1. Recall Property 1which states that the output secondary LLR of a CN is zeroas long as one of its input edges has the secondary LLR equalto zero. As the CN degree dc increases, the probability of thisevent (there exists one edge whose secondary LLR is zero)also increases. As a result, there will be more zero-secondaryLLR at the output of the CN nodes. This will restrain thepropagation of the secondary LLR from the CN to the ACCof the inner decoder. As the CN degree becomes very high,the propagation of the secondary LLR becomes minimal andModel I and Model II tends to be identical.

From [33], it is known that to minimize the area betweenthe EXIT functions of the component decoders, a capacityachieving IRA code tends to have a fairly large average CNdegree, i.e., dc > 2. In this circumstance, for inner componentdecoder, the upper bound (Model I) and lower bound (ModelII) of the approximation of the EXIT functions overlaps witheach other. Therefore, either Model I or II can be used toobtain the approximation of the EXIT function of the innercomponent. Similarly, we can also use Model I or Model IIto obtain the upper and lower bounds on the approximationof the EXIT function of the outer component decoder. Weremark that since Model II gives a lower bound on theapproximation of the EXIT function for either the inner or theouter component decoder, an optimal code based on Model IIcan always have its convergence guaranteed when the SNR isabove its designed threshold.

Example 1: We consider an IRA-PNC scheme with per-user code rate of R = 1/3. In particular, the IRA underconsideration has an average CN degree dc = 2.4 and anaverage VN degree dv = 7.2. The code parameters are givenin Table I. In Fig. 5, we plot the EXIT function obtainedby using the a priori information Model II and the actualdecoding trajectory obtained from simulation. We observe that

Fig. 5. Comparison between EXIT Model II and actual decoding trajectoryfor an IRA code at R = 1/3 and Eb/N0 = 4 dB.

using the a priori information Model II, the EXIT functionsof the component decoders of the IRA-PNC scheme can beaccurately characterized. In the sequel, we will focus on usinga priori information Model II for the design of the IRA-PNCscheme.

C. Code Optimization via Curve-fitting of the EXIT functions

Based on the developed EXIT functions of the componentdecoders of the IRA-PNC scheme, we now adopt the EXITchart curve-fitting technique to design optimal IRA codes. Thegoal is to find CN and VN degree distributions such that thegap between the EXIT curves of the inner component decoderand the outer component decoder is minimized. Similar to[33], we first select an appropriate CN degree distribution.Then, we fit the EXIT curve of VN decoder to that of theCN-ACC decoder, by optimizing the degree distribution ofthe VN decoder via linear programming.

We next show two examples of the code design via EXITcurve-fitting for the IRA-PNC scheme. To avoid redundancy,the implementation details of the curve-fitting are omitted andcan be found in [33]. In this paper, we restrict ourselves to thecommonly used concentrated check degree distributions [36,Section 3.17].

Example 2: We consider an IRA-PNC scheme where thecode rate of each user is R = 3/4. In a conventional single-user case, it is known that, for a non-systematic RA code,a non-zero fraction of the CNs should have degree one toensure that its decoder makes progress in the first iteration[33, 37]. From (12), we notice that in the IRA-PNC scheme,the CN degree distribution should contain a non-zero fractionfor degree one CNs, similar to the conventional single-user RAcodes case. In this design example, the choice of the portion ofdc = 1 CNs follows from the convention in [33]. The detailsare provided in Table I. In addition, to carry out EXIT curve-fitting, flexibility of the VN nodes are required so that theaverage VN degree cannot be too small, e.g., dv > 3. Then,for a code rate of 3/4, the average CN degree should be largeenough, e.g., dc > 2. In this setting, the EXIT function of thebi-regular code can be characterized by Model II.


Fig. 6. EXIT charts of the bi-regular coded PNC scheme and the optimizedIRA-PNC scheme, where R = 3/4.

Fig. 6 shows the EXIT chart of a bi-regular RA coded PNCscheme whose degree distributions are given in Table I. Thedecoding threshold for this benchmark scheme is found to beat Eb/N0 = 6 dB. The EXIT chart of our optimized IRA-PNC scheme is also shown in Fig. 6, whose degree distri-butions are given in Table I. The decoding threshold of ouroptimized IRA-PNC scheme is found to be at Eb/N0 = 3.4dB. This shows that our developed IRA-PNC scheme cansignificantly outperform the bi-regular RA coded PNC scheme.The performance improvement is obtained from fitting theEXIT functions, and we refer to this performance improvementas a curve-fitting gain. In this example, the curve-fitting gainis about 2.6 dB.

Example 3: We consider another case where R = 1/3. Fora regular RA coded PNC scheme, the threshold is found tobe at Eb/N0 = 2.2 dB, as shown in Fig. 7. We construct anIRA code for the CPNC scheme, using the curve-fitting tech-nique based on our developed EXIT functions. The decodingthreshold is found to be at Eb/N0 = 2.1 dB. The degreedistributions of our designed IRA code for the CPNC schemeare also given in Table I. We see that for the case of R = 1/3,the CPNC scheme with the designed IRA code only slightlyoutperforms that with a regular code. We next explain whythe performance improvement is slight in this case.

Consider a simplified computation approach in which thesecondary LLR ΩP is always set to zero in the iterativecomputation process. Then, from (14), the soft informationexchanged in the CPNC components can be completely spec-ified by [p0 + p2, p1], which has two elements. Here, werefer to this simplified approach as iterative computation withbinary information exchange. In contrast, we refer to theapproach utilizing both the primary and secondary LLRs asiterative computation with ternary information exchange, sincethe exchanged soft information has three elements, see (14).The performance improvement of using ternary informationexchange (which utilizes the secondary LLR), over that withbinary information exchange (which does not use the sec-ondary LLR), is referred to as the secondary LLR gain. For theCPNC scheme of per-user coding rate R = 1/3, it is shown

Fig. 7. EXIT charts of the regular coded PNC scheme and the optimizedIRA-PNC scheme, where R = 1/3.

that the secondary LLR gain is as much as 0.5 dB when aregular RA code is utilized (see Fig. 8). In the process ofoptimizing the degree distributions of the IRA code to obtainthe curve-fitting gain, the inner component decoder tends tohave a relatively large average CN degree. This results in areduced secondary LLR gain, as discussed in Section IV.B.For a relatively large CN degree, the secondary LLR gainvanishes. Finally, the combined effect of increased curve-fitting gain and reduced secondary LLR gain leads to onlya slight performance improvement.

In contrast, in Example 2 where R = 3/4, the averagedegree of the CN of the CPNC scheme with a regular/biregularRA code is already relatively large, e.g., dc ≥ 2. In this case,the secondary gain is already fairly small, as we can see inFig. 4. Therefore, as we carry out the curve-fitting, there isno loss in the secondary LLR gain and the curve-fitting gainleads to a significant performance improvement.

It is noteworthy that IRA codes are special LDPC codeswith a simpler encoder than general LDPC codes but with sim-ilar performance. The code optimization technique proposedin this paper may apply to general LDPC codes.

V. SIMULATION RESULTS

In the previous section, we have designed IRA-PNCschemes based on EXIT chart analysis and curve-fitting tech-niques. In this section, we present numerical results to showthe benefits of our designed IRA-PNC schemes for finitecode lengths. Specifically, we first compare the bit-error rate(BER) performance of our developed IRA-PNC schemes tothe existing CPNC schemes with regular (or bi-regular) RAcodes. Next, we compare the performance of our developedIRA-PNC scheme to the capacity limits, as well as to thecomplete decoding-based scheme.

In the simulations, we consider the BER performance ofcomputing the NC message bN at the relay. In all simulations,the length of the binary message sequence of each user is set tok = 32768. In the iterative computation process, the maximumnumber of iterations is set to 200.


1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Per−user Eb/N

0 (dB)

BER

Irregular, ternary message exchange

Irregular, binary message exchange

Regular/Bi−regular, ternary message exchange

Regular/Bi−regular, binary message exchange

R = 1/3 R = 3/4

Fig. 8. Simulation results of the designed IRA-PNC scheme where R =3/4, 1/3.

A. IRA-PNC Schemes versus Regular (or Bi-regular) codedPNC Schemes

1) Per-user Code Rate R = 3/4: The BER simulationresults of the CPNC scheme with this code rate are shownin Fig. 8. At a BER of 10−4, our developed IRA codedPNC scheme performs about 2.6 dB better than the bi-regularRA coded PNC scheme. This is in line with our EXITchart analysis. From this result, we can conclude that IRAcodes designed based on our EXIT analysis can significantlyimprove the performance of the CPNC scheme. We also noticethat there is no performance degradation when the iterativecomputation with ternary information exchange is replaced bythat with the binary information exchange. This behavior hasbeen explained in Section IV.C.

2) Per-user Code Rate R = 1/3: The BER simulationresults of the CPNC scheme with this code rate are shown inFig. 8. When the iterative computation with binary informationexchange is utilized, our IRA-PNC scheme is about 0.5 dBbetter than the existing PNC scheme with the regular code in[2]. The performance improvement is from the full realizationof the curve-fitting gain. When ternary information exchangeis utilized, the designed IRA-PNC scheme is about 0.1 dBbetter than that with the regular RA code. These results arealso in line with our EXIT chart analysis.

3) Other Code Rates: Fig. 9 shows the performance of theoptimized IRA-PNC scheme with various code rates, whereternary information exchange is utilized. For code rates of 1/2and 2/3, at a BER of 10−4, we observe that the performanceimprovement over regular RA-PNC schemes are 1.6 dB and1.9 dB, respectively. The code parameters are given in TableI.

B. CPNC Schemes Versus Complete Decoding-Based NetworkCoding Schemes

Now, we compare the performance of the CPNC schemeto the scheme which performs complete decoding to gen-erate the NC message at the relay. For a fair comparison,the optimized IRA-PNC scheme and the optimized complete

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 710−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Per−user Eb/N0 (dB)

BER R = 1/3, Irregular

R = 1/3, Regular

R = 1/2, Irregular

R = 1/2, Regular

R = 2/3, Irregular

R = 2/3, Bi−regular

R = 3/4, Irregular

R = 3/4, Bi−regular

Fig. 9. Performance of optimized IRA-PNC schemes at various code rates,where ternary information exchange decoding is utilized.

decoding-based scheme are considered. In particular, givena total power constraint, equal power allocation is the bestfor a PNC scheme, see [2, 38]. On the other hand, unequalpower allocation is optimal for the complete decoding-basedscheme [41, 42], as this facilitates the complete separation oftwo users’ codewords.

For the scheme with complete decoding, we employ iterativemulti-user detection and decoding (IDD) [41] to fully decodeboth user A and user B’s messages bA and bB , and thendetermine the NC message as bA⊕ bB . In an IDD algorithm,soft information is iteratively exchanged between multiplesingle user decoders and a multi-user detector; details for codeoptimization using IDD algorithm can be found in [42, 43]. Weemphasize that the “CNC1” scheme in [2] is equivalent to thecomplete decoding scheme (considered in this paper) with noiteration between the multi-user detector and the decoder. Theperformance loss of not using the IDD, however, can be upto several dB in power efficiency. For a fair comparison, inthe scheme with complete decoding at the relay, we use theoptimal power allocation between the two users [35, 42] andoptimize its IRA code for degree distributions. The optimizedcode is provided in Table I.

In Fig. 10, we compare the performance of our IRA-PNCscheme and the complete decoding-based scheme, with R =3/4. The optimized power allocation ratio for the completedecoding-based scheme is 3.2 at this code rate. The capacitylimit for the complete decoding-based scheme is found to beat Eb/N0 = 4.3 dB. The limit from the cut-set capacity upperbound4 [7] of a Gaussian TWRC with binary inputs is foundto be at Eb/N0 = 1.67 dB. Note that the capacity limits forboth schemes are for binary inputs with BPSK modulation. AtBER = 10−4, our developed IRA-PNC scheme is about 1.7dB away from the capacity upper bound. At BER = 10−4,the IRA-PNC scheme is about 2 dB better than the optimizedcomplete decoding-based scheme. Note that, for this case, ourdesigned IRA-PNC scheme clearly outperforms the capacity

4The actual capacity limit of a binary-input Gaussian TWRC is still anopen problem. Therefore, we use the upper bound as a reference.


Fig. 10. Comparison between the CPNC scheme and the complete decoding-based scheme where R = 3/4.

limit of the complete decoding-based scheme.In Fig. 11, we compare the performance of the IRA-

PNC scheme and the complete decoding-based scheme whereR = 1/3. The optimized power allocation ratio for thecomplete decoding-based scheme is 1.6 at this code rate. AtBER = 10−4, the optimized complete decoding-based schemewith IDD is about 0.6 dB better than the optimized IRA-PNCscheme. This shows that PNC with compute-and-forward maynot be a good choice when the code rate is low. This is in linewith the information theoretic result [7, 31], which shows thatcomplete decoding-based scheme can outperform the CPNCscheme in terms of their achievable rates, as the SNR orcoding rate becomes small. In Fig. 11, we also include theperformance of the CNC1 scheme discussed in [2], which isequivalent to the complete decoding-based scheme but withoutiteration between the detector and decoders. The CNC1 suffersfrom a loss of about 1 dB relative to the complete decoding-based scheme with IDD. Due to this loss, the CNC1 schemeperforms worse than the IRA-PNC scheme. We emphasizethat when the complete decoding-based scheme is properlydesigned, it outperforms the CPNC scheme at R = 1/3.

VI. DISCUSSION OF GENERALIZATIONS

Here, we briefly discuss how to extend our CPNC designmethod to cases without equal receive power and perfectsynchronization. For the case with unequal receive power forthe two users, the relay observes a quaternary signal, insteadof the ternary one as in the equal power case. In this case, wemay still use the primary LLR to characterize the convergencebehavior of the network coded bits. As opposed to the equalpower case, however, two secondary LLRs have to be analyzedregarding their impact on the convergence behavior of theprimary LLR. The optimization of the code parameters willbe similar. For BPSK modulation, imperfect synchronizationwill have further effects on the two users’ received power,which can be treated with additional secondary LLRs. In thiscase, similar approach to the unequal received power case maybe applied. For QPSK modulation, the approaches developedin [13, 16, 38] may be applied to deal with the asynchronous

Fig. 11. Comparison between CPNC scheme the complete decoding-basedscheme where R = 1/3.

phase problem. For example, in [16], an improved belief prop-agation algorithm, based on over-sampling of the continuoussignal, was introduced to deal with asynchronous carrier phase.This technique may be incorporated in our design of the CPNCscheme in the case with carrier phase asynchrony.

The work in this paper may be extended to higher-ordermodulation schemes, like 4-PAM and 16-QAM. This can bedone by employing non-binary RA coded high-level modula-tions. In this case, the code optimization approach proposed inthis paper needs to be modified for the non-binary codes. Forexample, for the non-binary RA coded high-level modulation,we need to define and track multiple primary LLRs andmultiple secondary LLRs, and evaluate the contribution of thesecondary LLRs to the primary LLRs in the iterative decodingprocess.

In this paper, we only focused on the symmetric case wherethe two users have the same code rate. For an asymmetriccase, e.g., where user A has a higher code rate than user B,we may consider that user B utilizes a subset of user A’scodewords. This will result in a nested code [39, 40]. Then theproposed code optimization technique can be applied to findthe optimum code parameters. These extensions are beyond thescope of this paper and will be considered in future works.

VII. CONCLUSION

In this paper, we developed an IRA coded PNC scheme forbinary input Gaussian TWRCs. We extended the EXIT charttechnique to analyze the convergence behavior of the iterativecomputation process of the IRA-PNC scheme. Based on that,we optimized the degree distributions of the componentsof the IRA-PNC scheme. Our optimized IRA-PNC schemesignificantly outperforms existing regular (or bi-regular) RAcoded PNC schemes. We also showed that a CPNC schemehas the most significant benefit when the code rate is high.Interestingly, in a high coding rate regime, the performanceimprovement of using our EXIT curve-fitting to design an IRAcoded PNC scheme is most significant. We also noted that,for a very low SNR or code rate, CPNC scheme is worse thanthe complete decoding based scheme with iterative multi-user


Λ(1)Q = log

(q(1)0 + q

(1)2

q(1)1

)= log

(4(dv−2) ·

∏dv

l=2 p(l)0 + 4(dv−2) ·

∏dv

l=2 p(l)2

2(dv−2) ·∏dv

l=2 p(l)1

)

= (dv − 2) log 2 +

dv∑l=2

Λ(l′)P +KVN. (23)

detection and decoding. This agrees with existing informationtheoretic analysis results.

ACKNOWLEDGMENT

This work has been supported by the Australian ResearchCouncil under the ARC Discovery Grant DP110104995 andDP0986089. The authors also like to thank the editor and theanonymous reviewers for their thorough review and valuablecomments.

APPENDIX ADERIVATION OF VN UPDATE RULE

Consider a VN with degree dv . Without loss of generality,we consider the update of the extrinsic information on the firstedge, based on the a priori information from the edges withindex 2 to dv .

Borrowing the result Eq. (13) of [2] and extending it todv − 1 edges, we have

q(1)0 = γ · 4(dv−2) ·

dv∏l=2

p(l)0 ,

q(1)1 = γ · 2(dv−2) ·

dv∏l=2

p(l)1 ,

q(1)2 = γ · 4(dv−2) ·

dv∏l=2

p(l)2 ,

where γ is for normalization purpose. The primary LLR Λ(1)Q

is calculated by (23) where

KVN = log

1 +

dv∏l=2

exp(Ω

(l′)P

)dv∏l=2

(1 + exp

(Ω

(l′)P

)) . (24)

The derivation of the secondary LLR Ω(1)Q can be carried

similarly.

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TABLE ICODE PARAMETERS

Scheme Code Type CN VNdc dv

Decoding Thresholddc ρ dv λ Eb/N0

CPNC

Regular R = 1/3 1 1 3 1 1 3 2.2 dBRegular R = 1/2 1 1 2 1 1 2 4 dB

Bi-regular R = 2/31 0.65 4 1

2.67 4 4.8 dB4 0.14447 0.2056

Bi-regular R = 3/41 0.2288 4 1

3 4 6 dB3 0.54245 0.2288

Irregular R = 1/3

1 0.30 2 0.1542

2.4 7.2 2.1 dB3 0.70 3 0.3353

7 0.13758 0.223721 0.1493

Irregular R = 1/2

1 0.30 3 0.3612

3.1 6.2 2.4 dB4 0.70 4 0.4282

16 0.177817 0.0328

Irregular R = 2/3

1 0.20 2 0.2243

3.4 5.1 2.9 dB4 0.80 3 0.4322

6 0.18237 0.107328 0.0539

Irregular R = 3/4

1 0.20 2 0.3221

4.2 5.6 3.4 dB5 0.80 3 0.3297

6 0.22727 0.047831 0.0732

Complete decoding-based scheme

Irregular R = 1/3

1 0.20 3 0.4963

2.6 7.8 1.5 dB

3 0.80 4 0.11449 0.082910 0.200429 0.087030 0.0190

Irregular R = 3/4

1 0.10 2 0.2672

2.8 3.73 5.6 dB3 0.90 3 0.5915

7 0.04938 0.061019 0.0310

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Tao Huang (S’09) received his B.Eng. degreein Electronic and Information Engineering fromHuazhong University of Science and Technology,Wuhan, China, in 2003. He received his M.Eng.(Advanced) degree in Sensor System Signal Process-ing from the University of Adelaide, Adelaide, Aus-tralia, in 2007. He is currently working toward thePh.D. degree in the School of Electrical Engineeringand Telecommunications, University of New SouthWales, Sydney, Australia, where he is a recipientof Australian Postgraduate Award, and Engineering

Research Award. His research interests include error control channel coding,iterative error correction, physical-layer network coding, and cooperativecommunications. He co-authored a Best Academic Paper Award of IEEEWireless Communications and Networking Conference (WCNC), Cancun,Mexico in 2011. He was a research engineer at the Southwest CommunicationResearch Institute of Chinese Electronics Technology Group Corporation(30th CETC), Chengdu, China, in 2003-2005. From 2007 to 2009, he wasa senior communication software engineer at the EMS SATCOM Pacific,Adelaide, Australia.

Tao Yang (S’07, M’10) received B.Sc. degree inelectronic engineering in 2003 from Beijing Uni-versity of Aeronautics and Astronautics (BeihangUniversity), Beijing, China. He received Master byresearch and Ph.D. degrees in electrical engineeringfrom the University of New South Wales, Sydney,Australia, in 2006 and 2010, respectively. He iscurrently a Research Fellow in the Wireless andNetworking Technologies Laboratory (WNTL) atCommonwealth Scientific and Industrial ResearchOrganization (CSIRO), Sydney, Australia. His re-

search expertise and interests include multi-user and MIMO communications,error-control coding, iterative signal processing and decoding, physical-layernetwork coding and network information theory. He was the recipient ofAustralian Postgraduate Award (APA), NICTA research project award (NRPA)and Supplementary Engineering Award (SEA) from the University of NewSouth Wales.

Jinhong Yuan received the B.E. and Ph.D. degreesin electronics engineering from Beijing Instituteof Technology, Beijing, China, in 1991 and 1997,respectively. From 1997 to 1999 he was a ResearchFellow at the School of Electrical Engineering, theUniversity of Sydney, Sydney, Australia. In 2000he joined the School of Electrical Engineering andTelecommunications, the University of New SouthWales, Sydney, Australia, where he is currently aProfessor for Telecommunications of the school.He has published two books, two book chapters,

over 200 papers in telecommunications journals and conference proceedingsand 40 industrial reports. He is a co-inventor of one patent on MIMOsystems and two patents on low-density-parity-check (LDPC) codes. He co-authored three Best Paper Awards and one Best Poster Award, includinga Best Paper Award of IEEE Wireless Communications and NetworkingConference (WCNC), Cancun, Mexico in 2011, and a Best Paper Awardof IEEE International Symposium on Wireless Communications Systems(ISWCS), Trondheim, Norway in 2007. His publication is available fromhttp://www2.ee.unsw.edu.au/wcl/JYuan.html. He serves as the IEEE NSWChair of joint Communications/Signal Processions/Ocean Engineering Chap-ter. His current research interests include error control coding and informationtheory, communication theory, and wireless communications.

Ingmar Land is Senior Research Fellow at theInstitute for Telecommunications Research (ITR),University of South Australia. Before joining ITR in2007, he was Assistant Professor for CommunicationTheory at Aalborg University, Denmark. He receivedhis Dr-Ing in 2004 from the University of Kiel,Germany, and he studied for his Dipl.-Ing. at theUniversity of Ulm, Germany, and at the University ofErlangen-Nurnberg, Germany. His research interestsare coding and information theory with applicationto cooperative communications, multiuser commu-

nication, distributive source coding and physical-layer security.

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