A Physical-Layer Rateless Code for Wireless Channels

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A Physical-layer Rateless Code for WirelessChannels

Shuang Tian,Member, IEEE, Yonghui Li, Senior Member, IEEE, Mahyar Shirvanimoghaddam,Student Member,IEEE, and Branka Vucetic,Fellow, IEEE

Abstract—In this paper, we propose a physical-layer ratelesscode for wireless channels. A novel rateless encoding schemeis developed to overcome the high error floor problem causedby the low-density generator matrix (LDGM)-like encodingscheme in conventional rateless codes. This is achieved byproviding each symbol with approximately equal protection inthe encoding process. An extrinsic information transfer (EXIT)chart based optimization approach is proposed to obtain arobust check node degree distribution, which can achieve near-capacity performances for a wide range of signal to noise ratios(SNR). Simulation results show that, under the same channelconditions and transmission overheads, the bit-error-rate (BER)performance of the proposed scheme considerably outperformsthe existing rateless codes in additive white Gaussian noise(AWGN) channels, particularly at low BER regions.

Index Terms—rateless codes, low-density parity-check (LDPC)codes, physical-layer channel coding

I. I NTRODUCTION

RATELESS codes [1] were initially developed to achieveefficient transmission in erasure channels. The Luby

transform (LT) code [2] is broadly viewed as the first practicalrealization of rateless codes. The LT encoded symbols aregenerated in accordance with a specific degree distributionandthere is potentially an unlimited number of encoded symbols.These symbols are generated on the fly and broadcasted tothe receiver until an acknowledgment (ACK) response ofsuccessful decoding is received at the transmitter.

Ideally, a rateless code should allow the receiver to recoverthe message symbols with high probability by receiving as fewencoded symbols as possible. Aiming to reduce the necessarynumber of encoded symbols, Shokrollahi proposed Raptorcodes in [3]. In Raptor codes, a precoding (e.g., based onlow-density parity-check (LDPC) codes) is applied to themessage symbols prior to the LT code. In [3], Raptor codesare constructed in systematic and non-systematic manners fordifferent erasure channel scenarios.

The initial work on rateless codes has mainly been limited toerasure channels with the primary application in multimediavideo streaming. Recently, the design of rateless codes forwireless channels, such as binary symmetric channels (BSC)[4], additive white Gaussian noise (AWGN) channels [5],[6], as well as fading channels [7], has attracted significant

The authors are with the Telecommunications Laboratory, Schoolof Electrical and Information Engineering, The Universityof Syd-ney, Australia (Email: shuang.tian, yonghui.li, mahyar.shirvanimoghaddam,branka.vucetic@sydney.edu.au).

This work was supported by the Australian Research Council (ARC) GrantsDP120100190, FT120100487, LP0991663, DP0877090.

attention. Rateless codes have a wide spectrum of applicationsin various modern wireless communication networks, suchas to enhance the transmission efficiency in IEEEad-hoc802.11b wireless networks [8] and cooperative relay networks[9], [10], and to control the peak-to-average power ratio inOFDM systems [11]. Rateless codes are particularly beneficialto wireless transmissions because, in contrast to traditionalfixed-rate coding schemes, the transmitter potentially does notneed to know the channel state information before sending itsencoded symbols, and the receiver can retain a resilient de-coding performance. This property is of particular importancein the design of codes for time varying wireless channels.

In general, there are two categories of physical-layer ratelesscodes for wireless channels: systematic [8], [12] and non-systematic [6], [13], [14], [15]. For systematic rateless codes,the systematic message symbols are first transmitted to thereceiver, followed by a number of encoded symbols. For non-systematic rateless codes, only encoded symbols are broad-casted. The receiver uses the classic belief propagation (BP)algorithm for decoding [16]. At the transmitter, the encoderof the existing rateless codes uses coding schemes that areakin to the low-density generator matrix (LDGM) codes [17].The LDGM-like rateless codes are of linear complexity forencoding and their encoded symbols can be simply calculatedon the fly.

A major drawback of the existing physical-layer ratelesscodes is the notably high error floor. This error floor ispresent mainly because, from the BP decoding perspective, theencoder creates an encoded symbol by selecting systematicmessage symbols only. As a result, the encoded symbolis connected to only one check node and cannot receiveupdated information as the decoding progresses. Therefore,an erroneous encoded symbol keeps propagating the errormessage through its corresponding check node to the sys-tematic message symbols, which are connected to that checknode. To improve the error floor performance, a considerableamount of research has been conducted in the past severalyears, which can be broadly divided into two directions. Onedirection applies serial concatenated coding schemes [13],[14], [18], where the LDGM-like rateless code is used asa component code in the concatenation structure. The otherdirection modifies the LDGM-like rateless code as a stand-alone code without concatenation [6], [15]. As reported in[6], the error performance of concatenation codes is largelydetermined by the stand-alone rateless code itself, and theconcatenation will inevitably increase the system complexity.Therefore, in this paper, we will focus on the design of the

2

stand-alone rateless code. In [15] and [19], it shows thatthe error floor in LDGM-like codes is closely related to thesystematic message symbols connected to a relatively smallnumber of check nodes. The error floor can be reduced to acertain level by reducing the number of message symbols ofa lower-degree. For this purpose, [15] proposed to connect allmessage symbols to an approximately equal number of checknodes. However, the method in [15] suffers from a high errorfloor problem especially at low signal to noise ratio (SNR)regime. This is because all encoded symbols still have degree-one, and thus cannot receive updated information from othercheck nodes in the BP decoder.

Another major drawback of the existing physical-layer rate-less codes is that different wireless channel conditions needdifferent check node degree distributions to achieve the near-capacity performance [6]. This is primarily because, for theLDGM-like rateless codes, the degree of systematic messagesymbols continues to increase as the transmission goes onto generate and transmit more encoded symbols. From theextrinsic information transfer (EXIT) chart perspective [20],increasing the degree of systematic message symbols willchange the EXIT curve of variable nodes. To achieve a lowbit-error-rate (BER) for a near-capacity transmission overhead,an open but minimum spacing EXIT chart tunnel is required.As a result, the inverted EXIT curve of check nodes needsto follow the change, which in turn causes the check nodedegree distribution to change. Obviously, to utilize multipledistributions and adjust them to fit different channel conditionswould complicate the transmission mechanism. A robust checknode degree distribution that can perform well over a widerange of SNRs would dramatically simplify the system. Someinitial studies [14], [21] have attempted to design a robustdegree distribution, but the progress has been limited so far.

In this paper, we first develop a rateless encoding approachto resolve the high error floor problem caused by the con-ventional LDGM-like rateless encoding schemes in wirelesschannels. For the proposed approach, the encoder creates anencoded symbol by selecting not only the systematic messagesymbols, but also the encoded symbols created prior to thepresent encoded symbol. By doing so, each encoded symbolwill be connected to multiple check nodes in the Tanner graph[22]. Moreover, to provide all the symbols with approximatelyequal protection in the BP decoder, we require each variablenode to connect to an approximately equal number of checknodes. To achieve this, we let the variable nodes with alower degree be selected with a higher priority when theencoder selects variable nodes for each check equation toform an encoded symbol. By using the proposed encodingapproach, the encoded symbol can exchange information withother symbols through multiple check nodes as the decodingprogresses. As a result, the error floor is lowered.

To provide satisfactory performance over a wide range ofSNRs, we develop an EXIT-chart-based optimization approachand achieve a near optimalrobust check node degree distri-bution for the proposed rateless code. This robust distributionis able to create a constant EXIT chart tunnel for differentSNRs. In general, for the conventional rateless codes undera particular channel condition, the shape of the EXIT curve

of variable nodes is determined by the variable node degreedistribution. This distribution depends on two factors: one isthe check node degree distribution, and the other is the numberof symbols transmitted [6]. To achieve a satisfactory decodingperformance, the lower the SNR is, the more symbols needto be transmitted. Therefore, when SNR changes, the EXITcurve inevitably changes. In contrast, for the proposed ratelesscode, we can keep the EXIT curve unchanged by dividing thevariable nodes involved in the BP decoder into two sets andmanipulating the sizes of the two sets. One set corresponds tothe received symbol sequence from the wireless channel. Thisreceived sequence consists of all the encoded symbols andthe message symbols that are actually transmitted. The otherset of variable nodes corresponds to the remaining messagesymbols that are yet to be transmitted. We adjust the numberof message symbols in the two sets in order to achieve a stableEXIT curve of variable nodes for a desired range of SNRs.Consequently, we can achieve a well-behaved and constantEXIT chart tunnel leading to superior performances. Simula-tion results show that, under the same channel condition andtransmission overhead, the BER performance of the proposedcode considerably outperforms the existing rateless codesinadditive white Gaussian noise (AWGN) channels, particularlyat low BER regions.

The rest of the paper is organized as follows. In Section II,the system model is introduced. In Section III, the conventionalrateless codes are discussed. In Section IV, the construction ofthe proposed rateless code is described first, followed by adiscussion of optimizing the check node degree distributionbased on the EXIT chart analysis. Simulation results arepresented in Section V. Some concluding remarks are madein Section VI.

II. SYSTEM MODEL

In this paper, we consider a point-to-point physical-layertransmission in AWGN channels. The system model is shownin Fig. 1. Let b = b1, ..., bj, ..., bK denote the binarymessage symbols, generated by the source, wherebj is thej-th symbol andK is the length of message. The messagesymbols are first encoded by a rateless channel encoder. Letvt represent thet-th binary rateless coded symbol, which isthen modulated and transmitted. We consider the binary phaseshift keying (BPSK) modulation. Letxt ∈ +1,−1 be themodulated signal ofvt. The received signal,yt, at the receiveris given by

yt =√

PTxt + nt (1)

wherePT is the transmission power, which is assumed to beof unity power, i.e.PT = 1, andnt is the real-valued additivenoise in AWGN channels with zero-mean and variance ofσ2

n.After receiving the rateless coded symbols, the receiver appliesthe BP decoding algorithm [16] to recover the source message.For the rateless code, the encoder generates an unlimitednumber of symbols and keeps transmitting these symbols untilit receives an ACK response from the receiver.

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Fig. 1. The considered digital communication system model

Fig. 2. The Tanner graph representing an instance of a systematic LDGMcode [17]. ‘©’ represents the systematic message symbols, ‘’ the encodedsymbols and ‘’ the check nodes.

III. C ONVENTIONAL PHYSICAL-LAYER RATELESSCODES

In the encoding phase, the encoder first draws a valueof d from a distribution called the check node degree dis-tribution. The encoder then randomly selectsd systematicmessage symbols, and performs binary summation (i.e. XOR)of these selected symbols to generate an encoded symbol.The relationship of systematic message symbols and encodedsymbols can be represented by a bipartite Tanner graph [22].Fig. 2 shows the Tanner graph of the conventional LDGM-likerateless codes. In the Tanner graph, the systematic messagesymbols are denoted by ‘©’ and the encoded symbols by ‘’.The vertices ‘’ are check nodes representing the parity-checkequation constraint.

At the receiver, to evaluate the BER performance of therateless codes in AWGN channels, Etesami and Shokrollahiintroduced a formal definition of the reception overheadδ ofa decoder in [6]. LetM denote the number of binary symbolscollected at the receiver to perform satisfactory decodinginthe BP decoder. We expressM as

M =K

Cσn

(1 + δ) (2)

where Cσnis the achievable code rate for binary-input

continuous-output AWGN channels with the noise level pa-rameterσn. This noise levelσn is often called theShannonlimit. From (2), we can see that the overheadδ represents thenumber of collected binary symbols away from the optimalnumber defined byK/Cσn

.For equiprobablext ∈ +1,−1, Cσn

is given by [23]

Cσn= −

∫ ∞

−∞

φσn(y) log2 φσn

(y)dy − 1

2log2 2πeσ

2n (3)

where

φσn(y) =

1√

8πσ2n

(

exp

(

− (y + 1)2

2σ2n

)

+ exp

(

− (y − 1)2

2σ2n

))

.(4)

0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Code Rate

Sha

nnon

Lim

it

Fig. 3. Shannon Limitσn vs. Code RateR

Since there is no closed form expression, the integral in (3)isevaluated by numerical integration techniques [24]. TableI andFig. 3 show the Shannon limits as a function of some specificcode ratesR. These values will be used later for the designof the proposed rateless code. The last column in Table I isthe Shannon limit expressed as the ratio (dB) of signal energyper symbol to noise power spectral density, and is defined as

Eb

N0= 10 log10

(

1

2Rσ2n

)

. (5)

IV. PROPOSEDPHYSICAL-LAYER RATELESSCODES

SCHEME

In this section, the construction of the proposed ratelesscode is described first. Aball-into-bin method is developed tohave the encoded symbol connected to multiple check nodes,and a full rank parity-check matrix tailored to the transmis-sion characteristics of wireless channels is constructed.TheEXIT-chart-based optimization program is then employed tooptimize the robust check node degree distribution.

A. Construction of the Proposed Coding Scheme

To overcome the high error floor problem in the LDGM-like encoding scheme, we propose a rateless coding schemewhere the encoder generates an encoded symbol by using notonly the systematic message symbols, but also the encodedsymbols created before the present one. Each encoded symbolwill be connected to multiple check nodes (check equations).Through these check nodes, each variable node and checknode can exchange information with other nodes as the BPdecoding progresses. In this way, the error floor can be loweredconsiderably.

Let us look at a simple example in Fig. 4, which illustratesthe proposed encoding representation by using a Tanner graph.In this example, the number of systematic message symbolsK = 6. The set of vertices on the left is called variable nodes,representing the systematic message symbols ‘©’ and theencoded symbols ‘’. The vertices ‘’ on the right are checknodes. Since each encoded symbol is generated by a check

4

R σnEbN0

(dB)0.501 0.977 0.19340.912 0.5 3.41040.999 0.2859 7.864

TABLE ISHANNON LIMITS FOR VARIOUS CODE RATESR.

Fig. 4. The Tanner graph representing an instance of the proposed ratelesscode. ‘©’ represents the systematic message symbols, ‘’ the encodedsymbols and ‘’ the check nodes.

Fig. 5. The parity-check matrix representing the instance of the proposedrateless code in Fig. 4

equation, each variable node corresponding to an encodedsymbol is connected to its corresponding check node (checkequation) in the graph.

An ideal rateless code commonly requires a check node tobe randomly connected to variable nodes. For a check node

of degreei in the LT-based encoding schemes, the checknode selectsi− 1 systematic message symbols in a uniformlyrandom manner to produce an encoded symbol as shown inFig. 2. This implies that the degree of the message symbols, orin other words, the number of check nodes a message symbolconnects to, can be very broadly distributed. Some of themessage symbols may not even have a connection to any checknode unless a sufficiently large number of check nodes areprovided. Consequently, even for very high SNRs, there is stilla great chance that the decoding error would occur due to theunconnected message symbols. Thus, from the transmissionefficiency point of view, a completely random selection wouldnot be suitable for high SNR regions in AWGN channels.To boost the transmission efficiency at high SNRs, we applya full rank constraint to the parity-check matrixH of theproposed rateless code. The structure ofH of the code inFig. 4 is shown in Fig. 5. Each row ofH corresponds toa parity-check equation and each column ofH correspondsto a symbol in the rateless code. We decomposeH into twoparts. The first part ofH corresponds to the encoded symbolsand consists of the (K + 1)-th to the rightmost columns ofH. This part is a sparse lower triangular matrix. The secondpart ofH corresponds to theK message symbols and consistsof the first K columns ofH. In the second part, a sparseupper triangular matrix is on top of a regular sparse matrix.The purpose of producing the upper triangular matrix is toguarantee a full rank matrix for the firstK rows of H. Thisfull rank design ensures that when SNR is sufficiently high,even though none of the message symbols are transmitted, allmessage symbols can be successfully decoded as soon as thedecoder receives the firstK encoded symbols. When SNRis lowered, more symbols have to be transmitted to achievesatisfactory decoding performance. In this case, we requireall the variable nodes to connect to an approximately equalnumber of check nodes, which ensures that all transmittedsymbols receive almost the same protection from the BPdecoder. In the following, aball-into-bin encoding scheme isdeveloped to construct a rateless code as discussed above.

Ball-into-bin encoding procedure: to better describe theencoding process, let us first look at an example withK = 6.Suppose there areK + L ‘variable balls’,v1, . . . , vK+L, cor-responding toK+L variable nodes. As shown in Fig. 6, thesevariable balls are classified into two groups. The first group,which we call thesystematic ball group, includes balls,v1, . . . , vK , representing theK systematic binary messagesymbols. The second group, which we call theencoded ballgroup, contains balls,vK+1, . . . , vK+L, representing theLencoded symbols, and the values for these encoded symbolsare to be determined in the encoding process. LetΩi denote

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Fig. 6. An instance of theball-into-bin encoding process forl = 0. ‘©’represents the variable balls and ‘’ the check nodes. (K = 6)

Fig. 7. An instance of theball-into-bin encoding process forl = 1. ‘©’represents the variable balls and ‘’ the check nodes. (K = 6)

the check node degree distribution, wherei = 1, 2, . . . , imax.Ωi represents the probability that a check node has degreei,and imax is the maximum check node degree. Similarly, wedenote bydmax the maximum variable node degree. There aredmax + 1 degree bins labeled as ‘Bin of Degree0’, ‘Bin ofDegree1’, ‘Bin of Degree 2’, . . . , ‘Bin of Degreedmax’. Aball vj sitting in ‘Bin of Degreed’ means thatvj is of degreedand has connections tod check nodes. We also introduce a binlabeled as ‘Buffer Bin’ to produce the upper triangular matrixin H. Only systematic balls can be put into ‘Buffer Bin’ by theencoder. If a ball is put into ‘Buffer Bin’, it will not be usedto generate encoded symbols unless drawn out of ‘Buffer Bin’and put into the degree bins. The detailed application of the‘Buffer Bin’ will be described later in this section. In addition,since there areL encoded symbols, we haveL check nodes,c1, . . . , cL in Fig. 6, correspondingly.

Denoting byl the index of the check nodes, the encodingprocedure progresses asl increases from one toL, and isdivided into two phases.Phase I, 1 ≤ l ≤ K, will generate thefirst K encoded symbols, which will form the upper triangularmatrix of H; Phase II, K + 1 ≤ l ≤ L, will generate the restL−K encoded symbols. Before the encoding process begins(l = 0, Fig. 6), all systematic balls sit in ‘Bin of Degree0’.

The 1-st encoding step (l = 1): as shown in Fig. 7,firstly,the encoder randomly produces a degree numberi accordingto the check node degree distributionΩi and assigns it to thecheck nodec1. In this example,i = 4. Sincec1 is alreadyconnected tov7, the encoder then picksi − 1 more variableballs out of ‘Bin of Degree0’ in a uniformly random manner(e.g.v1, v3 andv6) and calculates the value ofv7 by combiningthe values of thesei− 1 balls using modulo-2 addition. From

Fig. 8. An instance of theball-into-bin encoding process forl = 2. ‘©’represents the variable balls and ‘’ the check nodes. (K = 6)

the perspective ofH, the first, third, sixth and seventh columnsof the first row have entries of1, and the other columns of thesame row have entries of0. Secondly, the encoder moves thevariable balls involved in the calculation (i.e.v1, v3, v6 andv7) to ‘Bin of Degree1’, since they have had one connectionto the check nodes.Thirdly, the encoder randomly choosesone systematic ballvj′ (e.g. v3) from those systematic onespicked in the present encoding step, and moves it into ‘BufferBin’. At last, the ‘Buffer Bin’ will record the degree bin fromwhich the systematic ball comes.

The 2-nd encoding step (l = 2): as shown in Fig. 8, theencoder randomly produces a degree numberi for c2 (e.g.i = 5). In order to have each variable node connected to anapproximately equal number of check nodes, we require theencoder to select as many variable balls as possible in the binlabeled with the lowest degree. If there are not enough ballsinthis bin, the encoder will take all of them and randomly choosethe rest from the bin labeled with the second lowest degree.It means that balls with a lower degree have a higher priorityto be chosen by the check nodes than those with a greaterdegree; balls sitting in the same degree bins, i.e. of the samedegree, have the equal chance to be chosen. In this example,the degree number forc2 is i = 5, so the encoder firstly picksall the three variable ballsv2, v4 andv5 in ‘Bin of Degree0’and randomly chooses a variable ballv7 from ‘Bin of Degree1’ to calculate the value of ballv8. Next, the encoder movesballs v2, v4, v5 and v8 to ‘Bin of Degree1’, and ball v7 to‘Bin of Degree2’. Finally, the encoder randomly chooses thesystematic ballv5 and moves it into ‘Buffer Bin’.

In the remaining ofPhase I, a similar encoding procedure isperformed for3 ≤ l ≤ K. In Phase I, we require the encoderto pick at least one systematic ball in each encoding step. Eachtime l increases by one, the encoder chooses one systematicball, which it picks to form the encoded symbol in thel-thencoding step, and moves it into ‘Buffer Bin’. Systematic ballssitting in ‘Buffer Bin’ will not be picked again by the encoderto generate any other encoded symbols untilPhase Iends. Itmeans that, if a systematic ballvj′ is put into ‘Buffer Bin’ inthe l-th encoding step, the (l, j′)-th entry is ‘1’ and there willbe no more entries of ‘1’ from the (l+ 1)-th to theK-th rowfor the j′-th column. By doing so, as the encoding progressesfrom the1-st to theK-th step, we can obtain a matrix whichis equivalent to the upper triangular sub-matrix ofH in Fig. 5.

The (K + 1)-th encoding step (l = K + 1): as shown in

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Fig. 9. An instance of theball-into-bin encoding process forl = 7. ‘©’represents the variable balls and ‘’ the check nodes. (K = 6)

Fig. 9, Phase II of the encoding process starts. At this point,the ‘Buffer Bin’ has collected all theK systematic balls. Theencoder first puts all these systematic balls back into thosedegree bins from which they were moved to the ‘Buffer Bin’during Phase I. Next, the encoder performs the same processas inPhase Iexcept that the ‘Buffer Bin’ is not used, becausethe encoder does not move balls to ‘Buffer Bin’ inPhase II.The encoding process goes on asl increases until it reachesL.

By following the steps above, we obtain a rateless code thatcan be represented by a parity-check matrixH as shown inFig. 5. Next, we discuss the optimization of the robust checknode degree distribution for a desired range of SNRs.

B. Optimization of the Robust Check Node Degree Distribu-tion

The check node degree distribution is usually expressed inthe polynomial form [6] and is given by

Ωi(x) =

imax∑

i=2

Ωixi (6)

So we haveΩi(1) = 1. The check node degree distribution canbe defined with respect to the nodes themselves asΩi(x) orwith respect to the edges asωi(x). Let ωi, i = 1, 2, ..., imax,represent the check node degree distribution with respect to theedges, i.e. the probability that a given edge is connected toacheck node of degreei. The degree polynomial with respectto the edges is defined by

ωi(x) =

imax∑

i=2

ωixi−1 (7)

The formula for converting between the node-view and edge-view distributions is given as

ωi =Ωi · i

∑

j Ω · j =Ωi · iβ

(8)

whereβ is the average degree of check nodes.Degree distributions are similarly defined for the vari-

able nodes. We denoteΛd(x) =∑

d Λdxd and λd(x) =

∑

d λdxd−1 the polynomials for the variable nodes with re-

spect to the nodes and the edges, respectively. Assuming there

areL encoded symbols, and denoting byα the average degreeof variable nodes, we have

α(L+K) = βL (9)

Recall that, for the proposed rateless encoding scheme witha sufficiently largeL, the majority of variable nodes havea degree approaching the average degreeα. From (9), thevariable node distribution can be approximately given by adistribution form for variable nodes in regular LDPC codes as

Λd(x) ≈ xα = xβL

K+L , λd(x) ≈ xβL

K+L−1 (10)

Alternatively, in practice, to achieve a more accurate variablenode degree distribution, we can use computer simulationsby first generating the variable node degree distribution manytimes using a fixed check node degree distribution, and thencalculating the empirical mean.

The EXIT chart [20], [25] is a popular curve fitting toolwidely used to optimize degree distributions. This methodtracks the extrinsic mutual information between the transmit-ted BPSK symbols and the value of extrinsic log-likelihoodratios (L-value) in the BP decoder. Letq denote theq-thiteration of the BP decoder. At theq-th iteration, an extrinsicL-value on an edge outgoing from a variable nodev to a checknodec is given by

LLR(q)vc = LLRch +

∑

c′ 6=c

LLR(q−1)c′v (11)

whereLLRch is the L-value of the received symbols that aredetermined from the channel measurement.LLR

(q−1)cv is the

incoming extrinsic L-value, at the(q − 1)-th iteration, to thevariable nodev from a check nodec, which is in connectionto this variable node, and is given by

LLR(q−1)cv = ln

1 +∏

v′ 6=v tanh(LLR

(q−1)

v′c

2 )

1−∏

v′ 6=v tanh(LLR

(q−1)

v′c

2 )

(12)

For the initial phase whereq = 0 and from (1), we have

LLR(0)vc = LLRch = ln

p(y|x = +1)

p(y|x = −1)=

2

σ2n

yt, (13)

where p(y|x) is the conditional probability density function(pdf) of the signaly at the output of the AWGN channel giventhe input signalx ∈ ±1. Thus, we have the variance ofLLRch as

σ2ch =

4

σ2n

(14)

The EXIT chart analysis is carried out under two mainassumptions. Firstly, the code Tanner graph can be representedby a tree graph [6], [26], [27], so that all involved randomvariables are independent. The second assumption is that theextrinsic L-values passed between nodes have a symmetricGaussian distribution for meanσ2

LLR/2 and varianceσ2LLR

[20], [26]. Let function J(σLLR) denote the mutual infor-mation I(xt;LLR) between the transmitted BPSK symbols

7

and the extrinsic L-values with varianceσ2LLR, which can be

expressed as

J (σLLR) = H(xt)−H(xt|LLR)

= 1−∫ ∞

−∞

e−(µ−σ2LLR/2)2/2σ2

LLR

√

2πσ2LLR

· log2(1 + e−µ)dµ(15)

whereH(x) is the entropy function. LetIA be the average mu-tual information between the BPSK symbols and the L-valuesincoming to variable nodes, the average mutual informationoutgoing from variable nodesIV −out is then given by

IV −out =

dmax∑

d=1

λd · J(

√

(d− 1)[J−1(IA)]2 + σ2ch

)

(16)

On the side of check nodes, assume thatIE is the averagemutual information between the BPSK symbols and the out-going L-values calculated by check nodes, the average mutualinformation incoming to check nodesIC−in is given by

IC−in = 1−imax∑

i=1

ωi · J(

J−1(1 − IE)√i− 1

)

(17)

The derivation of (16) and (17) is detailed in [20]. In the EXITchart analysis, the sets of variable nodes and check nodes arerespectively referred to as the variable node decoder (VND)and check node decoder (CND) [20]. These two decoders formthe BP decoder. The curveIV −out vs. IA, (0 ≤ IA ≤ 1),is referred to as EXIT curve of the VND, and the curveIC−in vs. IE , (0 ≤ IE ≤ 1), is referred to as inverted EXITcurve of the CND. Three requirements need to be satisfied inorder to achieve a near optimal degree design. Firstly, boththe VND and inverted CND curves should be monotonicallyincreasing functions and reach the(1, 1) point on the EXITchart. Secondly, the VND curve should be always above theinverted CND curve. And thirdly, the VND curve has to matchthe shape of the inverted CND curve as accurately as possi-ble. Consequently, the VND and inverted CND curves buildan EXIT chart tunnel area. The EXIT chart analysis viewsiterative decoding as an evolution of the mutual informationin the EXIT tunnel, starting from point(0, IV −out(0)). Whenthe mutual information equals one, there are no bit errors.

As the transmission progresses, i.e. the code rate changes,for a fixed degree distribution of check nodes, the invertedCND curve will stay unchanged. It is desirable if we canobtain a suitable VND curve that is unchanged as well, sothat a stable EXIT chart tunnel can be obtained. From (16),it can be observed that the shape of the VND curve dependson two elements. The first element is the variable node degreedistributionλd, which is in turn determined by the check nodedegree distribution and the number of encoded symbols. Thesecond element for shaping the VND curve is the channelparameterσ2

ch. This parameterσ2ch determines the starting

point of the VND curves,(0, IV −out(0)), which is also thepoint that the evolution of the mutual information starts fromin the BP decoder.

We start the design of the robust degree distribution bytrying to maintain the quantity ofIV −out(0) unchanged asthe transmission goes on. Let us first consider a scenario

where the SNR is sufficiently high, so that the receiver cansuccessfully decode all the systematic message symbols assoon as it receivesK encoded symbols. The reason for sendingencoded symbols at first, rather than sending message symbolsthemselves, is that the encoded symbols can provide variablenodes with protection by using parity check equations in theBP decoder. In this case, the BP decoder has2K variablenodes. From Fig. 4, we can observe that these variable nodescan be divided into two sets. One set consists of the(K +1)-th to 2K-th variable nodes representingK encoded symbols.For this set, the variable nodes are initiated by L-values ofthereceived symbols from the AWGN channel. From (16) and letIA = 0, the quantity of the initial average mutual informationfor these variable nodes is given by

INZV −out = J (σch) (18)

As shown in [20],J (σch) is the capacity of the channel withσch, which in our case is the achievable code rate for theAWGN channel with BPSK modulation. For sufficiently highSNRs,J (σch) is close to one. The other variable node set con-sists of theK message symbols. Since the message symbolshave not been transmitted yet, the initial mutual informationfor these variable nodes is given byIZV −out = 0. Let K ′ andρ0 denote the number and the percentage, respectively, of thevariable nodes that have zero initial mutual information intheBP decoder. We have

ρ0 =K ′

K + L(19)

whereL is the number of encoded symbols. For the consideredscenario,K ′ = L = K, so ρ0 = 0.5. The quantity ofIV −out(0) is then given by

IV −out(0) = (1− ρ0) · INZV −out + ρ0 · IZV −out ≈ 0.5 (20)

When SNR is lower, more symbols are necessary to betransmitted to guarantee a reliable decoding. Let us consider ascenario where SNR is about0dB. From Table I, we know thatthe achievable code rateJ (σch) ≈ 0.5. From (20), we observethat it requires a near-to-zeroρ0 to maintainIV −out(0) ≈ 0.5.A near-to-zeroρ0 means that, either the BP decoder receivesno message symbols but an infinitely large number of encodedsymbols, i.e.K ′ = K andL → ∞ in (19), or the sender startsto supply message symbols at some point of the transmission,i.e. to decreaseK ′. The latter approach can effectively reduceρ0 to zero. In this paper, we refer to it as thereverse systematictransmission.

Let us represent this proposedreversemechanism by par-titioning the variable nodes into four sub-codes, namely Sub-codeA, B, C and D as shown in the example in Fig. 4.Sub-codeC contains all the systematic message symbolsrepresented by the firstK variable nodes. Sub-codeA is thefirst K encoded symbols. The(K +1)-th to ⌊K · (1+∆)⌋-thencoded symbols constitute Sub-codeB, where function⌊x⌋represents the nearest integer tox, and∆ is a positive valueto be determined in the optimization program. Sub-codeDconsists of encoded symbols from the(⌊K · (1 + ∆)⌋ + 1)-th encoded one to a potentially unlimited number due to therateless property. The encoded symbols in Sub-codesA andB

8

are firstly transmitted in sequence, followed by the messagesymbols in Sub-codeC, and finally more encoded symbolsin Sub-codeD are sent. The sender can transmit symbol-by-symbol. Alternatively, it can transmit block-by-block andeachblock consists of a fixed number of symbols. The transmissionwill be terminated when the sender receives an ACK responsefrom the receiver. Next, we address the problem of optimizingthe value of∆, together with the design of a robust check nodedegree distribution.

Similar to (20), the entire curve ofIV −out vs. IA, (0 ≤IA ≤ 1), can be given by

IV −out =

(1− ρ0) ·dmax∑

d=1

λd · J(

√

(d− 1)[J−1(IA)]2 + σ2ch

)

+ ρ0 ·dmax∑

d=1

λd · J(√

d− 1J−1(IA))

(21)

It shows that a tunableρ0 can adjust the shape of the VNDcurves. Thus, it is possible for us to keep the VND curveunchanged by adjustingρ0 while the transmission goes on. Asshown in Fig. 4, for the reverse systematic transmission, themessage symbols are transmitted after⌊K · (1+∆)⌋ encodedsymbols. When all the message symbols are received at thedecoder, we haveK ′ = 0, so ρ0 would always be zero and,consequently, the decoder will lose the ability to adjust theVND curves. In order to have a tunableρ0, ∆ is expected tobe large enough so that the decoding can be successful beforeall the message symbols are sent. In this case, for a singleSNR value ofξdB, the rate of the proposed rateless code canbe calculated as

RξdB =K

(K +K +∆ξdB ·K)(1− ρ0,ξdB)

=αξdB

(1 + ∆ξdB)(1− ρ0,ξdB)

imax∑

i=2

ωi

i

(22)

It can be observed that the actual code rate depends on thedegree distributions, as well as on∆ξdB and ρ0,ξdB. Basedon this observation, an optimization program is developed toobtain a robust check node degree distribution.

Let the SNR range for the robust distribution target on[SNRlow, SNRhigh]. We chooseτ SNR points within thisrange, whereSNRlow ≤ SNR1 < SNR2 < ... <SNRτ−1 < SNRτ ≤ SNRhigh. Let CSNRζ

denote the ca-pacity of code rate corresponding toSNRζ, where1 ≤ ζ ≤ τ .Further, denoteχSNRζ

as

χSNRζ=

CSNRζ

RSNRζ

(23)

whereRSNRζis given by (22). We wantχSNRζ

to be asclose to one as possible, which meansRSNRζ

is close tothe capacity. For the given∆SNRζ

, ρ0,SNRζandαSNRζ

, therobust degree distribution ofωi is to minimize the maximumelement of

χSNR1 , χSNR2 , ..., χSNRτ−1 , χSNRτ (24)

subject to four constraints

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

IA & I

E

I V−

out &

I C−

in

IV−out

vs. IA for 0.19dB

IV−out

vs. IA for 3.41dB

IC−in

vs. IE

Fig. 10. The EXIT chart forIV −out vs. IA (curve of VND) andIC−in vs. IE (inverted curve of CND)

1)∑imax

i=2 ωi = 1,2) ωi ≥ 0,3) |IV −out,SNRζ

− IV −out,SNRζ′| < ε, where ε → 0,

4) IV −out,SNRζ> IC−in for IA = IE .

This optimization program can be solved by using a lin-ear program with computer search. Condition (3) is usedto enforce each pair of the VND curves being as closeto each other as possible, so that a stable EXIT charttunnel is obtained. Condition (4) is used to enforce thecondition that the EXIT chart tunnel for everySNRζ

point stays open. Therefore, an arbitrarily low BER can beachieved with the near-capacity transmission overhead. Inaddition, to reduce the transmission complexity, it is idealto have one global∆. We propose to select that value as∆ = max∆SNR1 ,∆SNR2 , ...∆SNRτ−1 ,∆SNRτ

, coveringall ∆SNRζ

. Generally,∆ is determined by the lowest SNRpoint, because the lower the SNR is, the more transmittedsymbols are required. Finally, in practice, we found that itismore efficient to perform the degree optimization program bydealing with the lower SNR points first.

An example of the EXIT chart resulting from the computersearch for two SNR points0.19dB and 3.41dB is shownin Fig. 10. These two points were chosen to correspond toa lower code rateR(σn=0.977) = 0.501 and a higher coderate R(σn=0.5) = 0.912, respectively, as shown in Table I.In this computer search, we found the values∆ = 0.3,ρ0, 0.19dB = 0, ρ0, 3.41dB = 0.45, and the robust check nodedegree distribution as

Ω(x) = 0.475x3 + 0.525x6 (25)

In Fig. 10, we can observe that the two respective VND curvesobtained by using the robust check node degree distributionin (25) coincide with each other closely. It indicates that thesystem complexity can be reduced under different channelconditions.

9

0.1 0.2 0.3 0.4 0.5

10−5

10−4

10−3

10−2

10−1

Overhead

BE

R

Proposed, K=2000Proposed, K=3000Proposed, K=5000Non−Syst. [15], K=5000 Syst. [12], K=5000

Fig. 11. BER vs. Overhead forσn = 0.977

V. SIMULATION RESULTS

In this section, we will present the BER performance asa function of the transmission overhead for various ratelesscoding schemes. BPSK modulation is used in the simulationand the transmission signal power is set to unity. We firstconsider two scenarios in an AWGN channel, where the noisestandard deviations areσn = 0.977 and0.5, which correspondto 0.19 dB and 3.41 dB, respectively, as shown in Table I.Given that the current practical code-length requirement is afew thousands [28], [29] and with regards to the necessity ofsparseness in LDPC-like codes [30], the number of messagesymbolsK used in simulations is set to 2000, 3000 and5000. The comparison is given between the proposed scheme,the non-systematic rateless code developed in [15], and thesystematic rateless code in [12].

Fig. 11 shows the BER performance with respect to trans-mission overheadδ at the low SNR region forσn = 0.977.From (2) and Table I, the number of received symbols is givenby

M =K

0.501(1 + δ) (26)

We can observe that, for the proposed rateless code at a fixedBER level, the overhead decreases rapidly asK increases.For K = 5000, we can see that the schemes in [15] and [12]have an error floor at BER of10−3 and 10−5 respectively,while the proposed scheme achieves significant performanceimprovement and the error floor has not appeared at BER of10−5. Moreover, for BER of10−5, the proposed scheme saves

M ′ =5000

0.501× 0.18 ≈ 1796 (27)

redundant symbols on average, i.e. almost36% of K, com-pared to the existing rateless coding schemes. The improvedperformance brought by the proposed rateless coding schemeis the result of the elimination of the degree-one encodedsymbols and the application of the specially designed full rank

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

10−5

10−4

10−3

10−2

10−1

Overhead

BE

R

Proposed, K=2000Proposed, K=3000Proposed, K=5000Non−Syst. [15], K=5000Syst. [12], K=5000

Fig. 12. BER vs. Overhead forσn = 0.5

parity-check matrixH. We also compared the performance ofthese schemes forK = 2000 and3000, and obtained similarresults. To simplify the illustration and reduce the numberofcurves, we only show the comparisons forK = 5000 in thefigure.

Fig. 12 compares the BER performance at a higher SNRregion for σn = 0.5. Similarly, for the proposed code ata fixed BER level, the overhead decreases asK increases.It can also be observed that the systematic code gives theworst performance for BER lower than10−2. Although boththe proposed and the non-systematic coding schemes haveerror floors at BER of about10−5, the proposed scheme stilloutperforms the non-systematic one and saves about

M ′ =5000

0.912× 0.05 ≈ 274 (28)

redundant symbols on average, i.e. about5.5% of K.In Fig. 13, the BER performance forσn = 0.2859 is plotted.

We can observe that the proposed scheme has the best perfor-mance for BER better than10−4. The same robust check nodedegree distribution used in Fig. 11 and Fig. 12 is also usedhere. ForK = 2000, BER of 10−5 is reached with overheadless than0.1. The results are fairly satisfactory, although thecase ofσn = 0.2859 was not taken into consideration whenwe conducted the computer search in the optimization programin Section IV B.

In addition, we provide the experimental results for dif-ferent check node degree distributions to justify the use ofEXIT chart analysis in the proposed encoding method. Wefirst choose three given degree distributions, and then obtaintheir channel threshold parameters by using the EXIT chartapproach. Thirdly, we perform simulations for the decodingperformance. Results show that the degree distribution of agreater threshold does give a better decoding performance.This is consistent with the EXIT chart analysis.

We consider a code rateR = 0.8, and the three degreedistributions and their channel threshold valuesσth obtained

10

Index Degree DistributionΩ(x) Thresholdσth

DD1 0.475x3 + 0.525x6 0.53DD2 0.1x2 + 0.4x3 + 0.5x6 0.525DD3 0.6x3 + 0.4x6 0.44

TABLE IITHREE DEGREE DISTRIBUTIONS AND THEIR CORRESPONDING THRESHOLD VALUES .

0 0.05 0.1 0.15 0.2

10−6

10−5

10−4

10−3

10−2

10−1

100

Overhead

BE

R

Proposed, K=2000Proposed, K=3000Proposed, K=5000Non−Syst. [15], K=5000Syst. [12], K=5000

Fig. 13. BER vs. Overhead forσn = 0.2859

0.40.450.50.55

10−5

10−4

10−3

10−2

10−1

Noise Level σn

BE

R

DD1 K=5000DD1 K=3000DD2 K=5000DD2 K=3000DD3 K=5000DD3 K=3000DD1 σ

th=0.53

DD2 σth

=0.525

DD3 σth

=0.44

Fig. 14. BER vs. Noise Levelσn

by EXIT chart analysis are given in Table II. Next, we performsimulations for the three degree distributions, which are shownas DD1, DD2 and DD3 in Fig. 14, with code-length of 6250and 3750 symbols (i.e. the number of message symbolsK is5000 and 3000 respectively). The threshold valueσth of eachdegree distribution is also shown in Fig. 14.

From EXIT chart analysis, we know that the threshold

valuesσth of DD1 and DD2 are 0.53 and 0.525 respectively.Simulation results show that DD1 and DD2 have waterfallregions that closely match the threshold prediction by theEXIT chart analysis. As the code-length goes very long, wecan presume an even closer match in the threshold betweenthe simulations and the EXIT chart analysis. For DD3, it doesnot have clear waterfall and error floor regions, because itsthreshold is too far away from the Shannon limit for code rateR = 0.8, so DD3 is far from a good distribution choice for thiscode rate. Overall, we come to the conclusion from simulationresults that the degree distribution of a greater thresholdhasa better decoding performance, so the EXIT chart analysisis justified in the degree distribution design for the proposedrateless encoding method.

VI. CONCLUSIONS

In this paper, we propose a physical-layer rateless codingscheme for wireless channels. Aball-into-binmethod is firstlyintroduced to construct a particular parity-check matrixH.The aim of this method is to let both the systematic messagesymbols and the encoded symbols have an approximatelyequal degree, so that they can receive equal protection inthe BP decoder. Moreover, in order to obtain a robust degreedistribution of check nodes for a range of SNRs, a degreeoptimization program based on the EXIT chart analysis isproposed. The proposed rateless code with the robust degreedistribution achieves near-capacity performances with respectto the transmission overhead. Simulation results show thatthetransmission redundancy is reduced as the message lengthincreases. Moreover, under the same channel condition andtransmission overhead, the BER performance of the proposedcode outperforms the existing rateless codes in AWGN chan-nels, particularly at low BER regions. For example, for BERof 10−5 andEb/N0 of 0.19 dB, the proposed scheme savesmore than one-third of the message length compared to theexisting rateless schemes.

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Shuang Tian (M’13) received the B. Sc. degree(with highest honors) from Harbin Institute of Tech-nology, China, in 2005, the M. Eng. Sc. (Research)degree from Monash University, Australia, in 2008,and is currently working toward the Ph.D. degreeat The University of Sydney, Australia, all in elec-trical engineering. From 2008 to 2009, he workedas a system and standard engineer with HuaweiTechnologies Co. Ltd., China. His research interestsinclude coding techniques, cooperative communi-cations, 3GPP/WiMAX network protocols, OFDM

communications, and optimization of resource allocation and interferencemanagement in heterogeneous networks. He is a recipient of University ofSydney Postgraduate Awards.

Yonghui Li (M’04-SM’09) received his Ph.D. de-gree in November 2002 from Beijing University ofAeronautics and Astronautics. From 1999 - 2003,he was affiliated with Linkair Communication Inc,where he held a position of project manager withresponsibility for the design of physical layer solu-tions for the LAS-CDMA system. Since 2003, he hasbeen with the Centre of Excellence in Telecommu-nications, The University of Sydney, Australia. Heis now an Associate Professor in School of Electri-cal and Information Engineering, The University of

Sydney. He was the Australian Queen Elizabeth II Fellow and is currently theAustralian Future Fellow.

His current research interests are in the area of wireless communications,with a particular focus on MIMO, cooperative communications, codingtechniques and wireless sensor networks. He holds a number of patentsgranted and pending in these fields. He is an executive editorfor EuropeanTransactions on Telecommunications (ETT). He has also beeninvolved inthe technical committee of several international conferences, such as ICC,Globecom, etc.

Mahyar Shirvanimoghaddam received the B. Sc.degree with the 1’st Class Honour from TehranUniversity, Iran, in 2008, and the M. Sc. Degreewith the 1’st Class Honour from Sharif Univer-sity of Technology, Iran, in 2010, and is currentlyworking toward the Ph.D. degree at The Universityof Sydney, Australia, all in Electrical Engineering.His research interests include coding techniques,cooperative communications, compressive sensing,and wireless sensor networks. He is a recipientof University of Sydney International Scholarship

(USydIS) and University of Sydney Postgraduate Awards.

Branka Vucetic (M’83-SM’00-F’03) received theB.S.E.E., M.S.E.E., and Ph.D. degrees in 1972,1978, and 1982, respectively, in electrical engineer-ing, from The University of Belgrade, Belgrade,Yugoslavia. During her career, she has held vari-ous research and academic positions in Yugoslavia,Australia, and the UK. Since 1986, she has beenwith the Sydney University School of Electrical andInformation Engineering in Sydney, Australia. Sheis currently the Director of the Centre of Excellencein Telecommunications at Sydney University. Her

research interests include wireless communications, digital communicationtheory, coding, and multi-user detection.

In the past decade, she has been working on a number of industry sponsoredprojects in wireless communications and mobile Internet. She has taught awide range of undergraduate, postgraduate, and continuingeducation coursesworldwide. Prof. Vucetic has co-authored four books and more than twohundred papers in telecommunications journals and conference proceedings.