 Research Article
 Open Access
 Published:
Analysis and Construction of FullDiversity Joint NetworkLDPC Codes for Cooperative Communications
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 805216 (2010)
Abstract
Transmit diversity is necessary in harsh environments to reduce the required transmit power for achieving a given error performance at a certain transmission rate. In networks, cooperative communication is a wellknown technique to yield transmit diversity and network coding can increase the spectral efficiency. These two techniques can be combined to achieve a double diversity order for a maximum coding rate on the MultipleAccess Relay Channel (MARC), where two sources share a common relay in their transmission to the destination. However, codes have to be carefully designed to obtain the intrinsic diversity offered by the MARC. This paper presents the principles to design a family of fulldiversity LDPC codes with maximum rate. Simulation of the word error rate performance of the new proposed family of LDPC codes for the MARC confirms the full diversity.
1. Introduction
Multipath propagation (smallscale fading) is an important salient effect of wireless channels, causing possible destructive adding of signals at the receiver. When the fading varies very slowly, errorcorrecting codes cannot combat the detrimental effect of the fading on a pointtopoint channel. Space diversity, that is, transmitting information over independent paths in space, is a means to mitigate the effects of slowly varying fading. Cooperative communication [1–4] is a wellknown technique to yield transmit diversity. The most elementary example of a cooperative network is the relay channel, consisting of a source, a relay, and a destination [3, 5]. The task of the relay is specified by the strategy or protocol. In the case of coded cooperation [4], the relay decodes the message received from the source, and then transmits to the destination additional parity bits related to the message; this results in a higher information theoretic spectral efficiency than simply repeating the message received from the source [6]. The resulting outage probability [7] exhibits twice the diversity, as compared to pointtopoint transmission. However, the overall errorcorrecting code should be carefully designed in order to guarantee full diversity [8].
We focus on capacity achieving codes, more precisely, lowdensity paritycheck (LDPC) codes [9], because their word error rate (WER) performance is quasiindependent of the block length [10] when the block length is becoming very large.
Considering two users, and , and a common destination , a double diversity order can be obtained by cooperating. When no common relay is used, the maximum achievable coding rate that allows to achieve full diversity is (according to the blockwise Singleton bound [7, 11]). However, when one common relay for two users is used (a Multiple Access Relay Channel—MARC), it can be proven that the maximum achievable coding rate yielding full diversity is [12]. The increase of the maximum coding rate yielding full diversity from to is achieved through network coding [13] at the physical layer, that is, sends a transformation of its incoming bit packets to (only linear transformations over GF(5) are considered here). From a decoding point of view, this linear transformation can be interpreted as additional parity bits of a linear block code. Hence, the destination will decode a joint networkchannel code. Therefore, the problem formulation is how to design a fulldiversity joint networkchannel code construction for a rate .
Up till now, no family of fulldiversity LDPC codes with for coded cooperation on the MARC has been published. Chebli, Hausl, and Dupraz obtained interesting results on joint networkchannel coding for the MARC with turbo codes [14] and LDPC codes [15, 16], but these authors do not elaborate on a structure to guarantee full diversity at maximum rate, which is the most important criterion for a good performance on fading channels. A fulldiversity code structure describes a family of LDPC codes or an ensemble of LDPC codes, permitting to generate many specific instances of LDPC codes.
In this paper, we present a strategy to produce excellent LDPC codes for the MARC. First, we outline the physical layer network coding framework. Then, we derive the conditions on the MARC model and the coding rate necessary to achieve a double diversity order. In the second part of the paper, we elaborate on the code construction. A joint networkchannel code construction is derived that guarantees full diversity, irrespective of the parameters of the LDPC code (the degree distributions). Finally, the coding gain can be improved by selecting the appropriate degree distributions of the LDPC code [17] or using the doping technique [18] as shown in Section 7.2. Simulation results for finite and infinite length (through density evolution) are provided. To the best of authors' knowledge, this is the first time that a joint fulldiversity networkchannel LDPC code construction for maximum rate is proposed.
ChannelState Information is assumed to be available only at the decoder. In order to simplify the analysis, we consider orthogonal halfduplex devices that transmit in separate timeslots.
2. System Model and Notation
2.1. Multiple Access Relay Channel
We consider a Multiple Access Relay Channel (MARC) with two users and , a common relay , and a common destination . Each of the three transmitting devices transmits in a different timeslot: in timeslot 1, in timeslot 2, and in timeslot 3 (Figure 1). In this paper, we limit the scheme to two sources, but any extension to a larger number of sources is possible by applying the principles explained in the paper. We consider a joint networkchannel code over this network, that is, an overall codeword is received at the destination during timeslot 1, timeslot 2, and timeslot 3, which form together one coding block. The codeword is partitioned into three parts: , where , , and , and where and transmit bits (note that each user is given an equal slot length because of fairness), and transmits bits, so that . We define the level of cooperation, , as the ratio . Because the users do not communicate between each other, the bits , transmitted by , and the bits , transmitted by , are independent.
Since the focus in this paper is on coding, BPSK signaling is used for simplicity, so that the transmitters send symbols , where stands for the timeslot number and is the symbol time index in timeslot . The channel is memoryless with real additive white Gaussian noise and multiplicative real fading. The fading coefficients are only known at the decoder side where the received signal vector at the destination is
where , , and are the received complex signal vectors in timeslots , , and , respectively. The noise vector consists of independent noise samples which are real Gaussian distributed, that is, , where is the average signaltonoise ratio . The Rayleigh distributed fading coefficients , and are independent and identically distributed. (The average signaltonoise ratios on the , ; and  channels are the same.) The channel model is illustrated in Figure 2. In some parts of the paper, a block binary erasure channel (block BEC) [19, 20] will be assumed, which is a special case of block fading. In a block BEC, the fading gains belong to the set , where means the link is a complete erasure, while means the link is perfect.
We assume that no errors occur on the  and  channels. This simplifies the analysis and does not change the criteria for the code to attain fulldiversity, as will be shown in Section 3.2.
2.2. LDPC Coding
We focus on binary LDPC codes with block length and dimension , and coding rate . (We consider two sources each with information bits and an overall errorcorrecting code with codebits.) The code is defined by a paritycheck matrix , or equivalently, by the corresponding Tanner graph [7, 9]. Regular LDPC codes have a paritycheck matrix with ones in each column and ones in each row. For irregular LDPC codes, these numbers are replaced by the socalled degree distributions [9]. These distributions are the standard polynomials and [21]:
where (resp., ) is the fraction of all edges in the Tanner graph, connected to a bit node (resp., check node) of degree . Therefore, and are sometimes referred to as left and right degree distributions from an edge perspective. In Section 6, the polynomials and , which are the left and right distributions from a node perspective, will also be adopted:
where (resp., ) is the fraction of all bit nodes (resp., check nodes) in the Tanner graph of degree , hence and likewise with .
The goal of this research is to design a fulldiversity ensemble of LDPC codes for the MARC. An ensemble of LDPC codes is the set of all LDPC codes that satisfy the left degree distribution and right degree distribution .
In this paper, not all bit nodes and check nodes in the Tanner graph will be treated equally. To elucidate the different classes of bit nodes and check nodes, a compact representation of the Tanner graph, adopted from [22] and also known as protograph representation [9, 23, 24] (and the references therein), will be used. In this compact Tanner graph, bit nodes and check nodes of the same class are merged into one node.
2.3. Physical Layer Network Coding
The coded bits transmitted by are a linear transformation of the information bits from and , denoted as and , where both vectors are of length . (In some papers, the coded bits transmitted by are a linear transformation of the transmitted bits from and , which boils down to the same as the information bits, since the transmitted bits (parity bits and information bits) are a linear transformation of the information bits.) Let stand for a matrix multiplication in GF(5);
The matrix represents the network code, which has to be designed. Let us split into two matrices and such that , where is an matrix and is an matrix. Now we have the following relation:
Equation (5) can be inserted into the paritycheck matrix defining the overall errorcorrecting code. Instead of designing , we can design and using principles from coding theory.
3. Diversity and Outage Probability of MARC
3.1. Achievable Diversity Order
The formal definition of diversity order on a block fading channel is well known [25].
Definition 1.
The diversity order attained by a code is defined as
where is the word error rate after decoding.
However, in this document, as far as the diversity order is concerned, we mostly use a block BEC. It has been proved that a coding scheme is of full diversity on the block fading channel if and only if it is of full diversity on a block BEC [22]. The channel model is the same as for block fading, except that the fading gains belong to the set . Suppose that on the , , and  links, the probability of a complete erasure, that is, , is .
Definition 2.
A code achieves a diversity order on a block BEC if and only if [26]
where is the word error rate after decoding and means proportional to.
Therefore, it is sufficient to show that two erased channels cause an error event to prove that , because the probability of this event is proportional to . Consider, for example, that the  channel has been erased, as well as the  channel. Then, the information from can never reach , because does not communicate with . Therefore, the diversity order .
A diversity order of two is achieved if the destination is capable of retrieving the information bits from and , when exactly one of the , , or  channels is erased. The maximum coding rate allowing the destination to do so will be derived in Section 3.4.
3.2. Perfect SourceRelay Channels
Here, we will show that the achieved diversity at does not depend on the quality of the sourcerelay () channel. Therefore, in the remainder of the paper, we will assume errorless  channels to simplify the analysis.
Let us consider a simple block fading relay channel with one source , one relay , and one destination . All considered pointtopoint channels (, , ) have an intrinsic diversity order of one. In a cooperative protocol, where has to decode the transmission from in the first slot, two cases can be distinguished: () is able to decode the transmission from and cooperates with in the second slot, hence receives two messages carrying information from ; () is not able to decode the transmission from and therefore does not transmit in the second slot, hence receives only one message carrying information from , namely, on the  channel. Now, the decoding error probability, that is, the WER , at can be written as follows:
The probability is equal to the probability of erroneous decoding at . For large , we have and [25], where is a constant. The probability is equal to the probability of erroneous decoding on the  channel; hence for large , . Now, the error probability at large is proportional to
where is a positive constant. According to Definition 1, fulldiversity requires that at large , . We see that this only depends on the behavior of at large , because the second case where the relay cannot decode the transmission from the source in the first slot does automatically give rise to a double diversity order without the need for any code structure. This means that as far as the diversity order is concerned, it is sufficient to assume errorless  channels (yielding ). Furthermore, techniques [8] are known to extend the proposed code construction to nonperfect sourcerelay channels, so that, for the clarity of the presentation, perfect sourcerelay channels are assumed in the remainder of the paper.
3.3. Outage Probability of the MARC
We denote an outage event of the MARC by . An outage event is the event that the destination cannot retrieve the information from or , that is, the transmitted rate is larger than or equal to the instantaneous mutual information. The transmitted rate is the average spectral efficiency of user whereas is the overall spectral efficiency, so that . (The average spectral efficiency denotes the average number of bits per overall channel uses, including the channel uses of the other devices, that is, transmitted over the MARC channel.) We can interprete as the total spectral efficiency, that is, transmitted over the network. The MARC block fading channel has a Shannon capacity, that is, essentially zero since the fading gains make the mutual information a random variable which does not allow to achieve an arbitrarily small word error probability under a certain spectral efficiency. This word error probability is called information outage probability in the limit of large block length, denoted by
The outage probability is a lower bound on the average word error rate of coded systems [27].
The mutual information from user 1 to the destination is the weighted sum of the mutual informations from the channels from  and . (The transmission of corresponds to redundancy for and at the same time. From the point of view of , the transmission of contains interference from . By using the observations from , the decoder at the destination can at most cancel the interference from in the transmission from .) Hence the spectral efficiency is upper bounded as
where and are the fractions of the time during which and are active [25, Section ]. The same holds for user 2:
Combining (11) and (12) yields
However, there is a tighter bound for . Indeed, (11) and (12) both rely on the fact that the destination can cancel the interference from the other user on the relaytodestination channel, but therefore, the destination must be able to decode one of the users' information from their respective transmission. Hence, there exist two scenarios: () in the first scenario, decodes the information of from the transmission of (), so that it can cancel the interference from in the transmission from ((11) holds); () the second scenario is the symmetric case ( and (12) holds). Both scenarios lead to a tighter bound for :
Bound (14) can be verified when considering the instantaneous mutual information between the sources and the sinks in the network. We denote the instantaneous mutual information of the MARC as , which is a function of the set of fading gains and average SNR . The overall mutual information is
because the three timeslots behave as parallel Gaussian channels whose mutual informations add together. Of course, the timeslots timeshare a timeinterval, which gives a weight to each mutual information term [25, Section ]. The total transmitted rate must be smaller than , which yields (14).
From the above analysis, we can now write the expression of an outage event:
The three terms , , and are each the average mutual information of a pointtopoint channel with input , received signal with , conditioned on the channel realization , which is determined by the following wellknown formula [28]:
where is the mathematical expectation over given and . Therefore, three terms , , and are
Now, the outage probability can be easily determined through MonteCarlo simulations to average over the fading gains and to average over the noise. (Averaging over the noise can be done more efficiently using GaussHermite quadrature rules [29].)
3.4. Maximum Achievable Coding Rate for Full Diversity
In Section 3.1, we established that the maximum achievable diversity order is two. Here, we will derive an upper bound on the coding rate yielding full diversity, valid for all discrete constellations (assume a discrete constellation with M bits per symbol).
It has been proved that a coding scheme is of full diversity on the block fading channel if and only if it is of full diversity on a block BEC [22]. So let us assume a block BEC, hence , . The strategy to derive the maximum achievable coding rate is as follows: erase one of the three channels (see Figure 3), and derive the maximum spectral efficiency that allows successful decoding at the destination. (Another approach from a coding point of view has been made in [30].)
The criteria for successful decoding at the destination are given in the previous subsection see (11), (12), and (14). Because one of the three channels has been erased (see Figure 3), one of the mutual informations is zero. The channels that are not erased have a maximum mutual information (discrete signaling). A user's spectral efficiency allows successful decoding if and only if
It can be easily seen that (20) is a looser bound than (19) (), so that finally
which is maximized if , such that . The destination decodes all the information bits on one graph that represents an overall code with coding rate . Hence the maximum achievable overall coding rate is . It is clear that to maximize , the spectral efficiencies and should be equal, that is, all users in the network transmit at the same rate. In this case, (21) and (19) are equivalent and it is sufficient to bound the sumrate only. In our design, we will take , so that the maximum achievable coding rate can be achieved.
4. FullDiversity Coding for Channels with Multiple Fading States
In the first part of the paper, we established the channel model, the physical layer network coding framework, the maximum achievable diversity order, and the maximum achievable coding rate yielding full diversity. In a nutshell, if the relay transmits a linear transformation of the information bits from both sources during of the time, a double diversity order can be achieved with one overall errorcorrecting code with a maximum coding rate . Now, in the second part of the paper, this overall LDPC code construction that achieves full diversity for maximum rate will be designed. First, in this section, rootchecks will be introduced, a basic tool to achieve diversity on fading channels under iterative decoding [22]. Then, in the following section, application of these rootchecks to the MARC will define the network code, that is, and , such that a doublediversity order is achieved. Finally, these claims will be verified by means of simulations for finite length and infinite length codes.
4.1. Diversity Rule
In order to perform close to the outage probability, an errorcorrecting code must fulfil two criteria:

(1)
fulldiversity, that is, the slope of the WER is the same as the slope of the outage probability at ;

(2)
coding gain, that is, minimizing the gap between the outage probability and the WER performance at high SNR.
The criteria are given in order of importance. The first criterion is independent of the degree distributions of the code [22], hence serves to construct the skeleton of the code. It guarantees that the gap between the outage probability and the WER performance is not increasing at high SNR. The second criterion can be achieved selecting the appropriate degree distributions or applying the doping techniques (see Section 7.2). In this paper, the most attention goes to the first criterion.
In the belief propagation (BP) algorithm, probabilistic messages (loglikelihood ratios) are propagating on the Tanner graph. The behavior of the messages for determines whether the diversity order can be achieved [17]. However, the BP algorithm is numerical and messages propagating on the graph are analytically intractable. Fortunately, there is another much simpler approach to prove full diversity. Diversity is defined at . In this region the fading can be modeled by a block BEC, an extremal case of blockRayleigh fading. Full diversity on the block BEC is a necessary and sufficient condition for full diversity on the blockRayleigh fading channel [22]. The analysis on a block BEC channel is a very simple (bits are erased or perfectly known) but very powerful means to check the diversity order of a system.
Proposition 1.
One obtains a diversity order on the MARC, provided that all information bits can be recovered, when any single timeslot is erased.
This rule will be used in the remainder of the paper to derive the skeleton of the code.
4.2. Rootcheck
Applying Proposition 1 to the MARC leads to three possibilities (Figure 3).
Case 1.
The  channel is erased: , ,
Case 2.
The  channel is erased: , ,
Case 3.
The  channel is erased: , , .
Let us zoom on the decoding algorithm to see what is happening. We illustrate the decoding procedure on a decoding tree, which represents the local neighborhood of a bit node in the Tanner graph (the incoming messages are assumed to be independent). When decoding, bit nodes called leaves pass extrinsic information through a check node to another bit node called root (Figure 4). Because we consider a block BEC channel, the check node operation becomes very simple. If all leaf bits are known, the root bit becomes the modulo2 sum of the leaf bits, otherwise, the root bit is undetermined (P(bit=1)=P(bit=0)=0.5). Dealing with Case 3 is simple: let every source send its information uncoded and sends extra parity bits. If receives the transmissions of and perfectly, it has all the information bits. So the challenging cases are the first two possibilities. Let us assume that the nodes corresponding to the bits transmitted by , , and are filled red, blue, and white, respectively. Assume that all red (blue) bits are erased at . A very simple way to guarantee full diversity is to connect a red (blue) information bit node to a rootcheck (Figures 4(a) and 4(b)).
Definition 3.
A rootcheck is a special type of check node, where all the leaves have colors that are different from the color of its root.
Assigning rootchecks to all the information bits is the key to achieve full diversity. This solution has already been applied in some applications, for example, the cooperative multiple access channel (without external relay) [8]. Note that a check node can be a rootcheck for more than one bit node, for example, the second rootcheck in Figure 4.
4.3. An Example for the MARC
The sources and transmit information bits and parity bits that are related to their own information, and transmits information bits and parity bits related to the information from and . The previous description naturally leads to 8 different classes of bit nodes. Information bits of are split into two classes: one class of bits is transmitted on fading gain (red) and is denoted as , the other class is transmitted on (white) and denoted as ; similarly, red and white parity bits derived from the message of are of the classes and , respectively. Likewise, bits related to are split into four classes: blue bits and (transmitted on ), and white bits and (transmitted on ). The subscripts of a class refer to the associated user. In the remainder of the paper, the vectors , , , and collect the bits of the classes , , , and , respectively. A similar notation holds for . This notation is illustrated in Figure 5.
Above, we concluded that all information bits should be the root of a rootcheck. The class of rootchecks for is denoted as . Translating Figure 4 to its matrix representation renders
The identity matrix concatenated with a matrix of zeros assures that bits of the class are the only red bits connected to check nodes of the class . (Note that the identity matrix can be replaced by a permutation matrix. For the simplicity of the notation, in the rest of the paper will be used.) As the bits from and are independent, the matrix representation can be further detailed:
Hence, a fulldiversity code construction for the MARC can be formed by assigning this type of rootchecks (introducing new classes , , and ) to all information bits:
(The reader can verify that this is a straightforward extension of fulldiversity codes for the block fading channel [22].) transmits and , transmits and , and the common relay first transmits and and then transmits and , hence the level of cooperation is . The reader can easily verify that if only one color is erased, all information bits can be retrieved after one decoding iteration. Note that both sources do not transmit all information bits, but the relay transmits a part of the information bits. This is possible because if receives and perfectly it can derive (because of the rootchecks ) and consequently (after reencoding). (This code construction can be easily extended to nonperfect relay channels using techniques described in [8].) The same holds for . It turns out that splitting information bits in two parts and letting one part to be transmitted on the first fading gain and the other part on the second fading gain is the only way to guarantee full diversity for maximum coding rate [22]. This code construction is semirandom, because only parts of the paritycheck matrix are randomly generated. However, every set of rows and set of columns contains a randomly generated matrix and, therefore, can conform to any degree distribution. It has been shown that despite the semirandomness (due to the presence of deterministic blocks), these LDPC codes are still very powerful in terms of decoding threshold [22]. No network coding has been used to obtain the code construction discussed above. The aim of this subsection was to show that through rootchecks, it is easy to construct a fulldiversity code construction. However, when applying network coding, as will be discussed in Section 5, the spectral efficiency can be increased.
4.4. Rootchecks for Punctured Bits
In the previous subsection, we have illustrated that, through rootchecks, fulldiversity can be achieved. Another feature of rootchecks is to retrieve bits that have not been transmitted, which are called punctured bits. Punctured bits are very similar to erased bits, because both are not received by the destination. However, the transmitter knows the exact position of the punctured bits inside the codeword which is not the case for erased bits. Formally, we can state that from an algebraic decoding or a probabilistic decoding point of view, puncturing and erasing are identical, an erased/punctured bit is equivalent to an error with known location but unknown amplitude. From a transmitter point of view, punctured bits have always fixed position in the codeword whereas channel erased bits have random locations.
When punctured bits are information bits, the destination must be able to retrieve them. There are two ways to protect punctured bits.

(i)
The punctured bit nodes are connected to one or more rootchecks. If the leaves are erased or punctured, the punctured root bit cannot be retrieved after the first decoding iteration. The erased or punctured leaves on their turn must be connected to rootchecks, such that they can be retrieved after the first iteration. Then, in the second iteration the punctured root bit can be retrieved. These rootchecks are denoted as secondorder rootchecks (see Figure 6). Similarly, higherorder rootchecks can be used.

(ii)
The punctured bit nodes are connected to at least two rootchecks where both rootchecks have leaves with different colors (see Figure 6). If one color is erased, there will always be a rootcheck without erased leaves to retrieve the punctured bit node.
Combinations of both types of rootchecks are also possible.
5. FullDiversity Joint NetworkChannel Code
In this section, we join the principles of the previous section with the physical layer network coding framework. We will use the same bit node classes as in the previous section, hence transmits and , and transmits and . The bits transmitted by the relay are determined by (5) and are of the class . Adapting (5) to the classes of bit nodes gives
where the dimensions of are . Please note that , , , and are not transmitted anymore (these bits are punctured). The number of transmitted bits by the relay is determined by the coding rate. There are information bits. The sources and each transmit bits, hence to obtain a coding rate , the relay can transmit bits. We will include the punctured information bits and in the paritycheck matrix for two reasons:

(i)
without and , we cannot insert (25) in the paritycheck matrix;

(ii)
the destination wants to recover all information bits, that is, , , , and , so and must be included in the decoding graph.
(The matrices in the following of the paper correspond to codewords that must be punctured to obtain the bits actually transmitted.) The paritycheck matrix now has the following form:
Because the nodes and have been added, we have now columns and rows. rows are used to implement (25), while the other rows define in terms of the information bits and (used for encoding at ), and in terms of the information bits and (used for encoding at ). The first two set of rows and are rootchecks for and ; see Section 4. Now it boils down to design the matrices , , , , and , such that the set of rows and represent rootchecks of the first or second order for all information bits. There exist 8 possible paritycheck matrices that conform to this requirement; see Appendix A. With the exception of matrix (A.7), all matrices have one or both of the following disadvantages.

(i)
There is no random matrix in each set of columns, such that cannot conform to any degree distribution.

(ii)
There is an asymmetry wrt. and and/or wrt. and and/or and which results in a loss of coding gain.
Therefore, we select the matrix (A.7). The paritycheck matrix (A.7) of the overall decoder at shows that the bits transmitted by are a linear transformation of all the information bits , , , and . Furthermore, the checks represent rootchecks for all the information bits, guaranteeing full diversity. The checks are necessary because the bits are not transmitted. Note that the punctured bits have two rootchecks that have leaves with different colors. One of the rootchecks is a secondorder rootcheck. For example, the punctured bits of the class have two rootchecks, one of the class and one of the class . The rootcheck of the class has only red leaves, while the rootcheck of the class has white and blue leaves. All but one blue leaves are punctured such that the rootcheck of the class is a secondorder rootcheck.
6. Density Evolution for the MARC
In this section, we develop the density evolution (DE) framework, to simulate the performance of infinite length LDPC codes. In classical LDPC coding, density evolution [9, 24, 31] is used to simulate the threshold of an ensemble of LDPC codes. (Richardson and Urbanke [9, 31] established that, if the block length is large enough, (almost) all codes in an ensemble of codes behave alike, so the determination of the average behavior is sufficient to characterize a particular code behavior. This average behavior converges to the cyclefree case if the block length augments and it can be found in a deterministic way through density evolution (DE).) The threshold of an ensemble of codes is the minimum SNR at which the bit error rate converges to zero [31].
This technique can also be used to predict the word error rate of an ensemble of LDPC codes [22]. We refer to the event where the bit error probability does not converge to 0 by Density Evolution Outage (DEO). By averaging over a sufficient number of fading instances, we can determine the probability of a Density Evolution Outage . Now, it is possible to write the word error probability of the ensemble as
where is the word error rate given a DEO event and is the word error rate when DE converges. If the bit error rate does not converge to zero, then the word error rate equals one, so that . On the other hand, depends on the speed of convergence of density evolution and the population expansion of the ensemble with the number of decoding iterations [32, 33], but in any case , so that the performance simulated via DE is a lower bound on the word error rate. Finite length simulations confirm the tightness of this lower bound.
In summary, a tight lower bound on the word error rate of infinite length LDPC codes can be obtained by determining the probability of a Density Evolution Outage . Given a triplet , one needs to track the evolution of message densities under iterative decoding to check whether there is DEO. (Messages are under the form of loglikelihood ratios (LLRs).) The evolution of message densities under iterative decoding is described through the density evolution equations, which are derived directly through the evolution trees. The evolution trees represent the local neighborhood of a bit node in an infinite length code whose graph has no cycles, hence incoming messages to every node are independent.
6.1. Tanner Graph and Notation
The proposed code construction has 7 variable node types and 4 check node types. Consequently, the evolution of message densities under iterative decoding has to be described through multiple evolution trees, which can be derived from the Tanner graph. A Tanner graph is a representation of the paritycheck matrices of an errorcorrecting code. In a Tanner graph, the focus is more on its degree distributions. In Figure 7, the Tanner graph of matrix (A.7) is shown. The new polynomials and are derived in Proposition 2.
Proposition 2.
In a Tanner graph with a left degree distribution , isolating one edge per bit node yields a new left degree distribution described by the polynomial :
Proof.
Let us define as the number of edges connected to a bit node of degree . Similarly, the number of all edges is denoted . From Section 2, we know that expresses the left degree distribution, where is the fraction of all edges in the Tanner graph, connected to a bit node of degree . So finally . A similar reasoning can be followed to determine :

(a)
is equal to the number of edges that are removed which is equal to the number of bits.

(b)
is equal to the number of edges connected to a bit of degree .
Similarly, we can determine , where . It can be shown that is the same as applying the transformation two times consecutively, hence first on , and then on .
6.2. DE Trees and DE Equations
The proposed code construction has 7 variable node types and 4 check node types. But not all variable node types are connected to all check node types. Therefore, there are 14 evolution trees. But it is sufficient to draw only 7 of them because of symmetry. To write down the equations we adopt the following notation.
Let and be two independent real random variables. The density function of is obtained by convolving the two original densities, written as . The notation denotes the convolution of with itself times.
The density function of the variable , obtained through a check node with and at the input, is obtained through the Rconvolution [9], written as . The notation denotes the tangent hyperbolic function and denotes the convolution of with itself times.
To simplify the notations, we use the following definitions:
Next, we will use the following definitions:
The first definition is necessary because of the nonlinearity of the Rconvolution. Therefore, the first equation is not equal to .
The following message densities at the iteration are distinguished:
Proposition 3.
The DE equations in the neighborhood of , , , and for all are listed in (33)–(34),
where
Note that the message densities propagating from bits of the class do not contain a channel observation because these information bits are punctured.
Proof.
See Appendix B.
7. Numerical Results
7.1. FullDiversity LDPC Ensembles
We evaluated the finite length performance of fulldiversity LDPC codes and the asymptotic performance by applying DE on the proposed code construction. The paritycheck matrix (A.7) is used by the destination to decode the information bits. This paper focuses on full diversity, rather than coding gain. Therefore, one of the codes is a simple regular LDPC code. This means that all the random matrices in (A.7) are randomly generated satisfying an overall row weight of and an overall column weight of . This matrix corresponds to a coding rate of , but because are punctured, the actual coding rate is . The other code, that is, simulated and is denoted as code 2 is an irregular LDPC ensemble [22] with left and right degree distributions given by the polynomials
We studied the following scenario.

(i)
The , , and  links have the same average SNR.

(ii)
The  and  links are perfect.

(iii)
The coding rate is and the cooperation level is .
Figure 8 shows the main results: the word error rate (WER) of a regular LDPC ensemble and of an irregular LDPC ensemble, which are both of full diversity.
It is clear that the DE results are a lower bound on the actual word error rates (a tight lower bound for the regular code and a less tight lower bound for the irregular code). The word error rate of a regular LDPC code is only about worse than the outage probability. The irregular LDPC code is only slightly better than the regular LDPC code in terms of word error rate.
7.2. FullDiversity RA Codes with Improved Coding Gain
Another technique, suggested in [17], that improves the coding gain is called doping. For all the Rootcheck based LDPC codes the reliability of the messages exchanged by the belief propagation algorithm can be improved by increasing the reliability of parity bits (which are not protected by rootchecks). In fact the LLR values of the messages exchanged by the belief propagation algorithm are in the form [17]:
where are the fading coefficients, are positive constants, and represents the noise. The higher the coefficients , the more reliable are the LLR messages. Since the output messages of the check node are limited by the lowest LLR values of the incoming messages, that is, the messages coming from parity bits, the doping technique aims to increase those values. The least reliable variable nodes are the parity bits sent on a channel in a deep fade.
In case of block BEC, consider the parity bits sent on a channel with fading coefficient and suppose that all the other fading coefficients are with . Consider the paritycheck matrix (A.7). The doping technique consists in fixing the random matrix such that, under BP, all the variable nodes can be recovered after a certain number of iterations. This is equivalent to having reliable parity bits, that is, connected to rootchecks of a certain order, and it guarantees to increase the coefficients .
While the aforementioned doping technique has been proposed and investigated for infinite length LDPC codes, finite length rootcheck based LDPC codes that get advantage of the doping technique have not been published yet. Ongoing studies have revealed construction problems with doped finite length RootLDPC codes, so that their performance cannot be included here. An important issue, that is often not considered, is the encoding complexity. This suggests to embed the wellknown repeataccumulate (RA) structure in the paritycheck matrix, which results in lineartime encoding. Hence, regardless the degree distribution, we substitute the matrices , , and with staircase matrices (46)
Figure 9 reports the simulation results for a regular RA code that show a 0.5 dB improvement compared to the proposed regular (3,6) code. Together with the fact that this simple code is now lineartime encoding, this result is impressive because we have lowered the complexity and improved the performance at the same time. As a benchmark the outage probability has been plotted. We have also included the best known LDPC code for the MARC in literature: the rate network code proposed in [16]; it reports a loss of almost 2.5 dB wrt. the proposed fulldiversity RA code.
8. Conclusions and Remarks
We have studied LDPC codes for the multiple access relay channel in a slowly varying fading environment under iterative decoding. LDPC codes must be carefully designed to achieve full diversity on this channel and network coding must be applied to increase the achievable coding rate to a maximum rate . Combining network coding with of full diversity channel coding gave rise to a new family of semirandom fulldiversity joint networkchannel LDPC codes for all rates not exceeding . A code that is only away from the outage probability limit has been presented.
For a block fading channel with several fading states per codeword, it has been pointed out that the poor reliability of the parity bits in fulldiversity LDPC codes (where especially the information bits are well protected) causes the actual gap with the outage probability limit. We increased the reliability of the parity bits by using a repeataccumulate structure and have improved the coding gain of the presented code construction for the MARC.
Appendices
A. FullDiversity ParityCheck Matrices
The reader can find here a list of fulldiversity paritycheck matrices , that is, matrices where all information bits are assigned to a rootcheck in the last two set of rows and . Matrix (A.7) performs the best for reasons of symmetry and randomness,
B. Proof of Proposition 3
Equations (33)–(40) are directly derived from the local neighborhood trees (see e.g., Figures 11 and 12). The proportionality factors (35)–(40) can easily be determined by analyzing the Tanner graph. Let denote the total number of edges between the variable nodes and the check nodes . Figure 10 illustrates how and are obtained:

(a)
The fraction of check nodes connected to edges of is . A similar reasoning proves (B.2).

(b)
The fraction of edges connecting to is and the fraction of edges connecting to is .
Note that in the first iteration, , , , and are equal to , because the received messages come from check nodes where one of the leaves corresponds to a punctured information bit (so that their message density is a Dirac function on ). Therefore, the message densities coming from the check nodes are also Dirac functions on . (The output of a check node is determined through its inputs , via the following formula: . If one of the inputs is always zero because its distribution is a Dirac function on , then the output will always be zero, so that its distribution will also be a Dirac function on .) But and are different from a Dirac function on after the first iteration, so that the next iteration also becomes different from a Dirac function on .
The factor in (42) and (43) takes into account that counts variable nodes while and count only parity check equations. Solving together (40)–(43), it is possible to prove that for any degree distribution
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Acknowledgments
D. Capirone wants to acknowledge professor Benedetto for helpful and stimulating discussions. This work was supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++ (Contract no. 216715).
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Duyck, D., Capirone, D., Boutros, J. et al. Analysis and Construction of FullDiversity Joint NetworkLDPC Codes for Cooperative Communications. J Wireless Com Network 2010, 805216 (2010). https://doi.org/10.1155/2010/805216
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Keywords
 Outage Probability
 Degree Distribution
 Spectral Efficiency
 Network Code
 LDPC Code