# Analysis and Construction of Full-Diversity Joint Network-LDPC Codes for Cooperative Communications

- Dieter Duyck
^{1}Email author, - Daniele Capirone
^{2}, - JosephJ Boutros
^{3}and - Marc Moeneclaey
^{1}

**2010**:805216

https://doi.org/10.1155/2010/805216

© Dieter Duyck et al. 2010

**Received: **29 December 2009

**Accepted: **3 June 2010

**Published: **28 June 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 well-known 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 Multiple-Access 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 full-diversity 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 (small-scale fading) is an important salient effect of wireless channels, causing possible destructive adding of signals at the receiver. When the fading varies very slowly, error-correcting codes cannot combat the detrimental effect of the fading on a point-to-point 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 well-known 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 point-to-point transmission. However, the overall error-correcting code should be carefully designed in order to guarantee full diversity [8].

We focus on capacity achieving codes, more precisely, low-density parity-check (LDPC) codes [9], because their word error rate (WER) performance is quasi-independent 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 network-channel code. Therefore, the problem formulation is how to design a full-diversity joint network-channel code construction for a rate
.

Up till now, no family of full-diversity LDPC codes with for coded cooperation on the MARC has been published. Chebli, Hausl, and Dupraz obtained interesting results on joint network-channel 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 full-diversity 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 network-channel 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 full-diversity network-channel LDPC code construction for maximum rate is proposed.

Channel-State Information is assumed to be available only at the decoder. In order to simplify the analysis, we consider orthogonal half-duplex devices that transmit in separate timeslots.

## 2. System Model and Notation

### 2.1. Multiple Access Relay Channel

*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.

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 full-diversity, as will be shown in Section 3.2.

### 2.2. LDPC Coding

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 full-diversity 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

*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 parity-check matrix defining the *overall* error-correcting 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.

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.

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 Source-Relay Channels

Here, we will show that the achieved diversity at does not depend on the quality of the source-relay ( - ) channel. Therefore, in the remainder of the paper, we will assume errorless - channels to simplify the analysis.

where is a positive constant. According to Definition 1, full-diversity 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 source-relay channels, so that, for the clarity of the presentation, perfect source-relay channels are assumed in the remainder of the paper.

### 3.3. Outage Probability of the MARC

*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].

because the three timeslots behave as parallel Gaussian channels whose mutual informations add together. Of course, the timeslots timeshare a time-interval, which gives a weight to each mutual information term [25, Section ]. The total transmitted rate must be smaller than , which yields (14).

Now, the outage probability can be easily determined through Monte-Carlo simulations to average over the fading gains and to average over the noise. (Averaging over the noise can be done more efficiently using Gauss-Hermite 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).

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 sum-rate only. In our design, we will take , so that the maximum achievable coding rate can be achieved.

## 4. Full-Diversity 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 error-correcting 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 double-diversity order is achieved. Finally, these claims will be verified by means of simulations for finite length and infinite length codes.

### 4.1. Diversity Rule

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 (log-likelihood 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 block-Rayleigh fading. Full diversity on the block BEC is a necessary *and sufficient* condition for full diversity on the block-Rayleigh 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.

Case 2.

Case 3.

The - channel is erased: , , .

*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 modulo-2 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 reader can verify that this is a straightforward extension of full-diversity 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 parity-check 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 full-diversity 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, full-diversity 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.

- (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

*second*-*order rootchecks*(see Figure 6). Similarly, higher-order 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. Full-Diversity Joint Network-Channel Code

Therefore, we select the matrix (A.7). The parity-check 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 second-order 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 second-order 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 cycle-free 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].

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 log-likelihood 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

Proposition 2.

Proof.

- (a)
- (b)

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 *R-convolution* [9], written as
. The notation
denotes the tangent hyperbolic function and
denotes the
-convolution of
with itself
times.

The first definition is necessary because of the nonlinearity of the R-convolution. Therefore, the first equation is not equal to .

Proposition 3.

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. Full-Diversity LDPC Ensembles

*code 2*is an irregular LDPC ensemble [22] with left and right degree distributions given by the polynomials

- (i)
- (ii)
- (iii)

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. Full-Diversity RA Codes with Improved Coding Gain

*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 parity-check 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 .

## 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 full-diversity joint network-channel 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 full-diversity 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 repeat-accumulate structure and have improved the coding gain of the presented code construction for the MARC.

## Appendices

## A. Full-Diversity Parity-Check Matrices

## B. Proof of Proposition 3

- (a)
- (b)

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 .

## Declarations

### 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).

## Authors’ Affiliations

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