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Jointly optimized ratecompatible UEPLDPC codes for halfduplex cooperative relay networks
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 22 (2014)
Abstract
This paper addresses the design of lowdensity paritycheck (LDPC) codes for halfduplex cooperative relay networks. Structured ratecompatible codes with unequal error protection (UEP) are designed through joint optimization of the codes for the channels between source and relay and source and destination. The proposed codes clearly outperform simpler LDPC codes which are not optimized for relay channels and puncturingbased ratecompatible LDPC codes, and they show a significant performance improvement over the direct link communication depending on the position of relay. The optimization algorithm for the proposed codes is based on density evolution using the Gaussian approximation and optimal variable node degree distributions are found through iterative linear programming. Interestingly, they anyhow show performance which is almost comparable to the performance of codes optimized through a more complex nonlinear optimization algorithm. We analyze the performance of our proposed codes with short, medium and long block lengths, and with low and high rates under realistic assumptions, i.e., imperfect decoding of the codeword at relay and variant signaltonoise ratio within a single codeword.
1 Introduction
Cooperative relay networks have opened a new research field in wireless communication. They can be used in mobile cellular networks, wireless ad hoc networks, and wireless sensor networks. They provide spatial diversity because the terminals in a relay network form virtual multiple antenna systems. Like multiple antenna systems, relay networks increase channel capacity under the same bandwidth constraints and with no additional power consumption. A simple cooperative relay network is composed of three nodes, source (S), relay (R), and destination (D), as shown in Figure 1. S is transmitting data to D via a direct link (S to D) and a relay link (S to D through R). There are three relay channels in a simple relay network, i.e., the channels between S and D, between S and R, and between R and D. The relay is used to increase the reliability and data rate when the channel between S and D is bad due to fading and interference.
In this paper, we propose lowdensity paritycheck (LDPC) code design for halfduplex relay networks based on rate compatibility and unequal error protection (UEP) capabilities. The code is designed by optimizing the variable node degree distribution using density evolution under Gaussian approximation [1]. We design ratecompatible LDPC codes with UEP using subdegree distributions both for classes of bits with different protection (denoted by protection classes) and for classes of bits transmitted over different relay channels. Henceforth, the bits transmitted over one specific relay channel will be referred to as belonging to one channel class. The separation of the variable node degree distribution into subdegree distributions significantly increases the number of design parameters. Therefore, it is important to note that the code design is solved by iterative linear programming (LP), which enables an efficient optimization of the subdegree distributions. Our code design is based on the design method for UEPLDPC codes with higher order modulations suggested in [2], which employs iterative LP, and the ratecompatible code structure suggested in [3]. Higher order modulations cause different bits in the codeword to have different signaltonoise ratios (SNRs) in the same way as relay channels with different channel gains cause different codeword bits to be received with different SNRs. In our design, we use the strategy proposed in [2] to optimize the codes for the case where two different codeword segments experience two different SNRs due to different channel gains of the S to D and the R to D relay channels. That is, our proposed optimization algorithm considers different SNRs of the relay channels in the code design, like in [3].
The performance of the source to relay channel plays an important role in the overall performance of cooperative relay networks [4, 5]. It is pointed out in [3] that code extension techniques provide better performance than code puncturing in the source to relay channel. In code extension techniques, optimal codewords can be transmitted over the source to relay channel to get the best performance. In code puncturing techniques, only punctured codewords can be transmitted over the source to relay channel, which results in a certain performance loss. Therefore, we design the ratecompatible LDPC codes for relay networks based on the extension technique.
In [3], LDPC codes for relay networks are designed by finding optimal variable and check node degree distributions using differential evolution, which is a nonlinear code optimization method. In the design proposed here, linear programming is used to find an optimal variable node degree distribution. Our code optimization algorithm requires less computational effort than that of [3] and the performance is anyhow similar to the performance of the codes proposed in [3]. Like in [3], joint optimization of the codes for the source to relay and the source to destination channels is performed, and variant SNR within a single codeword is considered in the design. In [3], it is assumed that the codeword received at the relay from the source is perfectly decoded at the relay. However, in this paper, it is not assumed that the codeword is perfectly decoded at the relay, which is a more realistic assumption. In [6], the authors have compared two different LDPC coding schemes from [7] and [8] with their proposed puncturingbased ratecompatible LDPC codes. The code design proposed here is different from the designs proposed in [3] and [6] because it provides UEP properties to the code, i.e., more important bits are made more protected by allocating them higher variable node degrees. The coding scheme in [6] has simple code design as it requires the design of only one singleuser LDPC. In contrast to the coding scheme in [6], our proposed coding scheme requires the joint design of two codes. However, our proposed code shows better performance than the code proposed in [6] because the code in [6] has performance loss due to puncturing in the cooperative relay network.
Most of the papers in the literature on relay networks consider a theoretical approach of code design. However, we focus on code design with realistic assumptions, such as imperfect decoding of the codeword in the relay terminal, like in [9] and variant SNR within a single codeword, like in [3].
In [9], quasicyclic LDPC (QCLDPC) codes are used in relay networks through repetition coding schemes. In the code design, the SNR is considered invariant within a single codeword. The fact that [9] uses QCLDPC codes makes them easier to implement and thereby more realistic, but on the other hand, they also get worse performance. Using the same time slots, transmission power, and processing of the codewords in the relay terminal with the decode and forward strategy as in [9], the codes proposed here achieve up to 1.9dB gain over the QCLDPC codes depending on the location of the relay with respect to the source and the destination. The better performance is partly due to the ratecompatible coding scheme and partly due to the UEP properties of the code. The ratecompatible coding scheme provides a better tradeoff between bit error rate and data rate than the repetition coding scheme. The UEP provides high protection to the codeword bits transmitted over the source to relay channel, which results in a better performance of the important source to relay channel. Also, in contrast to [9], the proposed codes are optimized for each position of the relay. Depending on the location of the relay relative to the source and the destination, the proposed lowrate codes with short, medium and long block lengths show a performance improvement of up to 3.8dB gain over the direct link communication. The proposed highrate code with medium block length has a 1.3dB gain over the direct link communication.
The remainder of this paper is organized as follows. Section 2 presents an overview of the background and the system model of a halfduplex relay network and introduces the LDPC coding scheme. Section 3 gives the details of the optimization algorithm for the design of ratecompatible LDPC codes with UEP properties for halfduplex relay networks. Section 4 describes the selection of a good check node degree distribution and discusses the simulation results of our proposed code design. The results are also compared to the simulation results for QCLDPC codes [9] as well as the codes proposed in [3] and [6]. Section 5 concludes this paper.
2 Background and system model
2.1 Halfduplex operation of relay and types of relay
In this paper, the relay is assumed to operate in halfduplex mode. Fullduplex operation of the relay is hard to implement from a practical system point of view because it is difficult to isolate the several orders of magnitude higher transmitted signal from the received signal in the same frequency band. Halfduplex operation is easy to implement by isolating the transmitted and the received signal in time or frequency. We consider time division multiple access (TDMA) using two time slots. In the first time slot t (normalized), known as broadcast mode (BC), S transmits to both R and D. In the second time slot $\stackrel{\u0304}{t}=1t$, known as multiple access mode (MAC), either only R or both R and S transmit to D as shown in Figure 2. In this paper, only R transmits to D in MAC mode.
There exist different types of cooperative relay strategies based on how the received signal is processed by the relay node. The two most common types of cooperative relay strategies are amplify and forward (AF) and decode and forward (DF). The AF relay [10] amplifies the received signal and then retransmits it to the destination. The DF relay first decodes the signal received from the source and then retransmits the decoded signal after reencoding, either with the same code as used by the source or with a different code. The drawback of the AF relay is that noise is also amplified along with the desired signal. In power limited communication networks, the AF relay is preferred as it performs no decoding and reencoding of the received signal. The drawback of the DF relay is the large delay due to decoding and reencoding as compared to the AF relay. However, the DF relay can also be implemented without reencoding [11]. In this work, the DF relay is considered.
2.2 Relay channel model
Figure 3 shows the relay system model as suggested in [12]. The distance between S and D is normalized to unity. R is located at a distance of d from S and 1−d from D. The relative distance model in Figure 3 is applied with normalized distance d, where 0<d<1. Note that all the halfduplex relay channels are AWGN channels with fixed path loss. The signals received at D and R in BC and MAC modes are as follows:
The signals transmitted by S and R are denoted by^{a} x_{ Si } and x_{ Ri }, respectively, where i∈{1,2} and 1 and 2 stands for BC and MAC modes, respectively. The received signals at R and D are represented by y_{ Ri } and y_{ Di }, respectively. In the above equations, h_{ SR }, h_{ SD } and h_{ RD } are the channel realizations between {S,R}, {S,D} and {R,D}, respectively. n_{ Ri } and n_{ Di } are AWGN at the R and D receivers with zero means and variance σ^{2}=N_{0}/2 in each dimension. The normalized channel gains of the {S,R}, the {S,D}, and the {R,D} channels are given as
where τ is the path loss exponent which normally ranges from 2 to 5 [13]. Furthermore, BPSK modulation and full channel state information (CSI) available at each terminal of the relay network are assumed. In order to fairly compare the relay channel with the direct link, we assume that the sum of the R and S transmit powers in the relay channel is equal to the direct link transmit power of S. Therefore, the following global power constraint is assumed:
P_{S 1} is the source transmit power in BC mode, P_{ R } is the relay transmit power in MAC mode and P is the total system transmit power.
2.3 Coding strategy for halfduplex relays
The proposed relay coding strategy is similar to [3] except that it is assumed that S is silent in MAC mode and perfect decoding of the codeword at R is not assumed. According to this relay coding strategy, a codeword w_{1} of length N_{1} is transmitted from S to R and D in BC mode, i.e., ${\mathbf{x}}_{S1}=\sqrt{{P}_{S1}}{\mathbf{w}}_{1}$. R decodes the codeword w_{1} and generates the extra parity bits w_{ e } of length N_{2}−N_{1}. D stores $\stackrel{~}{{\mathbf{w}}_{1}}$ (received version of w_{1}) until it receives $\stackrel{~}{{\mathbf{w}}_{e}}$ from R. Note that w_{ e } transmitted from R may contain errors if $\stackrel{~}{{\mathbf{w}}_{1}}$ is not decoded correctly at R. In MAC mode S is silent, so only R transmits w_{ e } to D, i.e., ${\mathbf{x}}_{R2}=\sqrt{{P}_{R}}{\mathbf{w}}_{e}$. D concatenates $\stackrel{~}{{\mathbf{w}}_{1}}$ with $\stackrel{~}{{\mathbf{w}}_{e}}$ to get the extended codeword $\stackrel{~}{{\mathbf{w}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\stackrel{~}{{\mathbf{w}}_{1}}\phantom{\rule{1em}{0ex}}\stackrel{~}{{\mathbf{w}}_{e}}\right]$ of length N_{2} belonging to the extended code ${\mathcal{C}}_{2}$. $\stackrel{~}{{\mathbf{w}}_{2}}$ is then decoded at D to recover the original information. From (2) and (3), it follows that the signals received at D in BC and MAC modes can be expressed respectively as
2.4 Brief introduction to LDPC codes
LDPC are classified as linear block codes. They are described by parity check equations and are represented by a sparse paritycheck matrix H. LDPC codes are decoded by a message passing algorithm [14]. The decoding complexity increases linearly with codeword length and also depends on the sparseness of H. The paritycheck matrix can be represented in graphical form by a bipartite graph, called Tanner graph [15]. The operation of the message passing algorithm is readily explained using a Tanner graph. The two types of nodes in the Tanner graph are variable nodes and check nodes, which correspond to codeword bits and paritycheck equations, respectively. The number of variable nodes (codeword bits) is denoted by n and the number of check nodes (paritycheck equations) is denoted by (n−k) where k is the number of the information bits. A variable node is connected to a check node by an edge if the bit corresponding to the variable node is included in the paritycheck equation corresponding to the check node. Thus, the number of edges in the Tanner graph is equal to the number of 1s in the paritycheck matrix H. The code rate of LDPC codes is defined by k/n. LDPC codes are classified as regular or irregular based on the degrees of the nodes. The degree of a node is the number of edges connected to it. The regular codes have variable nodes of fixed degree and their check nodes have another similar fixed degree. The irregular codes have variable and check nodes with varying degrees. The variable and check node degree distributions of irregular codes from an edge perspective are represented by $\lambda \left(x\right)=\sum _{i=2}^{{d}_{{v}_{\text{max}}}}{\lambda}_{i}{x}^{i1}$ and $\rho \left(x\right)=\sum _{i=2}^{{d}_{{c}_{\text{max}}}}{\rho}_{i}{x}^{i1}$, respectively, where ${d}_{{v}_{\text{max}}}$ is the maximum variable node degree and ${d}_{{c}_{\text{max}}}$ is the maximum check node degree of the code [16]. The coefficients λ_{ i } and ρ_{ i } describe the fraction of edges that are connected to degreei variable and check nodes, respectively. The optimal degree distribution of irregular codes can be found by density evolution (DE) using the Gaussian approximation [1]. DE tracks mutual information exchange between variable and check nodes in the message passing decoding algorithm.
2.5 Ratecompatible UEPLDPC codes for halfduplex relays
From an implementation point of view, the simplest code design scheme for halfduplex relays is the repetition coding scheme, i.e., transmitting the same codeword w_{1} of code ${\mathcal{C}}_{1}$ in BC and MAC modes[9]. However, the repetition coding scheme is not preferred as it provides a poor tradeoff between bit error rate and data rate. One alternative is to use coding schemes based on incremental redundancy, i.e., transmitting the codeword w_{1} in BC mode and extra parity bits w_{ e } being extracted from w_{1}, in MAC mode [3]. This yields the extended codeword w_{2}= [ w_{1} w_{ e }] belonging to the extended code ${\mathcal{C}}_{2}$. In this case, a joint design of the mother code ${\mathcal{C}}_{1}$ and extended code ${\mathcal{C}}_{2}$ is required. The incremental redundancy could also be achieved using puncturing, i.e., deleting a number of parity bits from w_{2} to get w_{1}. Code design in this case is the joint design of the mother code ${\mathcal{C}}_{2}$ and the punctured code ${\mathcal{C}}_{1}$.
The ratecompatible structure of the paritycheck matrix H_{2} representing the code ${\mathcal{C}}_{2}$, which follows the structure proposed in [3], is illustrated in Figure 4. H_{2} is composed of a nonzero parity check matrix H_{1} corresponding to the code ${\mathcal{C}}_{1}$, a zero submatrix O and nonzero submatrices A and B. The zero submatrix preserves the paritycheck equations of the mother code ${\mathcal{C}}_{1}$. The ratecompatible design of ${\mathcal{C}}_{2}$ is different from [17–19] as w_{1} and w_{ e } are in general received with different SNRs. Received SNRs for decoding w_{1} and w_{ e } are given later by (8) and (9), respectively.
We introduce UEP along with ratecompatibility in LDPC codes in order to provide better protection to the codeword bits which are transmitted in BC mode than to the extra parity bits transmitted in MAC mode, since the overall performance of relay networks depends largely on the source to relay channel performance [4, 5]. Through UEP, variable nodes of higher degrees are assigned to the w_{1} segment of the code ${\mathcal{C}}_{2}$ in order to give it higher protection than its w_{ e } segment.
3 Design of RC UEPLDPC codes for relay networks
3.1 Notations
The codeword bits of code ${\mathcal{C}}_{z}$ where z∈{1,2} are divided into ${N}_{c}^{{\mathcal{C}}_{z}}$ protection classes in order to give it UEP properties. The proportions ${\mathbf{\alpha}}^{{\mathcal{C}}_{z}}=\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}^{{\mathcal{C}}_{z}},\xb7\xb7\xb7,{\alpha}_{{N}_{c}^{{\mathcal{C}}_{z}}1}^{{\mathcal{C}}_{z}}]$ denote the normalized lengths of each class corresponding to the information bits u_{ i }. ${\alpha}_{i}^{{\mathcal{C}}_{z}}$ equals the number of bits belonging to protection class C_{ i } divided by the total number of information bits k. $\mathbf{p}=\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}^{{\mathcal{C}}_{z}}{R}_{z},\xb7\xb7\xb7,{\alpha}_{{N}_{c}^{{\mathcal{C}}_{z}}1}^{{\mathcal{C}}_{z}}{R}_{z},(1{R}_{z}\left)\right]$ defines proportions of the bits in the codeword belonging to different protection classes and R_{ z } is the code rate of ${\mathcal{C}}_{z}$. ${N}_{s}^{{\mathcal{C}}_{z}}$ is the number of relay channels (channel classes) between source, relay and destination associated with ${\mathcal{C}}_{z}$. The bit segments of the codeword being received through different relay channels have different SNRs. The bit segment with a distinct SNR will be classified as belonging to one channel class ${M}_{j},j=1,\xb7\xb7\xb7,{N}_{s}^{{\mathcal{C}}_{z}}$. The proportions of bit segments in the codeword belonging to different channel classes are defined by ${\mathbf{\beta}}^{{\mathcal{C}}_{z}}=\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}{\beta}_{1}^{{\mathcal{C}}_{z}},\xb7\xb7\xb7,{\beta}_{{N}_{s}^{{\mathcal{C}}_{z}}}^{{\mathcal{C}}_{z}}]$.
The vectors ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$ and ${\mathbf{a}}^{{\mathcal{C}}_{z}}$ contain the overall variable node degree distribution from an edge perspective and a node perspective, respectively, both for different protection classes and different channel classes for a codeword corresponding to code ${\mathcal{C}}_{z}$. Let ${\lambda}_{{M}_{j,i}}^{{\mathcal{C}}_{z},{C}_{k}}$ and ${a}_{{M}_{j,i}}^{{\mathcal{C}}_{z},{C}_{k}}$ be the proportion of edges and nodes, respectively, connected to variable nodes of degree i that belong to channel class M_{ j }, protection class C_{ k }, and code ${\mathcal{C}}_{z}$. Similarly, ${b}_{i}^{{\mathcal{C}}_{z}}$ is the proportion of check nodes of degree i that belong to code ${\mathcal{C}}_{z}$. Define ${\mathbf{\lambda}}_{{M}_{j}}^{{\mathcal{C}}_{z},{C}_{k}}=\phantom{\rule{0.3em}{0ex}}{[\phantom{\rule{0.3em}{0ex}}{\lambda}_{{M}_{j,2}}^{{\mathcal{C}}_{z},{C}_{k}},\xb7\xb7\xb7,{\lambda}_{{M}_{j,{d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}}}^{{\mathcal{C}}_{z},{C}_{k}}]}^{T}$ and ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}=\phantom{\rule{0.3em}{0ex}}{[\phantom{\rule{0.3em}{0ex}}{{\mathbf{\lambda}}_{{M}_{1}}^{{\mathcal{C}}_{z},{C}_{1}}}^{T},\xb7\xb7\xb7,{{\mathbf{\lambda}}_{{M}_{1}}^{{\mathcal{C}}_{z},{C}_{{N}_{c}^{{\mathcal{C}}_{z}}}}}^{T},\xb7\xb7\xb7,{{\mathbf{\lambda}}_{{M}_{{N}_{s}^{{\mathcal{C}}_{z}}}}^{{\mathcal{C}}_{z},{C}_{1}}}^{T},\xb7\xb7\xb7,{{\mathbf{\lambda}}^{{\mathcal{C}}_{z},{C}_{{N}_{c}^{{\mathcal{C}}_{z}}}}}_{{M}_{{N}_{s}^{{\mathcal{C}}_{z}}}}^{T}]}^{T}$ where (·)^{T} denotes the transpose. ${\mathbf{\lambda}}_{{M}_{j}}^{{\mathcal{C}}_{z},{C}_{k}}$ is a column vector of length ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}1$ and ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$ is a column vector of length $({d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}1){N}_{s}^{{\mathcal{C}}_{z}}{N}_{c}^{{\mathcal{C}}_{z}}$. The vectors ${\mathbf{\rho}}^{{\mathcal{C}}_{z}}=\phantom{\rule{0.3em}{0ex}}{[\phantom{\rule{0.3em}{0ex}}{\rho}_{2}^{{\mathcal{C}}_{z}},\dots ,{\rho}_{{d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{z}}}^{{\mathcal{C}}_{z}}]}^{T}$ and ${\mathbf{b}}^{{\mathcal{C}}_{z}}$ describe the check node degree distribution from an edge and a node perspective, respectively. Note that ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}$ and ${d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{z}}$ are the maximum variable and check node degrees respectively in the code ${\mathcal{C}}_{z}$. The vector with allones is defined by 1 with suitable length.
3.2 Protection and channel classes
Figure 5 shows a diagram of the proposed coding scheme. The RC UEPLDPC code has ${N}_{c}^{{\mathcal{C}}_{z}}$ protection classes at its output. The information bits u_{ i } are divided into ${N}_{c}^{{\mathcal{C}}_{z}}1$ protection classes ${C}_{1}\xb7\xb7\xb7{C}_{{N}_{c}^{{\mathcal{C}}_{z}}1}$. The parity bits are assigned to ${C}_{{N}_{c}^{{\mathcal{C}}_{z}}}$. The protection classes are ordered with descending order of protection, i.e., C_{1} has the highest protection and ${C}_{{N}_{c}^{{\mathcal{C}}_{z}}}$ has the lowest protection. The bits from the different protection classes are remultiplexed and assigned to channel classes ${M}_{1},\xb7\xb7\xb7,{M}_{{N}_{s}^{{\mathcal{C}}_{z}}}$ corresponding to the different relay channels. In the halfduplex cooperative relay networks with three terminals, there are two relay channels through which destination receives the signals from the relay and the source, one between source and destination M_{1}, and the other between relay and destination M_{2}, see Figure 1.
3.3 SNRs of the relay channels
This subsection describes computation of SNRs of the relay channels in BC and MAC modes. The signals received at D in BC and MAC modes have different received SNRs. The SNRs of the signals received in BC and MAC modes are defined, respectively, as
In [2], unity signal power is assumed for simplicity, and equivalent noise variances are calculated for the different bits in the higher order constellation. To follow the code design proposed in [2], the equivalent noise variances of the relay channels in BC and MAC modes are required for the optimization of the variable node degree distributions. Therefore, the equivalent noise variances of the relay channels in BC and MAC modes are determined from different SNRs by means of an equivalent channel with unit received power, i.e.,
The equivalent noise variance is expressed as
The equivalent noise variances of the relay channels in BC and MAC modes are
The noise variances in (12) and (13) are required for density evolution in the proposed optimization algorithm in [2]. For the proposed optimization algorithm in [2], we define the noise vector ${\stackrel{~}{\mathbf{\sigma}}}^{2}=[\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{\sigma}}_{1}^{2},\xb7\xb7\xb7,{\stackrel{~}{\sigma}}_{{N}_{s}^{{\mathcal{C}}_{z}}}^{2}]$ to be a vector that contains the equivalent noise variances of each relay channel in BC and MAC modes, ordered with the highest variance first.
3.4 Optimization algorithm
The optimization algorithm presented here is the adaptation of the optimization algorithm proposed in [2] for halfduplex cooperative relay networks. We add an additional constraint to the optimization algorithm of [2] for the design of ratecompatible LDPC codes as described in [3]. In the optimization algorithm, the code ${\mathcal{C}}_{2}$ for halfduplex relay network is designed to be ratecompatible with UEP based on the given degree distributions of the UEP capable code ${\mathcal{C}}_{1}$. Code ${\mathcal{C}}_{1}$ is first designed without the ratecompatible constraint in the optimization algorithm. Then, the ratecompatible code ${\mathcal{C}}_{2}$ is designed given the variable and check node degree distributions of ${\mathcal{C}}_{1}$.
The target of the optimization algorithm proposed in [2] is to find a variable node degree distribution for the whole code that maximizes the average variable node degree of the class being optimized. UEP properties of the code are achieved by sequential execution of the optimization algorithm of [2], one protection class at a time and starting with the best protected class for an E_{ b }/N_{0} slightly higher than the threshold. In every sequential execution step, the optimization algorithm keeps the degree distributions of lower protection classes fixed and may change only the degree distributions of higher protection classes [2]. The objective function for protection class C_{ k } can be expressed as
The optimization algorithm with objective function (14) maximizes the average variable node degree. For maximization of the minimum variable node degree, the optimization is initiated with a high minimum variable node degree and decreased to a lower value until a variable node degree distribution for which density evolution convergence is found, see [2] for details.
3.4.1 Ratecompatible constraints
The optimization algorithm in [3] designs the code ${\mathcal{C}}_{2}$ corresponding to H_{2}, by considering the ratecompatible constraints for variable and check node degree distributions given the degree distributions of the code ${\mathcal{C}}_{1}$. However, the optimization algorithm in this paper designs the code ${\mathcal{C}}_{2}$ by considering only the ratecompatible constraint for the variable node degree distribution [3] (expressed by (19)) given the degree distributions of the code ${\mathcal{C}}_{1}$. The check node degree distribution of ${\mathcal{C}}_{2}$ is assumed to be given and must satisfy the ratecompatible constraints for the check node degree distribution [3] (expressed by (17)) before using it in the optimization algorithm to determine an optimal variable node degree distribution for ${\mathcal{C}}_{2}$. The ratecompatible constraints for the variable node degree distribution from [3] have been adapted to cover the UEP properties of our proposed code. For ${\mathcal{C}}_{2}$, the overall edge distributions ${\mathbf{\lambda}}^{{\mathcal{C}}_{2}}$ and ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}$ and the overall node distributions ${\mathbf{a}}^{{\mathcal{C}}_{2}}$ and ${\mathbf{b}}^{{\mathcal{C}}_{2}}$ must satisfy the following general constraints from [3] which have been adapted to cover the UEP properties of our proposed code:
The check node degree distributions of ${\mathcal{C}}_{1}$ from edge and node perspective $\{{\mathbf{\rho}}^{{\mathcal{C}}_{1}},{\mathbf{b}}^{{\mathcal{C}}_{1}}\}$ become part of the check node degree distributions of ${\mathcal{C}}_{2}$ from edge and node perspective $\{{\mathbf{\rho}}^{{\mathcal{C}}_{2}},{\mathbf{b}}^{{\mathcal{C}}_{2}}\}$, see Figure 4. The check node degree distributions from node perspective of ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ should satisfy the following constraints:
where W_{ z } is the number of rows in paritycheck matrix H_{ z } and $\Omega =\frac{(1{R}_{1}){N}_{1}}{(1{R}_{2}){N}_{2}}=\frac{(1{R}_{1}){R}_{2}}{(1{R}_{2}){R}_{1}}$; the first equality is derived from $1{R}_{z}=\frac{{W}_{z}}{{N}_{z}}$ and the second equality is derived from the ratecompatible feature of H_{2}, i.e., R_{1}N_{1}=N_{1}−W_{1}=N_{2}−W_{2}=R_{2}N_{2}. Converting the check node perspective constraint (16) into edge perspective using (15) gives
Similarly, the variable node degree distributions from the node perspective of ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ should satisfy the following constraints:
Converting the node perspective constraint (18) into edge perspective using (15) gives
3.4.2 Convergence constraint
The optimal variable node degree distributions are found through density evolution using the Gaussian approximation [1]. The mutual information messages from a check node to a variable node (x_{ cv }) and from a variable node to a check node (x_{ vc }) at iteration l are given by
and
respectively. Here, J(·) is the mutual information, given by
for a Gaussian random variable $q\sim \mathcal{N}(m,2m)$ with mean m and variance 2m. The mutual information evolution can be written as a combination of (20) and (21)
The ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$ and ${\mathbf{\rho}}^{{\mathcal{C}}_{z}}$ describe the degree distributions of the code for a given σ^{2} when the condition ${x}_{\mathit{\text{vc}}}^{\left(l\right)}>{x}_{\mathit{\text{vc}}}^{(l1)}$ for any ${x}_{\mathit{\text{vc}}}^{(l1)}$ is satisfied.
3.4.3 Proportion distribution constraints
The variable node degree distribution is constrained to have the total sum of the fractions equal to one, i.e.,
is also constrained by the proportion vectors ${\alpha}^{{\mathcal{C}}_{z}}$ and ${\beta}^{{\mathcal{C}}_{z}}$. The total number of variable nodes ${n}_{{C}_{k}}$ and ${n}_{{M}_{j}}$ belonging to protection class C_{ k } and to channel class M_{ j } is
and
respectively. The code rate can be expressed as
Furthermore, ${\lambda}^{{\mathcal{C}}_{z}}$ can be related to ${n}_{{C}_{k}}$ and ${n}_{{M}_{j}}$ by
and
respectively.
3.4.4 Stability condition
The optimal degree distribution must satisfy the stability condition [16]. For the direct link channel with BPSK, all the codeword bits experience the same noise variance σ^{2}. The stability condition for the direct link channel is
In (30), P_{0}(x) is the received message density and ${\lambda}^{{\mathcal{C}}_{{z}^{\prime}}}\left(x\right)$ and ${\rho}^{{\mathcal{C}}_{{z}^{\prime}}}\left(x\right)$ are the derivatives of the degree distributions. The stability condition gives an upper bound on the number of degree2 variable nodes. In case of cooperative relay networks, the codeword bits experience different noise variances ${\sigma}_{j}^{2}$ due to different SNRs of the relay channels and there are different densities P_{0,j}(x). The derivatives are ${\lambda}^{{\mathcal{C}}_{{z}^{\prime}}}\left(0\right)={\sum}_{j=1}^{{N}_{s}^{{\mathcal{C}}_{z}}}{\sum}_{k=1}^{{N}_{c}^{{\mathcal{C}}_{z}}}{\lambda}_{{M}_{j,2}}^{{\mathcal{C}}_{z},{C}_{k}}$ and ${\rho}^{{\mathcal{C}}_{{z}^{\prime}}}\left(1\right)={\sum}_{m=2}^{{d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{z}}}{\rho}_{m}^{{\mathcal{C}}_{z}}(m1)$. The stability condition for the cooperative relay network can be expressed as
Similar to [2], the optimization is performed at E_{ b }/N_{0}=δ+ε in dB, where δ is the lowest possible threshold for the given ${\mathbf{\rho}}^{{\mathcal{C}}_{z}}$ and ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}$, and ε is an offset from the lowest threshold that offers more freedom in the selection of ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$. The optimization algorithm is decomposed into an inner and an outer loop. For a fixed E_{ b }/N_{0}, the outer loop finds an optimal variable node degree distribution for each protection class and thus finds an optimal variable node degree distribution for the whole code. The inner loop performs the maximization of the minimum variable node degree of a protection class. The optimal ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$ is found by executing the optimization algorithm for a given check node degree distribution ${\mathbf{\rho}}^{{\mathcal{C}}_{z}}$, E_{ b }/N_{0}=δ+ε, a maximum variable node degree ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}$, a code rate R_{ z }, and proportion vectors ${\mathbf{\alpha}}^{{\mathcal{C}}_{z}}$ and ${\mathbf{\beta}}^{{\mathcal{C}}_{z}}$.
The operation of the optimization algorithm is summarized as follows.

1.
For any given E _{ b }/N _{0}=δ+ε compute ${\stackrel{~}{\mathbf{\sigma}}}^{2}$.

2.
Find ${\mathbf{\lambda}}^{{\mathcal{C}}_{z}}$ by executing the iterative linear programming optimization algorithm.
The structure of the optimization algorithm with its constraints is defined as follows.

1.
Initialization: ${d}_{{v}_{\text{min}}}^{\left(k\right){\mathcal{C}}_{z}}={d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}$

2.
While optimization failure:

(a)
Optimize
$$\begin{array}{l}\underset{{\mathbf{\lambda}}^{{\mathcal{C}}_{z}}}{max}\sum _{j=1}^{\underset{s}{\overset{{\mathcal{C}}_{z}}{N}}}\sum _{i=2}^{\underset{{v}_{\text{max}}}{\overset{{\mathcal{C}}_{z}}{d}}}\underset{{M}_{j,i}}{\overset{{\mathcal{C}}_{z},{C}_{k}}{\lambda}}\end{array}$$(32)
under the constraints [ C_{1}]−[ C_{7}].
[ C_{1}] Rate constraint, from [2] (see (27))
[ C_{2}] Proportion distribution constraints

i.
from[2] (see (24))
$$\begin{array}{l}\sum _{j=1}^{{N}_{s}^{{\mathcal{C}}_{z}}}\sum _{k=1}^{{N}_{c}^{{\mathcal{C}}_{z}}}{{\lambda}_{{M}_{j}}^{{\mathcal{C}}_{z},{C}_{k}}}^{T}\mathbf{1}=1\end{array}$$(34) 
i.
$$\forall k\in \{1,\xb7\xb7\xb7,{N}_{c}^{{\mathcal{C}}_{z}}1\}$$
, from [2] (see (25) and (28))
$$\phantom{\rule{10.0pt}{0ex}}\begin{array}{c}\sum _{j=1}^{{N}_{s}^{{\mathcal{C}}_{z}}}\sum _{i=2}^{{d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{z}}}\frac{{\lambda}_{{M}_{j,i}}^{{\mathcal{C}}_{z},{C}_{k}}}{i}={\alpha}_{k}^{{\mathcal{C}}_{z}}\frac{{R}_{z}}{1{R}_{z}}\sum _{i=2}^{{d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{z}}}\frac{{\rho}_{i}^{{\mathcal{C}}_{z}}}{i}\end{array}$$(35)i.
$$\forall j\in \{1,\xb7\xb7\xb7,{N}_{s}^{{\mathcal{C}}_{z}}\}$$
, from [2] (see (26) and (29))
[ C_{3}] Convergence constraint, from[2] (see (23))
[ C_{4}] Stability condition, from[2] (see (30) and (31))
[ C_{5}] Ratecompatible constraint, from[3] (see (19))
[ C_{6}] Minimum variable node degree constraint from [2]
[ C_{7}] Previous optimization constraints from [2]
(a) If failure,
End (While)
When optimal variable node degree distributions for code ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ are found, we construct the parity check matrices H_{1} and H_{2}, as shown in Figure 4, using the ACE algorithm[20]. The ACE algorithm is modified to first construct H_{1} and then H_{2}, while keeping H_{1} fixed.
4 Simulation results
In the following subsections, the performance of the proposed code design has been evaluated for low and highrate codes of medium block. The performance of the codes has also been evaluated for short and long block length codes. The proposed codes’ performance has been compared to QCLDPC codes[9], puncturingbased RC LDPC codes [6] and extensionbased RC LDPC codes [3]. The proposed codes are simulated with equal transmission power setting at the source and the relay, i.e., P_{S 1}=P/2 and P_{ R }=P/2. The channel gains γ_{ SD }, γ_{ SR }, and γ_{ RD } from (4) are dependent upon path loss propagation only and we consider τ=2. Using the relay system model shown in Figure 3, simulations have been performed for different positions d of the relay relative to the source. Note that in the simulations for relay network, 20 decoding iterations at the relay and 20 decoding iterations at the destination are considered. In all the direct link simulations, 40 decoding iterations at the destination are considered.
4.1 Lowrate medium block length codes
This subsection describes the simulation results of the proposed codes of lowrate and medium block length for cooperative relay networks. The results are also compared to the results for quasicyclic LDPC codes (QCLDPC) from [9]. The code ${\mathcal{C}}_{1}$ for BC mode, described by the paritycheck matrix H_{1}, is designed by optimizing the variable node degree distributions for R_{1}=1/2, ${N}_{c}^{{\mathcal{C}}_{1}}=2$, ${N}_{s}^{{\mathcal{C}}_{1}}=1$, ${\alpha}^{{\mathcal{C}}_{1}}=1$, ${\beta}^{{\mathcal{C}}_{1}}=1$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{1}}=11$, and ${\rho}^{{\mathcal{C}}_{1}}={x}^{6}$. The code ${\mathcal{C}}_{2}$ for the halfduplex relay network is extended from the code ${\mathcal{C}}_{1}$ by appending extra parity bits to the codewords of the code ${\mathcal{C}}_{1}$. Variable node degree distributions of ${\mathcal{C}}_{2}$ are optimized for R_{2}=1/4, ${N}_{c}^{{\mathcal{C}}_{2}}=3$, ${N}_{s}^{{\mathcal{C}}_{2}}=2$, ${\alpha}^{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}0.5\phantom{\rule{1em}{0ex}}0.5\right]$, ${\beta}^{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}0.5\phantom{\rule{1em}{0ex}}0.5\right]$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{2}}=15$ and check node degree distribution ${\rho}^{{\mathcal{C}}_{2}}$. Suitable ${\rho}^{{\mathcal{C}}_{2}}$ for ${\mathcal{C}}_{2}$ are chosen by evaluating the performance of ${\mathcal{C}}_{2}$ for different check node degree distributions for different positions of relay as explained in the following subsection.
Like in [3], the code ${\mathcal{C}}_{1}$ in this paper is designed by considering the SNR of the source to destination channel, therefore there is only one channel class ${N}_{s}^{{\mathcal{C}}_{1}}=1$. Note that the SNR of the source to relay channel is always higher than the SNR of the source to destination channel as h_{ SR }^{2}>h_{ SD }^{2}, therefore the successful decoding of the codeword w_{1} of the code ${\mathcal{C}}_{1}$ is possible at the relay even when it fails at the destination. The codeword w_{1} is composed of equally protected information bits and less protected parity bits, by choosing ${N}_{c}^{{\mathcal{C}}_{1}}=2$. The code ${\mathcal{C}}_{2}$ is designed by taking into account the SNRs of the source to destination and the relay to destination channels, i.e., ${N}_{s}^{{\mathcal{C}}_{2}}=2$. The codeword w_{2} of ${\mathcal{C}}_{2}$, which is decoded at the destination, is composed of w_{1} and w_{ e }. We let ${N}_{c}^{{\mathcal{C}}_{2}}=3$ for w_{2} assuming that it is composed of equally important information bits and less important parity bits of w_{1} and least important extra parity bits w_{ e }. w_{1} is received at the destination from the source, while w_{ e } is received at the destination from the relay. In relay networks, the source to destination channel is worse than the relay to destination channel. Therefore, the equally important information bits and less important parity bits of w_{1} in w_{2} are assigned to protection classes C_{1} and C_{2}, respectively, in order to give them good protection against the channel conditions of the source to destination channel and the least important bits w_{ e } in w_{2} are assigned to protection class C_{3}, as the relay to destination channel has a better SNR than the source to destination channel.
We compare the performance of the proposed codes for relay networks with an LDPC code designed for the direct link. The design of the code for the direct link is the same as for ${\mathcal{C}}_{1}$ except that the code is designed with code rate 1/4 the same as for ${\mathcal{C}}_{2}$, a suitable check node degree distribution ρ=0.1350x^{2}+0.3695x^{3}+0.0684x^{4}+0.2812x^{5}+0.1459x^{6} (from LOPT [21]), full transmission power P and ${d}_{{v}_{\text{max}}}=15$. In the simulations, the lengths of the codewords w_{1} and w_{2} are N_{1}=1296 and N_{2}=2592, respectively. In the simulations of the direct link, an LDPC codeword of length N_{2} is used.
4.1.1 Selection of good check node degree distributions
In this subsection, we analyze the performance of the proposed codes for different check node degree distributions and for different positions d of the relay. Optimization of the check node degree distributions is outside the scope of this paper. The aim of this analysis is to select a good check node degree distribution for each d.
Code ${\mathcal{C}}_{1}$ is designed with a concentrated check node degree distribution ${\mathbf{\rho}}^{{\mathcal{C}}_{1}}={x}^{6}$, since concentrated check node degree distributions have been shown to perform well[16]. The code ${\mathcal{C}}_{2}$ is designed with a nonconcentrated check node degree distribution because it is shown in [3] that cooperative relay networks show better performance with LDPC codes designed with nonconcentrated check node degree distributions. The maximum check node degrees are chosen as ${d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{1}}={d}_{{c}_{\text{max}}}^{{\mathcal{C}}_{2}}=7$ in order to keep the same maximum check node degree as the code in [9] to simplify the comparison.
For the design of code ${\mathcal{C}}_{2}$, optimal variable node degree distributions are found by choosing a check node degree distribution for which code ${\mathcal{C}}_{2}$ gives good performance in terms of bit error rate for a specific position d of relay. The performance of code ${\mathcal{C}}_{2}$ is analyzed for four different check node degree distributions for different positions d. The four nonconcentrated check node degree distributions for ${\mathcal{C}}_{2}$ satisfy the ratecompatible constraint for the check node degree distribution as defined by (17). It is observed that our designed codes have error floor at high bit error rate if the four check node degree distributions are chosen with a minimum check node degree less than 4 and have the waterfall region at high SNR if the four check node degree distributions are chosen with a minimum check node degree greater than 4. Therefore, to design the codes with the error floor at lower bit error rate and the waterfall region at lower SNR, the four check node degree distributions are chosen with a minimum check node degree equal to 4 but with different fraction of edges of the minimum degree. Furthermore, 4 is the highest possible minimum degree with which the four check node degree distributions still satisfy (17). The optimal variable node degree distribution at each d is found for each one of the four check node degree distributions.
We observe that for d<0.6, the code designed with the check node degree distributions, which have the lowest fraction of edges of the minimum degree, has its waterfall region at high SNR as compared to the codes with check node degree distributions with a higher fraction of edges of the minimum degree. Figure 6 shows the performance of the proposed code with four different check node degree distributions for d=0.2. It is noticed in Figure 6 that the proposed code design gives good performance with ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.4177{x}^{3}+0.5823{x}^{6}$. This holds also for d=0.3 and 0.5, but simulation results are omitted here. Figure 7 shows the performance of the proposed codes with four different check node degree distributions for d=0.4 and good performance is observed with ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.5{x}^{3}+0.5{x}^{6}$. This holds also for d=0.1 but simulation results are omitted here. For d=0.6,0.7,0.8,0.9 it is observed in Figure 8 that the choice of check node degree distribution does not have a large impact on the performance.
4.1.2 Results for lowrate medium block length codes
This subsection describes the simulation results of the proposed codes with lowrate and medium block length in a relay network and compares their performance with a standard LDPC code for the direct link. The proposed codes at each d are designed with the suitable check node degree distributions found in the previous subsection. The simulation results for the proposed codes are shown in Figure 9 for different positions of the relay. The results show that the relay network achieves better performance with the proposed codes than the direct link with a LDPC code optimized for this link. At a BER of 10^{−5}, the relay network has best performance for d=0.4. At a BER of 10^{−4}, the proposed code achieves 3.8dB gain over the direct link transmission for the position of relay d=0.4, as shown in Figure 9. A degradation in the performance of the proposed codes in the waterfall region can be observed when the relay is located closer to the destination than the source, i.e., for relay positions d>0.5, as shown in Figure 9. Note that for all d, both the source and the relay have fixed transmit power, i.e., the total power P is divided in half for the source and the relay for all d. The performance of the cooperative relay network depends mostly on the performance of the source to relay channel. At d>0.5, the signal received at the relay has higher path loss than at d≤0.5, so the probability of erroneous decoding of $\stackrel{~}{{\mathbf{w}}_{1}}$ in the relay increases with increasing d. The erroneously decoded $\stackrel{~}{{\mathbf{w}}_{1}}$ is then used to calculate w_{ e }, which in turn is transmitted to the destination and increases the probability of erroneous decoding of the entire codeword w_{2}. This causes the performance degradation of the proposed codes at d>0.5. At d≥0.7, the relay network has worse performance than the direct link, at least for some SNRs.
4.2 Highrate medium block length codes
The performance of the proposed code design has also been analyzed for a highrate code of medium block length. The code ${\mathcal{C}}_{1}$ is designed by optimizing the variable node degree distributions for R_{1}=0.9, ${N}_{c}^{{\mathcal{C}}_{1}}=2$, ${N}_{s}^{{\mathcal{C}}_{1}}=1$, ${\mathbf{\alpha}}^{{\mathcal{C}}_{1}}=1$, ${\mathbf{\beta}}^{{\mathcal{C}}_{1}}=1$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{1}}=7$, and ${\mathbf{\rho}}^{{\mathcal{C}}_{1}}={x}^{29}$. Variable node degree distributions of ${\mathcal{C}}_{2}$ are optimized for R_{2}=0.6, ${N}_{c}^{{\mathcal{C}}_{2}}=3$, ${N}_{s}^{{\mathcal{C}}_{2}}=2$, ${\alpha}^{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}0.9\phantom{\rule{1em}{0ex}}0.1\right]$, ${\beta}^{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}2/3\phantom{\rule{1em}{0ex}}1/3]$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{2}}=50$ and check node degree distribution ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}$. The lengths of the codewords w_{1} and w_{2} are N_{1}=1,730 and N_{2}=2,592, respectively.
Good ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}$ for ${\mathcal{C}}_{2}$ at each d has been chosen by using the procedure adopted in Section 4.1.1 for finding good ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}$ for medium block length codes of low rate. We have analyzed the performance of ${\mathcal{C}}_{2}$ with four different nonconcentrated check node degree distributions ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.25{x}^{10}+0.75{x}^{29},{\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.5{x}^{10}+0.5{x}^{29},{\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.4177{x}^{10}+0.5823{x}^{29}$, and ${\mathbf{\rho}}^{{\mathcal{C}}_{2}}=0.1{x}^{10}+0.9{x}^{29}$ for different positions d. All the check node degree distributions for ${\mathcal{C}}_{2}$ satisfy the ratecompatible constraint for the check node degree distribution as defined by (17). In the same way as for the code with low rate in Section 4.1.1, the code ${\mathcal{C}}_{2}$ with high rate is also designed such that it has the error floor at lower bit error rates and the waterfall region at a lower SNR by keeping the minimum check node degree in all the check node degree distributions equal to 11. Note that 11 is the highest possible minimum degree with which the four check node degree distributions satisfy (17). The good check node degree distributions being found for d=0.1,0.2,0.4 and 0.5 are ${\rho}^{{\mathcal{C}}_{2}}=0.25{x}^{10}+0.75{x}^{29}$, ${\rho}^{{\mathcal{C}}_{2}}=0.25{x}^{10}+0.75{x}^{29}$, ${\rho}^{{\mathcal{C}}_{2}}=0.5{x}^{10}+0.5{x}^{29}$, and ${\rho}^{{\mathcal{C}}_{2}}=0.25{x}^{10}+0.75{x}^{29}$, respectively. As for the lowrate code proposed in Section 4.1.1, the choice of check node degree distribution does not have a large impact on the performance for d=0.6,0.7,0.8 and 0.9. The performance of the proposed code has been evaluated for d=0.1,0.2,0.4,0.5,0.7 and 0.9 as shown in Figure 10. It is shown that the relay network has the best performance for d=0.4, like the lowrate code in Section 4.1.2. It is reported in [9] that codes with different code rates have similar sensitivity to the relay position for a given transmit power distribution of the source and the relay; hence, for both cases of low and highrate codes, the optimal position where the proposed code achieves best performance is d=0.4. The performance of the proposed highrate code is compared to the performance of an LDPC code designed for the direct link. The design of the code for the direct link is the same as for ${\mathcal{C}}_{1}$ except that the code is designed with code rate 0.6, a suitable ρ=0.5143x^{10}+0.0986x^{18}+0.0717x^{24}+0.2339x^{26}+0.0816x^{29} (from LOPT [22]), full transmission power P and ${d}_{{v}_{\text{max}}}=50$. In the simulations of the direct link, the codeword length is 2,592 bits. At a BER of 10^{−5}, the proposed highrate code shows 1.3dB gain over the direct link transmission for d=0.4, as shown in Figure 10.
4.3 Short and long block length codes
In this subsection, the performance of the proposed code design has been analyzed for short and long block lengths. The codeword lengths for the short block length are N_{1}=512 and N_{2}=1,024 bits, and for the long block length, we consider N_{1}=5,344 and N_{2}=10,688 bits. Note that the proposed code optimization algorithm is independent of the code length. The parameters used for the design of the proposed code with short and long block lengths are the same as those used for medium block length in Section 4.1. For comparison, codes for the direct link with codeword lengths of 1,024 and 10,688 bits have been constructed, according to the degree distributions found for the medium length code in Section 4.1.
It is shown in [9] that the optimal position of the relay depends on the power distribution of the source and the relay in the relay network. For the proposed code with medium block length, it has been shown in the previous subsections that the optimal position is d=0.4. We have analyzed the performance of the proposed code with short and long block lengths at the optimal position of the relay, i.e., at d=0.4. At a BER of 10^{−4}, the proposed code with short and long block lengths have 3.8dB gain over the direct link transmission for d=0.4, as shown in Figure 11.
4.4 Comparison
This subsection provides a performance comparison of the codes proposed in this paper to QCLDPC codes [9], extensionbased RC LDPC codes [3] and puncturingbased RC LDPC codes [6].
4.4.1 Comparison to QCLDPC codes
In this subsection, we compare the performance of the RC UEPLDPC codes to the QCLDPC codes proposed in [9], which are implemented through a repetition coding scheme in a halfduplex cooperative relay network. The simulation results for the QCLDPC codes of [9] are reproduced in this paper, as shown in Figure 12. The proposed RC UEPLDPC codes, like the QCLDPC codes, are structured and irregular. They are however implemented in the halfduplex cooperative relay network through a ratecompatible coding scheme. We assume, like in [9], that the source is silent in MAC mode and use the same codeword length. In contrast to [9], the relay terminal in our case transmits the extra parity bits instead of retransmitting the decoded codeword in MAC mode. The RC UEPLDPC codes are more complex from a practical implementation point of view compared to the QCLDPC codes in [9], but they require a lower E_{ b }/N_{0} to achieve the same bit error probability.
Equal time slots are assumed in BC and MAC modes as well as equal transmit powers at the source and the relay, i.e., P_{S 1}=P/2 and P_{ R }=P/2. In [9], the length of each codeword being transmitted in BC and MAC modes is 1,296 bits and the rate is 1/2. For the RC UEPLDPC codes, the length of w_{1} is equal to 1,296 with rate 1/2. Additional parity bits w_{ e } of length 1,296 are transmitted in MAC mode. Hence, the total number of bits received at the destination, both for the QCLDPC codes and for the RC UEPLDPC codes, is equal to 2,592 bits.
In [9], the LDPC code is decoded by the layered belief propagation (LBP) algorithm as defined in [23]. LBP is a variation of the standard belief propagation [1] which is designed for architecture aware (AA) LDPC codes with quasicyclic properties. It is shown that with LBP, the decoding of AA LDPC codes is improved by two times in the number of iterations required for a given error rate [24]. However, we have used standard belief propagation decoding to reproduce the results of [9] and to analyze the performance of the proposed LDPC codes in cooperative relay networks. To get the equivalent performance of LBP using standard belief propagation decoding, the number of decoding iterations is set to twice the number of iterations defined in [9]. In [9], ten decoding iterations at the relay and ten decoding iterations at the destination are used. We use 20 decoding iterations at both the relay and the destination.
From the comparison of Figures 9 and 12, the RC UEPLDPC codes give better performance than the QCLDPC codes in [9]. The QCLDPC codes also show best performance for d=0.4. Under the same time slots, transmission power and using the DF strategy for the relay, the proposed codes outperform the codes in [9] with a gain of 1.9 dB at a BER of 10^{−4} for d=0.4, as shown in Figure 12. For the positions of the relay other than d=0.4, the proposed codes have different gains over the codes in [9], which can be observed by comparing the results for each d in Figures 9 and 12. The improved performance of the proposed codes over the QCLDPC codes in [9] is due to separate optimization of the proposed codes for each relay position. Furthermore, the proposed code design is based on the ratecompatible coding scheme, which provides a better tradeoff between bit error rate and data rate than the repetition coding scheme.
4.4.2 Comparison to extensionbased RC LDPC Codes
It is observed from Figure 9 that the performance of the proposed codes is close to the performance of the code proposed in [3] for the range of d between 0.2 and 0.3 (see the simulation of the optimized code of length 20,000 in [3] for a comparison). We perform the optimization of the code through an iterative linear programming algorithm in contrast to the code design in [3], which is achieved through differential evolution, a complex nonlinear constrained optimization algorithm. There are three reasons why the RC UEPLDPC codes are not exactly approaching the performance of the codes in [3]. First, the codeword length considered in [3] is almost ten times higher than the codeword length considered for the RC UEPLDPC codes. Second, in [3], it is assumed that each codeword is perfectly decoded at the relay. Third, in [3], it is assumed that the source is active also in MAC mode.
4.4.3 Comparison to puncturingbased RC LDPC codes
The performance of the proposed codes is also compared to puncturingbased RC LDPC codes proposed in [6] for halfduplex cooperative relay networks. The puncturingbased ratecompatible coding scheme proposed in [6] requires the design of only one singleuser LDPC code as a mother code . The mother code with code rate R_{ c } is first designed either by linear programming or by using an optimized LDPC degree distribution from [25]. Then, a proportion of the parity bits is punctured randomly, which raises the code rate of to R_{ p }. In BC mode, the punctured codeword of is transmitted to the destination and the relay. The destination stores the punctured codeword until it receives the punctured parity bits from the relay. The punctured parity bits are recovered at the relay and are sent as extra parity bits to the destination in MAC mode. Thus, the destination receives the entire codeword corresponding to the mother code with different SNRs for the punctured codeword and the extra parity bits as expressed by (8) and (9). The coding scheme proposed in [6] only requires the design of a mother code and the code design proposed in [6] is therefore simpler than the code design proposed here, which requires joint design of two codes.
We use the linear programming optimization algorithm defined in this paper without the constraints for UEP and ratecompatibility to find an optimized variable node degree distribution for the code with a suitable ρ=0.2465x^{3}+0.2990x^{5}+0.1307x^{6}+0.1774x^{7}+0.1464x^{8} (from LOPT [25]), R_{ c }=1/4 and ${d}_{{v}_{\text{max}}}=33$. The code is punctured at the source and the code rate is raised to R_{ p }=0.5. The performance of the coding scheme proposed in [6] is shown in Figure 13 for a codeword length of 2,048 bits and d=0.4. It is assumed in [6], as in the coding strategy proposed here, that the source is silent in MAC mode, the codeword is imperfectly decoded at the relay and that extra parity bits from the relay are sent without encoding to the destination.
To compare the performance of the RC UEPLDPC code proposed here to the coding scheme proposed in [6], the simulations of the RC UEPLDPC codes are set with codeword lengths N_{1}=1,024 and N_{2}=2,048 for w_{1} and w_{2}, respectively, code rates R_{1}=0.5 and R_{2}=0.25 and d=0.4. Other parameters that are specific to the code design proposed here are as follows. Code ${\mathcal{C}}_{1}$ is designed by optimizing the variable node degree distributions for ${N}_{c}^{{\mathcal{C}}_{1}}=2$, ${N}_{s}^{{\mathcal{C}}_{1}}=1$, ${\mathbf{\alpha}}^{{\mathcal{C}}_{1}}=1$, ${\mathbf{\beta}}^{{\mathcal{C}}_{1}}=1$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{1}}=30$, and ${\mathbf{\rho}}^{{\mathcal{C}}_{1}}={x}^{8}$. Code ${\mathcal{C}}_{2}$ is designed by optimizing the variable node degree distributions for ${N}_{c}^{{\mathcal{C}}_{2}}=3$, ${N}_{s}^{{\mathcal{C}}_{2}}=2$, ${\mathbf{\alpha}}^{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}0.5\phantom{\rule{1em}{0ex}}0.5\right]$, ${\mathbf{\beta}}_{{\mathcal{C}}_{2}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}0.5\phantom{\rule{1em}{0ex}}0.5\right]$, ${d}_{{v}_{\text{max}}}^{{\mathcal{C}}_{2}}=33$, and check node degree distribution ${\rho}^{{\mathcal{C}}_{2}}$. The performance of ${\mathcal{C}}_{2}$ has been analyzed for four different check node degree distributions ${\rho}^{{\mathcal{C}}_{2}}=0.5{x}^{4}+0.5{x}^{8},{\rho}^{{\mathcal{C}}_{2}}=0.25{x}^{4}+0.75{x}^{8},{\rho}^{{\mathcal{C}}_{2}}=0.4177{x}^{4}+0.5823{x}^{8}$, and ${\rho}^{{\mathcal{C}}_{2}}=0.1{x}^{4}+0.9{x}^{8}$ at d=0.4 by using the procedure adopted in Section 4.1.1 to find a good ${\rho}^{{\mathcal{C}}_{2}}$ for ${\mathcal{C}}_{2}$. The good check node degree distribution for ${\mathcal{C}}_{2}$ at d=0.4 is ${\rho}^{{\mathcal{C}}_{2}}=0.5{x}^{4}+0.5{x}^{8}$. The code proposed in [6] shows worse performance than the code proposed here, as shown in Figure 13. The reason for this may be that the code proposed in [6] uses punctured codewords in BC mode, which results in a certain performance loss in the source to relay channel. Note that the overall performance of the cooperative relay network depends largely on the performance of the source to relay channel [4, 5]. Our proposed coding scheme uses an optimized code in BC mode to achieve the best performance in the source to relay channel.
5 Conclusions
Using a relay node increases the reliability and the data rate when the direct link between the source and the destination is bad due to fading and interference. This paper proposes an LDPC code design algorithm for halfduplex cooperative relay networks, which is based on density evolution using linear programming. The proposed codes are ratecompatible with unequal error protection and the codes are designed through optimization of the variable node degree distribution. The proposed codes are jointly optimized for the source to relay and the source to destination channels.
The performance of the proposed codes is evaluated with short, medium and long block lengths, and with low and high rates under realistic assumptions covering imperfect decoding at the relay and variation of SNR within a single codeword. The proposed codes of rate 0.25 outperform the direct link communication for short, medium and long block lengths with 3.8dB gain at a BER of 10^{−4}, when the relay is located at position d=0.4. The proposed code of medium block length and highrate outperforms the direct link communication with 1.3dB gain at a BER of 10^{−5} for d=0.4. In contrast to the QCLDPC codes proposed in [9], the proposed codes are optimized for each position of the relay relative to the source by considering the variation of SNR within a codeword due to the transmission over different relay channels. Under the same time slots, transmission power and processing of the codewords at the relay terminal with the decode and forward strategy, the proposed codes achieve a 1.9dB gain over the repetition coding schemebased QCLDPC codes in [9] for the position of the relay d=0.4 at a BER of 10^{−4}. In contrast to the puncturingbased RCLDPC codes proposed in [6], the proposed code uses an optimized LDPC code in BC mode to achieve the best performance in the source to relay channel. Although the proposed code design is based on a lowcomplexity linear programming algorithm, its performance is close to the performance of codes proposed in [3] being optimized by the more complex differential evolution algorithm and using much longer codewords.
Endnote
^{a} In this paper, vectors are represented by bold letters.
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Acknowledgements
This study was supported by the Department of Computer Science, Electrical and Space Engineering, Luleå University of Technology, Sweden.
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Keywords
 LDPC code design
 Halfduplex relay
 Linear programming