Spatially coupled LDPC coding in cooperative wireless networks
Abstract
This paper proposes a novel technique of spatially coupled lowdensity paritycheck (SCLDPC) codebased soft forwarding relaying scheme for a twoway relay system. We introduce an arraybased optimized SCLDPC codes in relay channels. A more precise model is proposed to characterize the residual noise on the soft symbols, using a precalculated lookup table at the destination. This requires less signaling overhead compared to previous soft noise modeling techniques. We also introduce a variance correction factor to provide a rectification to the equivalent total noise variance at the destination. Finally, we propose an appropriate log likelihood ratio (LLR) former at the destination which is tailored to the proposed soft parity generation technique. Simulation results demonstrate that the proposed relay protocol yields an improved bit error rate (BER) performance compared to competing schemes proposed in the literature.
Keywords
Relay channels LDPC codes Decode and forward Network coding1 Introduction
Cooperative communications in wireless networks provides improved transmit diversity and spectral efficiency [1]. Judiciously designed signal forwarding at the relay and an accurate detection at the destination can greatly enhance the system performance. Popular relay protocols are amplifyandforward (AF) and decodeandforward (DF) [2]. In the AF protocol, the relay transmits an amplified version of the received signal to the destination. In the AF, no noise suppression is performed; therefore, that protocol suffers from severe noise propagation and power inefficiency under poor channel conditions. By using a detector/decoder at the relay, the DF protocol allows for regenerating the transmitted signal, such that the noise propagation can be avoided. However, any decoding error in the regenerated signal can cause a performance degradation at the destination.
A promising relay protocol called soft information relaying (SIR) has recently gained significant attention [3, 4, 5, 6, 7, 8, 9, 10]. In [3], the authors studied the implementation of SIR in conjunction with distributed turbo coding (DTC). A soft forwarding technique based on symbolwise mutual information (SMI) was investigated in [4] using physical layer network coding (PLNC) in the twoway relay channel. Recently, a soft decodecompressforward scheme was proposed in [5]; that work featured a new model, referred to as the soft scalar model, in order to facilitate the log likelihood ratio (LLR) computation at the destination. In [6], the authors suggested a “soft fading” model for the overall channel, experienced by a BPSK symbol, when transmitted from the source to relay to the destination. That idea was extended to soft twostep reencoding together with perscaling at the relay in [7]. In order to decode the SIR signals, the destination uses an independent Gaussian approximation, which models the residual noise in the estimated symbols as in [3, 5].
Errorcorrecting code is a pivotal ingredient of the DF cooperative relay system. The right choice of the code can greatly improve the performance. Lowdensity paritycheck (LDPC) codes, which became extremely popular in pointtopoint communications due to their excellent performance, are a natural candidate for use in such systems.
It was observed in [12] that the socalled “convolutional LDPC codes” outperform classical LDPC block codes. Recently, it has been shown that regular spatially coupled lowdensity paritycheck (SCLDPC) achieve capacity of the binary erasure channel (BEC) [13] and of general binaryinput memoryless outputsymmetric channel [14] under slidingwindow messagepassing decoding. Therefore, SCLDPC codes can be viewed as excellent candidates for use in the relay schemes.
In [15], it was shown that bilayer SCLDPC codes can achieve the Shannon limit of a DF relay system with orthogonal BEC links. As the SCLDPC code ensembles are regular, the design complexity is low compared to the schemes based on irregular LDPC code ensembles. However, the idea of using the SCLDPC codes in the relay systems over the additive white Gaussian noise (AWGN) and Rayleigh fading channels was not well exploited yet. In [16], protographbased SCLDPC codes were employed in threeterminal erasure relay channel. It was empirically observed that spatially coupled protographbased MacKayNeal LDPC codes approach the theoretical performance limits. In [17], SCLDPC codes were used in a decodeandforward scheme for a network consisting of two sources, one relay and one destination, connected by binary erasure channels. Network coding was used in the relay node. Analytical bounds on the achievable rate region were derived, and it was shown, in particular, by using density evolution, that the scheme based on the SCLDPC codes achieves the theoretical performance limits. The theoretical results were further justified by the experimental observations.
In another related work [18], the author considers arraybased SCLDPC codes and studies the minimum distance of the corresponding spatially coupled codes, obtained by various “unwrapping” vectors. A number of bounds on the minimum distance of the corresponding codes for special selections of parameters are obtained by mathematical analysis.
In this paper, we propose a new framework for designing a SCLDPC codebased soft SDF protocol over AWGN and Rayleigh/fading channels in the twoway relay channels. The paper proposes a more realistic soft noise approximation model using an estimated lookup table at the destination. The impacts of the different noise modelings are compared by the simulations. Also, a variance coefficient at the destination (referred to as a variance correction factor) is proposed to improve the accuracy of the Gaussian approximation made initially in the previous work in [5] on the equivalent noise modeling at the destination. In this paper, we also study the design of arraybased SCLDPC block codes for relay channels. The code construction is based on coupling of arraybased LDPC block codes akin [18, 19, 20]. We experimentally compare the performance of the proposed system for different selections of the “unwrapping” vectors in the employed quasicycle lowdensity paritycheck (QCLDPC) matrix code, decoded by messagepassing algorithm. The present work can be viewed as a set of simple but effective enhancements to the SIR scheme.
2 System model
In the first time slot, user A encodes a bit vector u _{ A } of length K using an LDPC encoder of rate R=K/N to produce the codeword c _{ A } of length N; this LDPC code is defined by a (N−K)×N paritycheck matrix H, i.e., we have \(\boldsymbol {H}\boldsymbol {c}_{A}^{T}=\boldsymbol {0}\). Here, 0 denotes the zero matrix, whose size is clear from the context. The bit vector c _{ A } is mapped to a BPSK symbol vector x _{ A }, via the mapping 0↦+1, 1↦−1, before the transmission takes place. We assume that all nodes have only one antenna working in a halfduplex mode. In the second time slot, the process at user B is similar to that at user A in the first time slot.
where n _{ iR } and \(\boldsymbol {n}_{i\bar {i}}\) are vectors having independent and identically distributed (i.i.d.) real Gaussian (noise) entries with zero mean and variance \( {\sigma}_{iR}^2 \) and \(\sigma _{i\bar {i}}^{2}\), respectively (both of which are here assumed to be equal to N _{0}/2, where N _{0} denotes the channel noise power spectral density). Here, \(i,\,\bar {i}\,\in \{A,\, B\}\) with \(i\neq \bar {i}\). Also, P _{ i } is the transmit power constraint at node i, and h _{ ik } stands for the Rayleigh fading coefficient between nodes i and k where \(\,k\,\in \{\bar {i},\, R\}\). In the third time slot, the relay aids the destination by transmitting a networkcoded message based on the signals received in the first and second time slots, i.e., c _{ R }=c _{ A }⊕c _{ B } where ⊕ denotes the XOR operation. This is equivalent to the multiplication of the corresponding BPSK symbols, i.e., \({x_{R}^{\,j}}=\hat {x}_{A}^{\,j}\hat {x}_{B}^{\,j}\), where \(\hat {x}_{i}^{j}\) is the (BPSK symbol) hard decision corresponding to \({x_{i}^{\,j}}\) at the relay.
In the low signaltonoise (SNR) regime, errors may occur in the decoding process at the relay and error propagation may be introduced. As a remedy, in the proposed scheme, the relay transmits a “soft” version of these networkcoded parity symbols. This process will be elucidated in Section 4.
3 Code design
LDPC block codes were discovered in the sixtieth but were almost forgotten [11]. They became widely used in the practical applications in the last two decades due to their excellent performance at rates close to the theoretical limit, when decoded by the messagepassing algorithms. LDPC codes possess a sparse paritycheck matrix.
Convolutional LDPC codes, or QCLDPC codes, outperform classical LDPC block codes, when decoded by using slidingwindow messagepassing algorithms. The use of sliding window allows for low latency, since the decoding can be successfully performed on a small part of a codeword, without waiting for the remaining part of the codeword. At the same time, for carefully chosen decoding window size, the error performance of the slidingwindow decoder is similar to that of the standard messagepassing decoder. Therefore, QCLDPC codes with slidingwindow decoders are an interesting alternative to classical block LDPC codes with messagepassing decoders.
In this paper, we consider arraybased QCLDPC codes, as described below.
Definition 1 (see, for example, [19]).

The row i+1 of the matrix H _{ l } contains δ _{ i } first entries as in the row i+1 of the matrix H _{ γ,p }, for all 0≤i≤γ−1. The remaining entries in H _{ l } are set to zeros.

The row i+1 of the matrix H _{ u } contains γ p−δ _{ i } last entries as in the row i+1 of the matrix H _{ γ,p }, for all 0≤i≤γ−1. The remaining entries in H _{ u } are set to zeros.
This procedure is illustrated by the following example.
Example 1.
Take p=5, γ=3, δ=(1,3,4). Then,
It is possible to define the γ p×κ p submatrix H _{ γ,p,κ } of H _{ γ,p }, for general κ<p, by taking its first κ p columns. The corresponding code is LDPC, and the same unwrapping procedure can be applied to it.
It is worth mentioning that the matrix H(γ,p, L,δ) has the same row and column weights as the original matrix H _{ γ,p } (except for the row weights of the first p rows and the last p rows of H(γ,p, L,δ)).
The matrix H(γ,p, L,δ) might not have full rank. Its rank, in particular, depends on the decomposition of H _{ γ,p } and the internal structure of the terms in that decomposition. The optimal choice of δ is not obvious.
The different cutting vectors δ to form spatially coupled LDPC codes
Patten index  δ 

a  (2,3,5) 
b  (3,3,3) 
c  (1,2,3) 
d  (4,3,2) 
e  (5,2,2) 
f  (5,3,2) 
g  (5,4,2) 
3.1 Channelnetworkcoded cooperative scheme
In each time slot, the destination (source i) receives a signal; one signal comes from the opposite source \( \bar {i} \) in the uplink transmission and another signal comes at the downlink transmission. In the case of errorfree decoding at the relay, the forwarded packet by the relay is viewed as redundancy produced by the channel coding of the networkcoded sequence. Hence, the channelnetworkcoded cooperative scheme can be considered as a networkcoded distributed LDPC (NCLDPC) coding scheme. Note that in contrast to the channel codes, the network code combines the information bits of A and B.
4 Soft information relaying scheme
The intuition behind (9) is as follows. Since \(\tilde {x}_{R}^{\,j}\) is an estimate of \({x_{A}^{\,j}}{x_{B}^{\,j}}\) from \(\tilde {x}_{A}^{\,j}\) and \(\tilde {x}_{B}^{\,j}\), its sign should be equal to the sign of \(\tilde {x}_{A}^{\,j}\tilde {x}_{B}^{\,j}\). On the other hand, the magnitude of the signal does not change significantly.
where n _{ Ri } is a vector having i.i.d. real Gaussian entries each having zero mean and variance \( {\sigma}_{Ri}^2={N}_0/2 \).
5 Calculation of LLR at the destination
where \(\tilde {n}^{j}\) is called the soft error variable and the constant η is called the soft scalar. We compute the value of η which minimizes the meansquare value of the soft error, i.e., \(\eta =\mathbb {E}(x_{R}\tilde {x}_{R})\) (c.f. [5]). In general, the soft scalar may be computed offline as \(\eta = \frac {1}{N}\sum _{i=1}^{N}[x_{R,i}\tilde {x}_{R,i}]\) for any desired sourcerelay SNR. We identified some properties of the parameter η as follows:
Property 1.
In general, the soft scalar has value −1≤η≤1. For the high values of sourcerelay SNR, \( \tilde {x}_{R}^{\,j} \approx 1 \), and this gives \( \tilde {x}_{R}^{\,j} \approx {x}_{R}^{\,j} \approx 1 \). In that case, η=1. In the event that \( \tilde {x}_{R}^{\,j} \) gives a hard decision error, \( \tilde {x}_{R}^{\,j} \approx {x}_{R}^{\,j} \approx 1 \). In that case, η=−1.
Property 2.
The relay does not forward when η=0. Note that the value of η=0 occurs in the very low SNR regime of the sourcerelay channel. In this regime, the a priori LLR will be very small in magnitude, and in turn, the generated soft symbols will have nearzero amplitude. This scenario is also similar to the case where the relay does not transmit any signal, i.e., the relay stays silent.
The effect of η is similar to that of a fading coefficient. We apply this model to the networkcoded symbol and not to the “softmodulated” symbols (an advantage of the current approach is that the soft symbol may have higher amplitudes when compared with the existing methods. This is one of the benefits from the network coding operation). The soft error variance may also be calculated as \(\sigma _{\tilde {n}}^{2}=(\sigma _{\tilde {x}_{R}}^{2}\eta ^{2})\), where \(\sigma _{\tilde {x}_{R}}^{2}\) is the variance of \(\tilde {x}_{R}\). By the use of these statistics, soft error variance can be estimated as \( \beta = \sqrt {\frac {1}{\eta ^{2}+\sigma _{\tilde {n}}^{2}}} \;. \) By invoking symmetry of the channel, BPSK modulation, and LDPC decoding process, we also have \(\mathbb {E}(\tilde {x}_{R}) = 0\); it follows that \(\mathbb {E}(\tilde {n}) = 0\).
Next, we describe the standard soft relay protocol as in [3]. The relationship between the correct symbols x _{ R } and the soft symbols \(\tilde {\boldsymbol {x}}_{R}\) is modeled in [3] by \(\tilde {x}_{R}^{\,j}={x_{2}^{\,j}}(1\bar {n}^{\,j})\), where \(\bar {n}^{\,j}\) is a soft noise variable whose mean and variance can be measured offline as \(\mu _{\bar {n}}=\frac {1}{N}\left ({\displaystyle \Sigma _{l=1}^{N}} \tilde {x}_{R,l}x_{R,l}\right)\) and \(\sigma _{\bar {n}}^{2}=\frac {1}{N}\left [\Sigma _{l=1}^{N}\left (1\tilde {x}_{R,l}x_{R,l}\mu _{\bar {n}}\right)^{2}\right ],\) respectively. Note that this model was used in [3] for soft BPSK but here we will apply it as a reference model for the soft networkcoded symbols.
where \(\bar {n}_{R{i}}^{\,j} = n_{R{i}}^{\,j}  \sqrt {P_{R}} h_{R{i}} \beta {x_{R}^{\,j}} (\bar {n}^{\,j}\mu _{\bar {n}})\) is the equivalent (zeromean) noise at the destination with variance \( {\bar{\sigma}}_{Ri}^2={\sigma}_{Ri}^2+{P}_R{h}_{Ri}^2{\beta}^2{\sigma}_{\bar{n}}^2 \) —this equivalent noise is modeled as having Gaussian distribution.
At the destination, the received LLR’s from the relay and the source i will be added as they are referring to the same underline source ith transmission where i∈{A,B}.
5.1 Equivalent onehop link model
In order to analyse the performance of the proposed scheme, we give a closed form expression of the virtual onehop link to represent the twohop links in the twoway relay channels in terms of equivalent SNR and other statistics [22]. We define the instantaneous SNR at links i to R and R to \(\bar {i} \) as: Γ _{ iR } and \( \Gamma _{R\bar {i}} \), respectively, where \( {\varGamma}_{iR}={\left{h}_{iR}\right}^2{P}_S/{\sigma}_{iR}^2 \), \( {\varGamma}_{R\bar{\i}}={\left{h}_{\bar{\i}R}\right}^2{P}_R/{\sigma}_{iR}^2 \), i∈{A,B}, and \( \bar {i} \in \{B,A\}\).
where the Qfunction is formally defined as \( Q(x)= \frac {1}{\sqrt {2\pi }} \int _{x}^{\infty } \exp \left (\frac {u^{2}}{2}\right)du\), Q ^{−1}(x) is the inverse function of Q(x), and Δ is a constant depending on the modulation scheme; for the BPSK scheme, Δ=2. This is an approximation of SNR of the twoway relay channel by the onehop SNR link.
The parameter Δ depends on the modulation scheme. For QPSK, we have Δ=4, then, from (20) and (21) we can verify that (22) holds. Therefore, the analysis in this work can be extended to higherorder modulation schemes.
5.2 Estimation of model parameters
where the pairs {(a _{ k },b _{ k })}_{ k } are estimated during the training phase and stored in the lookup table.
Thus, we can readily compute \( \sigma _{\tilde {n}}^{2} = \eta ^{2}/ \Gamma {}_{{\text {out}}}\) where η can be taken from the lookup table in Fig. 4, and \(\sigma _{\tilde {n}}^{2}=P_{\hat {n}} \) as \(\mathbb {E}(\tilde {n}) = 0\). Similarly, we can also compute β. In conventional SIR schemes in [3, 4, 5], statistical parameters \( \sigma _{\tilde {n}}^{2} \) and β (η in [5] as well) need to send from the relay to the destination for LLR computation. In our approach, the relay only has to send the received SNR to the destination, which decreases extra signaling overhead.
5.3 Variance correction factor
where α is a scalar, which is used for correction of \( {\widehat{\sigma}}_{Ri}^2 \) at the destination. We call it a variance correction factor and compute its value via simulations. Finding an analytical form for α is not straightforward since the probability distribution of the soft symbols is not known. In fact, by using α, we obtain a correction criterion to \( {\widehat{\sigma}}_{Ri}^2 \), the variance of the received signal at the destination. From Fig. 7, we obtain that the decoder is sensitive to the variance when α≤2. In the next section, we present the simulation results to demonstrate the impact of the different values of α. We observe that α=2 yields very good performance (i.e., the freepropagation attenuation is considered). From this study, we observe that the soft noise varince \( \sigma _{\tilde {n}}^{2} \) is underestimated in the previous section when assuming that it is an approximated Gaussian distribution. Such an inaccurate variance can significantly degrade the system performance. To obtain a satisfactory BER performance, we have to increase its value by a factor of two.
6 Simulation results and discussion
In this section, the simulation results are provided to demonstrate the effectiveness of the proposed system. All channels are assumed to exhibit quasistatic fading, i.e., the channel coefficients h _{ AD }, h _{ BD }, h _{ RD }, h _{ AR }, and h _{ BR } are constant for each transmission phase and change independently from one phase to the next. First, we consider a symmetric relay setup where the two sources A and B have the same distance to the destination, and where the two sources, the relay, and the destination are aligned in the same horizontal line. The distances between the sources and the destination are normalized to unity, i.e. d _{ AD }=d _{ BD }=1. The attenuation exponent was chosen to be γ=2.
Figure 5 demonstrates the BER performance of different arraybased spatially coupled LDPC codes, i.e., different cutting vectors δ using Rayleigh fading AWGN channels. As we can see, δ=(5,4,2) shows good BER performance. In the following simulation setup, therefore, we use spatially coupled LDPC codes based on δ=(5,4,2). As a competing scheme, we also simulate block arraybased LDPC code with parameters K _{1}=111, N _{1}=185, and rate \( \mathbb {R}=0.6 \) in the same figure. This LDPC code serves as the building block for the arraybased SCLDPC codes. It should be noted, that the latency of decoding block LDPC code using messagepassing algorithm is comparable to the latency of decoding QCLDPC codes constructed from that block LDPC code using slidingwindow messagepassing decoding, with appropriately selected size of the decoding window.
When the source to relay channel is poor, there will be decoding errors at the destination. This creates a huge error propagation if hard decisions are forwarded from the relay to the destination. That is why we obtain relatively poor performance in hard DF. The proposed scheme does not make any premature decision at the relay and forwards the soft symbols to the destination. As the destination decoder also works in soft domain, its performance is improved. For completeness, we also simulate (AF) relaying. As expected, it shows degraded performance due to the noise amplification at the relay.
In the compressandforward (CF) strategy, the received signal vector sent via sourcerelay link is quantized and compressed at a relay before being forwarded to the destination [24]. The CF strategy is usually referred to as estimateandforward in the literature [25]. In fact, the received signals at the relay and the destination are correlated due to the nature of broadcasting. We simulated CF as well wih the simulation setup discussed in this paper.
As it can be observed in Fig. 6, the proposed lookup table method provides slightly better BER performance (0.4 dB at 10^{−3}) as compared to the SIR scheme in [5] with SCLDPC codes. In the case of [3] and [5], the relay has to forward all parameters required for LLR modeling to the destination. Additionally, the proposed scheme offers a reduced signaling overhead since the relay transmits Γ _{in} only. By using the piecewise linear interpolation, the destination can create any combination of (γ _{in},γ _{out},η). The proposed SCLDPC scheme together with the piecewise linear interpolation method and modified soft network coding outperforms the other schemes under comparison. The proposed modified soft network method improves on the amplitude of the soft networkcoded symbol for the checked parameter settings. In the conventional approach, due to the product of two networkcoded symbols, \( \tilde {x}_{R}^{\,j} \) could have a very small amplitude and is susceptible to channel noise. Then, it will deteriorate the joint error performance at the destination.
In the simulation in Fig. 7, we set SNR_{ iR }=1 dB and SNR_{ iD }=SNR_{ Ri }. The results show that the BER performance of the proposed SIR scheme verses various α values. In this scheme, we observe that the BER performance improves as α increases up to α=2. In the region of α>2, the BER performance remains almost invariant and does not contribute to the overall system performance. This observation suggests that the choice α=2 is close to the optimum. This also highlights that \( {\widehat{\sigma}}_{Ri}^2 \) actually underestimates the actual variance of \( {\widehat{\boldsymbol{n}}}_{Ri} \) when assuming that it is an approximated Gaussian distribution.
We also observe that the performance for α=2 is superior by around 0.8 dB when compared to α=1 in Figs. 7 and 8. It is natural to notice that as SNR_{ Ri } increases, the BER performance improves. Finding an analytical solution for α is involved as the complete PDF for \( \tilde {n} \) is not known, and therefore, our conclusions are based on the simulations only. Based on the simulations in Fig. 7 and by using the quantiles of the theoretical Gaussian distribution against the empirical data points of the total equivalent noise at the destination, we conclude that α=2 is close to optimal.
7 Conclusions
We have developed a novel optimized soft information relaying scheme based on cooperative network coding in a twoway relay system by means of spatially coupled LDPC codes. Instead of forwarding hard decisions, the proposed scheme forwards modified soft networkcoded symbols. The error correction is achieved by using spatially coupled arraybased LDPC codes. We also modify our previously proposed soft error model, such that the scaling factor and the soft error variance of the residual noise can be easily computed. It reduces the signaling overhead and achieves better performance. Finally, we introduce a correction factor parameter in order to model the soft error variance at the destination. We compare the proposed scheme with its counterparts. The proposed scheme shows significant performance improvement in terms of BER, when compared to the competing schemes.
Notes
Acknowledgment
The authors wish to thank Eirik Rosnes, University of Bergen, Norway, and Michael Lentmaier, Lund University, Sweden, for the helpful discussions. This work is supported (in part) by the NorwegianEstonian Research Cooperation Programme through the grant EMP133, by the Estonian Research Council through the research grants PUT405 and IUT21, and by the European Regional Development Fund through the Estonian Center of Excellence in Computer Science, EXCS.
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