Latticecoded cooperation protocol for the halfduplex Gaussian twoway relay channel
 Shahab GhasemiGoojani^{1} and
 Hamid Behroozi^{1}Email author
https://doi.org/10.1186/s1363801504832
© GhasemiGoojani and Behroozi. 2015
Received: 25 November 2014
Accepted: 15 November 2015
Published: 2 December 2015
Abstract
This paper studies the Gaussian twoway relay channel (GTWRC), where two nodes exchange their messages with the help of a halfduplex relay. We investigate a cooperative transmission protocol, consisting of four phases: multiple access (MA) phase, broadcast (BC) phase, and two cooperative phases. For this setup, we propose a new transmission scheme based on superposition coding for nested lattice codes, random coding, and jointly typical decoding. This scheme divides the message of each node into two parts, referred to as satellite codeword and cloud center. Depending on the phase type, the encoder sends a linear combination of satellite codewords or cloud centers. For comparison, a rate region outer bound using a cutset bound is provided. We show that the proposed scheme can achieve the capacity region in the high signaltonoise ratio (SNR) regime. In addition, the achievable rate region is within 0.5 bit of the outer bound, regardless of all channel parameters. Using numerical examples, we show that our proposed scheme achieves a larger rate region than the best known 4phase transmission strategy so far, called the Hybrid Broadcast (HBC) protocol by Kim et al. Our proposed scheme not only improves upon previous 2 and 3 and 4phase protocols but also can perform superior at some cases to the introduced 6phase protocol by Gong, Yue, and Wang (which has more complexity relative to our 4phase protocol).
1 Introduction
In the last couple of years, cooperative communication and relaying has attracted great interest in wireless networks and some scenarios have been studied from information theory perspective. The first model to study this problem, which consists of 3 nodes, is introduced by Van der Meulen [1]. Cover and El Gamal presented two coding strategies for this model [2]. In addition to oneway relaying, twoway communication between two nodes or bidirectional relaying is of great interest. In the twoway relay channel (TWRC), there exists a relay that facilitates exchange of messages between two nodes. In the full duplex mode, each node is able to transmit and receive simultaneously but in a halfduplex communication, each node can either receive or transmit data at each time slot.
The basic protocol for the TWRC, which consists of two phases (phase 1 and 2 of Fig. 1), is Multiple Access and Broadcast (MABC) protocol. In the first phase, which is referred to as the MA phase, two nodes simultaneously transmit to the relay. In the second phase, i.e, the BC phase, the relay broadcasts a signal to both nodes. There are several practical coding schemes that investigate this protocol, see e.g., [9–13]. In the BC phase, the relay combines the data from both nodes and broadcasts the combined data back to both nodes. For this phase, there exist several strategies for the processing at the relay node, e.g., an amplifyandforward (AF) strategy [5], a decodeandforward (DF) strategy [5, 14], or a compressandforward (CF) strategy [15]. The AF protocol is a simple scheme, which amplifies the signal transmitted from both nodes and retransmits it to them, and unlike the DF protocol, no decoding at the relay is performed. In the twoway AF relaying strategy, the signals at the relay are actually combined on the symbol level. Due to amplification of noise, its performance degrades at low signaltonoise ratios (SNRs). The twoway DF relaying strategy was proposed in [5], where the relay decodes the received data bits from the both nodes. Since the decoded data at the relay can be combined on the symbol level or on the bit level, there has been different datacombining schemes at the relay for the twoway DF relaying strategy: superposition coding, network coding, and lattice coding [16]. In the superposition coding scheme, applied in [5], the data from the two nodes are combined on the symbol level, where the relay sends the linear sum of the decoded symbols from both nodes. Shortly after the introduction of the twoway relay channel, its connection to network coding [17] was observed and investigated. The network coding schemes combine the data from nodes on the bit level using the XOR operation, see, e.g., [10, 18–22]. Latticebased coding uses modulo addition in a multidimensional space and utilizes nonlinear operations for data combining. Applying lattice codes over twoway relaying systems was considered in, e.g., [7, 12, 13, 23–25]. In general, as in CF or partial DF relaying strategies, the relay node does not need to decode the source messages but only needs to pass sufficient information to the destination nodes. A strategy based on symbolwise network coding, in which two modulated symbols with different modulation types directly being mapped to a transmitted signal at the relay is investigated in [9]. A combination of bitwise network coding and channel decoding is considered in [10]. A coding scheme based on distributed lineardispersion spacetime codes is considered in [11]. Nested lattice codes for the GTWRC in the symmetric and asymmetric case are considered in [12, 13], respectively. For the asymmetric case, based on a lattice partition chain, [12] shows that the achievable rate region is within 0.5 bit from the capacity region for each user. Note that in [12] all nodes operate in fullduplex mode and there is no direct channel between the source nodes. Using a compressandforward strategy based on nested lattice codes, some new achievable rate regions for the GTWRC are provided [7] where it is assumed that all nodes operate in halfduplex mode without any direct link between the communication nodes. In the proposed scheme in [7], a layered coding is applied: a common layer is decoded by both receivers and a refinement layer is recovered only by the receiver which has the better channel condition. In [24] the GTWRC which operates in fullduplex mode is considered. Based on decoding noninteger linear combination of lattice codewords (instead of decoding linear combination of them), it is shown that the capacity region of the GTWRC under MABC protocol is partially achieved [24]. However, it is shown that the MABC protocol may not perform well when the channel gains are asymmetric [3]. Thus, to improve performance and to achieve a larger rate region, protocols with more phases are proposed in the literature, e.g., [3, 6,26–28].
The capacity region of relay channel with state information at the sources or at the relay is investigated in [29–36]. The relay channel and the cooperative relay broadcast channel controlled by random parameters are studied in [29]. It is shown that when the state is noncausally known to transmitter and intermediate nodes, the decodeandforward can achieve the capacity region under some cases. The relay channel with state known noncausally at the relay is investigated in [30, 31]. Using GelfandPinsker coding, ratesplitting and decodeandforward, a lower bound on channel capacity is obtained for this channel and it is shown that for the degraded Gaussian channels, the lower bound meets the upper bound and thus the capacity region is achievable. The relay channel when the state is available only at the source is studied in [32–34]. By obtaining lower and upper bounds, it is shown that in a number of special cases, the capacity region is achievable. A partially cooperative relay broadcast channel (PCRBC) with state is studied in [35] where two situations including availability of the state noncausally at both the source and the relay and only at the source are analyzed. The relay interference channel with cognitive source where only the source knows (noncausally) the interference from the interferer is considered in [36] and some achievable rate regions are obtained.
In [26], the GTWRC with four phases (phases 1, 2, 5, and 6 of Fig. 1) is considered. It is shown that for both full and halfduplex modes, partial decodeandforward can achieve a rate region strictly larger than the time shared region of pure decodeandforward and direct transmission. Two different decodeandforward protocols with three and four phases, which have a better performance than MABC under some constraints on the asymmetric model, are considered in [3]. These protocols are referred to as Time Division Broadcast (TDBC) (phases 2, 5, and 6 of Fig. 1) and Hybrid Broadcast (HBC) (phases 1, 2, 5, and 6 of Fig. 1). If channel coefficients tend to symmetricchannel coefficients, then TDBC protocol has a poor performance compared with the MABC. But it is shown that at some cases, the achievable sum rate of HBC protocol contains set of points that are outside the outer bounds of the MABC and TDBC protocols.
To achieve a larger rate region than the HBC protocol in [3, 37], a protocol that uses all possible 6 phases (shown in Fig. 1) is proposed in [27, 28]. Although increasing the number of transmission phases results in improvement of achievable rate region with respect to the HBC protocol, it has more complexity relative to a 4phase protocol. In [8], by obtaining achievable rate regions and outer bounds for a 6phase protocol, it is shown that it can achieve a larger rate region compared with other protocols at some cases. In [6], two protocols are investigated: MABC protocol and TDBC protocol. Using decodeandforward, compressandforward, amplifyandforward, and a new mixedforward scheme, some achievable rate regions for these protocols are obtained. A 3phase protocol, which is called Cooperative Multiple Access Broadcast (CoMABC) and consists of phases 1, 2, and 4 of Fig. 1, is proposed in [38]. Using doubly nested lattice codes, an achievable rate region for this scheme is obtained and it is shown numerically that the CoMABC outperforms the MABC and TDBC protocols in terms of sum rate in asymmetric channel conditions. The twoway relaying in a Gaussian diamond channel is considered in [39] and it is shown that lattice codes under certain conditions can achieve rate regions close to the outer bound.
In this paper, in contrast to [27], instead of increasing the number of phases in order to achieve a larger rate region than the HBC protocol, we propose a new 4phase cooperative MABC protocol for the halfduplex GTWRC, including phases 1, 2, 3, and 4 of Fig. 1. First, consider the MABC protocol, which includes two phases, MA and BC phases. It is well known that lattice codes can achieve the capacity region of the MABC protocol within 0.5 bit [6, 12]. Thus, it seems that we do not need to consider GTWRC with more transmission phases (i.e., three or four or six phases). But, suppose that the link from the relay node to node 2 (1) is very weak (noisy) such that node 2 (1) can correctly decode the message of node 1 (2) at a very low rate. Thus, we require defining other phases to increase the rate. Now, consider the CoMABC protocol [38]. The CoMABC protocol consists of three phases: phases 1 and 2 are similar to the MABC protocol and at the third phase, by cooperating between node 1 and the relay node, we send information to node 2. To explain why we must use the CoMABC protocol, suppose that we are at the MABC protocol. At the end of second phase, the relay sends information bits to both nodes. Node 1 can recover its data while node 2, since has a weak link, can only decode the message of node 1 at a very low rate. Thus, to increase the data transmission rate, we use another phase to send data to node 2. Now, to explain our proposed scheme, again, consider the CoMABC protocol. Suppose that at the end of phase 2, node 1 can recover message of node 2 at a very low rate. Since at phase 3, we send no data to node 1, thus, we must define other phases to send extra data to node 1 to decode the message of node 2 at a higher rate. In our proposed scheme, at phases 3 and 4, we send some data to node 1 and node 2 to decode the message of the other node at a higher rate.
Our proposed protocol is denoted by 2CoMABC. In phase 1 and phase 2, the protocol is similar to the MABC. In these phases, both nodes cannot completely recover the message of each other. Thus, we introduce two other phases. In phase 3 and phase 4, each node with the help of the relay and the other node tries to recover the message of the other node. These two phases are referred to as the cooperative phases. For the first time in this paper, we propose a scheme based on a “superposition coding for nested lattice codes” for the GTWRC. In superposition coding, we divide the message of each node into two parts using nested lattice codes: satellite codeword and cloud center. Thus, if we want to recover a message, we must recover the satellite codeword and the cloud center simultaneously. In phase 1, based on the idea of computation coding [40], we recover a linear combination of messages. Due to structured codes, we can calculate the satellite codeword and cloud center for linear combination of messages. Then, we send the satellite codewords to nodes in phase 2 using “random coding”. In phase 3 and 4, we send the cloud centers to both nodes. At the end of phase 4, we can recover both messages. Although we apply the superposition coding using nested lattice codes to the 2CoMABC protocol, but one can use it to achieve better or same rate regions which are obtained at other papers. For example, we can apply it to the CoMABC protocol proposed in [38] and show that our proposed scheme includes the CoMABC scheme as a special case.
Finally, by examining many numerical examples (out of which some are presented here) and comparing the achievable rate region of the proposed scheme with that of the HBC protocol, it can be observed that our proposed scheme has a better performance than the HBC protocol. In addition, this scheme not only improves upon previous 2 and 3 and 4phase protocols but also can perform superior at some cases to the 6phase protocol, proposed in [27] (which has more complexity relative to a 4phase protocol).

Proposing a new transmission scheme based on “superposition coding for nested lattice codes” and “random coding”.

Analyzing the proposed protocol in a new cooperative transmission scheme and showing that our proposed scheme can achieve the capacity rate region in the high SNR regime and within 0.5 bit in general.

Improving the rate region of the 4phase HBC protocol without increasing any complexity (in contrast to the proposed scheme at [27]).
The remainder of the paper is organized as follows. We present the channel model and the preliminaries on lattice codes in Section 2. In Section 3, first, we present the superposition coding for nested lattice codes and then we introduce and analyze our proposed scheme. In Section 4, an achievable rate region as well as a rate region outer bound based on the cutset bound are provided. Using numerical examples, achievable rate regions of different cooperative protocols are compared in Section 5. Section 6 concludes the paper.
Notations: Let \(C\,(x)=\frac {1}{2}\log \left (1+x\right)\). Logarithms are of base two. The random variables (RV) and their realizations are denoted by capital and small letters, respectively. x stands for a vector of length n, (x _{1},x _{2},…,x _{ n }). [x]^{+}= max{x,0} for \(x\in \mathbb {R}\).
2 Preliminaries: channel model and lattices
2.1 Channel model
In the following, similar to [6], we define encoders, decoders, and associated probability of errors: let \(W_{S,T}:=\left \{ W_{i,j}i\in S,\,j\in T,\,S,T\subset \mathcal {M}\right \} \) denote the set of messages from nodes in set S to nodes in set T. Note that if node i does not have a message for node j, then we have W _{ i,j }=Ø. At node i, the encoder at channel use k is a function \({X_{i}^{k}}\left (W_{\left \{ i\right \},\mathcal {M}},{Y_{i}^{1}},{Y_{i}^{2}},\ldots,Y_{i}^{k1}\right)\in \mathcal {X}_{i}\); the decoder at node i after all n channel uses produces an estimate of the message W _{ j,i } using function \(\widehat {W}_{j,i}\left ({Y_{i}^{1}},{Y_{i}^{2}},\ldots,{Y_{i}^{n}},W_{\left \{ i\right \},\mathcal {M}}\right)\). The error event to decode the message W _{ i,j } at the end of the block of length n is defined by \(E_{i,j}:=\left \{ W_{i,j}\neq \widehat {W}_{i,j}\left (.\right)\right \} \), and the error event at node j in which node j wants to find w _{ i } at the end of phase m is denoted by \(E_{i,j}^{(m)}\). For a protocol with phase durations {t _{ m }}, a set of rates R _{ i,j } is said to be achievable if there exist encoders/decoders of block length n=1,2,… with both P[E _{ i,j }]→0 and t _{ m,n }→t _{ m } as n→∞ for all i, j, m. An achievable rate region is the closure of a set of achievable rate tuples for fixed {t _{ m }}. The set of all achievable rate tuples is the capacity region of the TWRC.
In addition, g _{ ij } is the channel gain between transmitter i and receiver j. We assume that the channel is reciprocal such that g _{ ij }=g _{ ji } and each node is fully aware of g _{1r }, g _{2r }, and g _{12} (i.e., full CSI). Considering channel reciprocity, the channel coefficient between nodes 1 and r is denoted collectively as g _{1}, i.e., g _{1r }=g _{ r1}=g _{1}. Similarly, we have g _{2r }=g _{ r2}=g _{2} and g _{12}=g _{21}=g _{3}.
2.2 Lattice definitions
Here, we provide some necessary definitions on lattices and nested lattice codes. Interested readers can refer to [40–42] and the references therein for more details.
Definition 1.
Definition 2.
Definition 3.
Definition 4.
Definition 5.
Definition 6.
The sequence is indexed by the lattice dimension n. The existence of such lattices is shown in [44, 45].
Definition 7.
Definition 8.
Erez et al. show that there exists a sequence of lattices that are simultaneously good for packing, covering, source coding (Rogersgood), and channel coding (Poltyrevgood). In the following, we present a key property of dithered lattice codes.
Lemma 1.
[ 41 ] The Crypto Lemma Let V be a random vector with an arbitrary distribution over \(\mathbb {R}^{n}\). If D is independent of V and uniformly distributed over \(\mathcal {V}\), then (V+D)mod Λ is also independent of V and uniformly distributed over \(\mathcal {V}\).
Proof.
See lemma 1 in [41].
3 Latticecoded cooperation protocol
In this section, based on nested lattice codes, we propose an achievable rate region on the capacity region of the GTWRC. First, we present the superposition scheme for nested lattice codes that is a key to our code construction.
3.1 Superposition coding for nested lattice codes
The mesolattice point V _{ b,i } determines the cloud center in which V _{ i } resides, while V _{ a,i } identifies its location within the clouds (i.e., the individual codewords within the clouds). This scheme is similar to superposition coding in the broadcast channel [47].
The following theorem presents the main result of this paper.
Theorem 1.
where \(R_{i,r}^{*}\overset {\triangle }{=}\left [\frac {1}{2}\log \left (\frac {{g_{i}^{2}}P_{i}}{{g_{1}^{2}}P_{1}+{g_{2}^{2}}P_{2}}+{g_{i}^{2}}P_{i}\right)\right ]^{+}\) and [ x]^{+}= max{0,x}.
In the following, the steps of the proof are presented. First, we provide a brief explanation of our coding and then present our scheme in more details. Without loss of generality, we assume that R _{1}≥R _{2}. Since we need two codebooks, three nested lattices for generating these codebooks are required. One of the lattices, \(\Lambda _{c}^{(n)}\), constructs the codewords while the other two lattices (shaping lattices) satisfy the channel power constraints (\(\Lambda _{s1}^{(n)}\) and \(\Lambda _{s2}^{(n)}\)). Based on the idea of computation coding [40], at the end of phase 1, we decode two linear combinations of messages. In order to decompose these linear combinations, which are points of \(\Lambda _{c}^{(n)}\), we define another lattice \(\left (\Lambda _{m}^{(n)}\right)\) that partitions \(\Lambda _{c}^{(n)}\) into clouds. Based on this coding strategy, both linear combinations have the same satellite codeword (i.e., an individual codeword in \(\mathcal {V}_{m}\)) but different cloud centers (i.e., an individual codeword in \(\mathcal {V}_{s1}\) or \(\mathcal {V}_{s2}\)).
In phase 2, we send the satellite codeword of the linear combinations to both nodes while in phase 3 and 4, we communicate the cloud center associated with the linear combination of codewords to node 1 and 2, respectively. Thus, at the end of phase 4, based on having the cloud center associated with a nested lattice and the individual codeword in that cloud, we can fully find the linear combination of messages at both nodes. Using the later decoding, we can decode the message of each node at the opposite node. In the following, we present our scheme in more details.
3.2 Phase 1 (MA phase)

Encoding:
By calculating the optimum phase durations, t _{1}, t _{2}, t _{3}, and t _{4}, we can determine the codeword length in each phase as \(n_{1}=\frac {t_{1}}{T_{s}}\), \(n_{2}=\frac {t_{2}}{T_{s}}\), \(n_{3}=\frac {t_{3}}{T_{s}}\), and \(n_{4}=\frac {t_{4}}{T_{s}}\), where T _{ s } is the sampling interval. In the following, without loss of generality, we assume that \({g_{1}^{2}}P_{1}\geq {g_{2}^{2}}P_{2}\). In order to apply the rate splitting, we choose a chain of lattices as (2), such that \(\Lambda _{s1}^{(n_{1})}\), \(\Lambda _{s2}^{(n_{1})}\) and \(\Lambda _{m}^{(n_{1})}\) are Rogersgood and Poltyrevgood while \(\Lambda _{c}^{(n_{1})}\) is Poltyrevgood. The generation of these lattices is fully explained in [45].

Decoding:
(10) shows that the estimation of V _{ r,1} is incorrect if the effective noise Z _{eff} leaves the Voronoi region surrounding the true codeword, i.e., \(P_{e}=\text {Pr}\left (\boldsymbol {Z}_{\text {eff}}\notin \mathcal {V}_{c}\right).\)
where (15) follows from \(\Lambda _{s1}^{(n_{1})}\subseteq \Lambda _{s2}^{(n_{1})}\) and the distributive law of the modulo operation.
Now, the relay node decomposes the linear combinations of messages, V _{ r,1} and V _{ r,2}, as the following:
Note that \(\boldsymbol {L}_{a,i}\in \mathcal {C}_{a}^{(n_{1})}\) and \(\boldsymbol {L}_{b,i}\in \mathcal {C}_{b,i}^{(n_{1})}\) for i=1,2. As we showed in (17) and (19), due to the structure of nested lattice codes, we can determine L _{ a,i } and L _{ b,i } for i=1,2 using V _{ r,i }. Our coding strategy sends the linear combination of satellite codewords (associated with V _{1} and V _{2}), i.e., L _{ a,1} to both nodes in phase 2. In phase 3 and 4, we communicate the cloud center associated with the linear combination of codewords, i.e., L _{ b,2} and L _{ b,1} to node 1 and 2, respectively.
3.3 Phase 2 (broadcast phase)

Encoding:

Decoding:
3.4 Phase 3 (first cooperative phase)

Encoding:

Decoding:
3.5 Phase 4 (second cooperative phase)
Encoding and decoding at nodes
Phase 1  Phase 2  Phase 3  Phase 4  

(a) Encoding at nodes  
Node 1  \(\boldsymbol {X}_{1}=\frac {1}{g_{1}}\left [\boldsymbol {V}_{1}+\boldsymbol {D}_{1}\right ]\textrm {mod }\Lambda _{s1}^{(n_{1})}\)  −  −  \(\boldsymbol {X}_{1}^{(4)}\left (\boldsymbol {L}_{b,1}\right)\) 
Node 2  \(\boldsymbol {X}_{2}=\frac {1}{g_{2}}\left [\boldsymbol {V}_{2}+\boldsymbol {D}_{2}\right ]\textrm {mod }\Lambda _{s2}^{(n_{1})}\)  −  \(\boldsymbol {X}_{2}^{(3)}\left (\boldsymbol {L}_{2}^{(3)}\right)\)  − 
Relay  −  \(\boldsymbol {X}_{r}^{(2)}\left (\boldsymbol {L}_{a,1}\right)\)  \(\boldsymbol {X}_{r}^{(3)}\left (\boldsymbol {L}_{r}^{(3)}\right)\)  \(\boldsymbol {X}_{r}^{(4)}\left (\boldsymbol {L}_{b,1}\right)\) 
(b) Decoding at nodes  
Node 1  −  V _{ a,2}  V _{ b,2}  − 
Node 2  −  V _{ a,1}  −  V _{ b,1} 
Relay  \(\begin {array}{lcl} \boldsymbol {V}_{r,1} & = & \left [\boldsymbol {V}_{1}+\boldsymbol {V}_{2}\right.\\ & & \left.\mathcal {Q}_{\Lambda _{s2}}\left (\boldsymbol {V}_{2}+\boldsymbol {D}_{2}\right)\right ]\textrm {mod }\Lambda _{s1}^{(n_{1})}\\ \boldsymbol {V}_{r,2} & = & \left [\boldsymbol {V}_{1}+\boldsymbol {V}_{2}\right ]\textrm {mod }\Lambda _{s2}^{(n_{1})} \end {array}\)  −  −  − 
3.6 Achievable rate region
4 The rate region outer bound and capacity results
4.1 The rate region outer bound
In this subsection using the cutset bound, we obtain an outer bound over the rate region of the halfduplex GTWRC. This bound can be derived from the halfduplex cutset bound in [52].
Lemma 2.
where all t _{ m } are nonnegative subject to \(\overset {4}{\underset {m=1}{\sum }}t_{m}=1\).
Proof.
Now, by minimizing over two cuts, i.e., minimizing (43) and (44), we get the desired bound in (41). Similarly, we can conclude the bound on R _{2} which is given by (42).
4.1.1 \thelikesubsubsection Linear resource allocation problem
We can easily transform this problem to a standard form and solve this optimization problem.
4.2 Capacity results
Corollary 1.
The capacity region of the halfduplex Gaussian twoway relay channel via the 2CoMABC protocol, as shown in Fig. 2, is achievable within 0.5 bit.
Proof.
where (48) is based on the fact that the maximum gap occurs at \(\frac {{g_{1}^{2}}P_{1}}{{g_{1}^{2}}P_{1}+{g_{2}^{2}}P_{2}}+{g_{1}^{2}}P_{1}=1\). Now, from a simple inequality min(a _{1},a _{2})− min(b _{1},b _{2})≤ max(a _{1}−b _{1},a _{2}−b _{2}), the RHSs of (5) and (45) differ by at most \(\frac {1}{2}\) bit. The same holds for (6) and (46), and thus the achievable rate region which is given by (5) and (6) is within 0.5 bit of the outer bound for each user regardless of channel parameters.
Now, we investigate the achievable rate region of the GTWRC via 2CoMABC protocol in the high SNR regime.
Corollary 2.
where o(1)→0 as \({g_{1}^{2}}P_{1},{g_{2}^{2}}P_{2}\rightarrow \infty \).
By comparing this region with the outer bound in (45) and (46) for \({g_{1}^{2}}P_{1}\gg 1\) and \({g_{2}^{2}}P_{2}\gg 1\), we can see that the capacity region is achievable at high SNRs.
5 Numerical results

MABC protocol (outer bound): The MABC protocol is a twophase protocol (phases 1 and 2 of Fig. 1) where both users simultaneously transmit during the first phase and the relay alone transmits during the second. The outer bound of the MABC protocol is given by [3]:$$\begin{array}{@{}rcl@{}} R_{1} & \leq & \min\left\{ t_{1}C\left({g_{1}^{2}}P_{1}\right),t_{2}C\left({g_{2}^{2}}P_{r}\right)\right\},\\ R_{2} & \leq & \min\left\{ t_{1}C\left({g_{2}^{2}}P_{2}\right),t_{2}C\left({g_{1}^{2}}P_{r}\right)\right\}. \end{array} $$

TDBC protocol (outer bound): The second protocol considers sequential transmissions from the two users followed by a transmission from the relay:$${\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} R_{1} & \leq \min\left\{ t_{1}C\left(P_{1}\left({g_{1}^{2}}+{g_{3}^{2}}\right)\right),t_{1}C\left({g_{3}^{2}}P_{1}\right)+t_{3}C\left({g_{2}^{2}}P_{r}\right)\right\},\\ R_{2} & \leq \min\left\{ t_{2}C\left(P_{2}\left({g_{2}^{2}}+{g_{3}^{2}}\right)\right),t_{2}C\left({g_{3}^{2}}P_{2}\right)+t_{3}C\left({g_{1}^{2}}P_{r}\right)\right\}.\end{aligned}}} $$

In [38], using doubly nested lattice codes, an achievable rate region for three phases CoMABC protocol is obtained. The achievable rate region for this protocol is given by [38]$$\begin{array}{@{}rcl@{}} R_{1} & \leq & \min\left\{ t_{1}R_{1,r}^{*}+t_{3}C\left({g_{3}^{2}}P_{1}\right),t_{2}C\left({g_{2}^{2}}P_{r}\right)\right.\\&&+\left. t_{3}C\left({g_{2}^{2}}P_{r}+{g_{3}^{2}}P_{1}\right)\right\},\\ R_{2} & \leq & \min\left\{ t_{1}R_{2,r}^{*},t_{2}C\left({g_{1}^{2}}P_{r}\right)\right\}. \end{array} $$

HBC protocol (achievable rate region): The HBC protocol contains four phases (phases 1, 2, 5, and 6 of Fig. 1) which starts with the broadcast phases (5 and 6) followed by the MABC phases (1 and 2). In [3], it is shown that for the HBC protocol, the following rate region is achievable:$$\begin{array}{@{}rcl@{}} {}R_{1} & \leq & \min\left\{ t_{1}C\left({g_{1}^{2}}P_{1}\right)+t_{3}C\left({g_{1}^{2}}P_{1}\right)\right., \\ &&t_{1}C\left({g_{3}^{2}}P_{1}\right) + \left.t_{4}C\left({g_{2}^{2}}P_{r}\right)\right\}, \end{array} $$(51)$$\begin{array}{@{}rcl@{}} {}R_{2} & \leq & \min\left\{ t_{2}C\left({g_{2}^{2}}P_{2}\right)+t_{3}C\left({g_{2}^{2}}P_{2}\right)\right., \\ &&t_{2}C\left({g_{3}^{2}}P_{2}\right) +\left. t_{4}C\left({g_{1}^{2}}P_{r}\right)\right\}, \end{array} $$(52)$$\begin{array}{@{}rcl@{}} {}R_{1}+R_{2} & \leq & t_{1}C\left({g_{1}^{2}}P_{1}\right)+t_{2}C\left({g_{2}^{2}}P_{2}\right)\\&&+t_{3}C\left({g_{1}^{2}}P_{1}+{g_{2}^{2}}P_{2}\right). \end{array} $$(53)

HBC protocol (outer bound)$$\begin{array}{@{}rcl@{}} R_{1} & \leq & \min\left\{ t_{1}C\left({g_{1}^{2}}P_{1}+{g_{3}^{2}}P_{1}\right)+t_{3}C\left({g_{1}^{2}}P_{1}\right),\right.\\ && \left.t_{1}C\left({g_{3}^{2}}P_{1}\right)+t_{4}C\left({g_{2}^{2}}P_{r}\right)\right\},\\ R_{2} & \leq & \min\left\{ t_{2}C\left({g_{2}^{2}}P_{2}+{g_{3}^{2}}P_{2}\right)+t_{3}C\left({g_{2}^{2}}P_{2}\right),\right.\\&&\left. t_{2}C\left({g_{3}^{2}}P_{2}\right)+t_{4}C\left({g_{1}^{2}}P_{r}\right)\right\}. \end{array} $$

6phase protocol (achievable rate region) [27]$$\begin{array}{@{}rcl@{}} {}R_{1} & \leq & R_{1,2}^{(4)}+\min\left\{\vphantom{\left.t_{2}C\left({g_{2}^{2}}P_{r}\right)+R_{r,2}^{(4)}\right\}}\left(t_{1}+t_{5}\right)C\left({g_{1}^{2}}P_{1}\right),t_{5}C\left({g_{3}^{2}}P_{1}\right)\right.\\&&+\left.t_{2}C\left({g_{2}^{2}}P_{r}\right)+R_{r,2}^{(4)}\right\},\\ {}R_{2} & \leq & R_{2,1}^{(3)}+\min\left\{\vphantom{\left.t_{2}C\left({g_{2}^{2}}P_{r}\right)+R_{r,2}^{(4)}\right\}} \left(t_{1}+t_{6}\right)C\left({g_{2}^{2}}P_{2}\right),t_{6}C\left({g_{3}^{2}}P_{2}\right)\right.\\&&+\left.t_{2}C\left({g_{1}^{2}}P_{r}\right)+R_{r,1}^{(3)}\right\},\\{} R_{1}+R_{2} & \leq & t_{5}C\left({g_{1}^{2}}P_{1}\right)+R_{1,2}^{(4)}+t_{6}C\left({g_{2}^{2}}P_{2}\right)\\&&+t_{1}C\left({g_{1}^{2}}P_{1}+{g_{2}^{2}}P_{2}\right)+R_{2,1}^{(3)},\\ {}R_{1,2}^{(4)} & \leq & t_{4}C\left({g_{3}^{2}}P_{1}\right),\:R_{r,2}^{(4)}\leq t_{4}C\left({g_{2}^{2}}P_{r}\right),\:R_{1,2}^{(4)}\\&&+R_{r,2}^{(4)}\leq t_{4}C\left({g_{2}^{2}}P_{r}+{g_{3}^{2}}P_{1}\right),\\ {}R_{2,1}^{(3)} & \leq & t_{3}C\left({g_{3}^{2}}P_{2}\right),\:R_{r,1}^{(3)}\leq t_{3}C\left({g_{1}^{2}}P_{r}\right),\:R_{2,1}^{(3)}\\&&+R_{r,1}^{(3)}\leq t_{3}C\left({g_{1}^{2}}P_{r}+{g_{3}^{2}}P_{2}\right). \end{array} $$
Kim et al. [3] show that the achievable rate region for the HBC protocol contains points that are outside the outer bounds of the MABC and TDBC protocols. In [27], it is shown that by using a 6phase protocol, we can achieve a better rate region than the obtained rate region of the HBC protocol in [3]. Note that this improvement is due to increasing number of transmission phases from 4 to 6 (which induces higher complexity). Here, we numerically compare 2CoMABC with the abovementioned protocols. When we compare the sum rate outer bounds or achievable sum rates for different protocols, linear programming is used to optimize the portion of time allocated to each phase. In the following, for all nodes, we assume that the power constraint equals to P and we define \(\text {SNR}_{i}={g_{i}^{2}}P\) for i∈{1,2,3}.
We compare the sum rate in an environment with path loss. We assume the channel gains to be \(g_{1}=\left (1+d\right)^{\frac {\gamma }{2}}\), \(g_{2}=\left (1d\right)^{\frac {\gamma }{2}}\) and \(g_{3}=2^{\frac {\gamma }{2}}\), where d is the position of the relay and γ is the path loss exponent.
6 Conclusions
In this paper, the Gaussian twoway relay channel in the halfduplex mode, which operates in four phases, is studied. By using superposition coding, a scheme which achieves the outer bound within 0.5 bit is proposed. In this scheme, both structured codes and random coding have been used. In phase 1 (MA phase), we decompose the message of each user into two parts due to structured codes. In phase 2 (BC phase) and phase 3 and 4 (cooperative phases), random coding is applied. In the high SNR regime, the proposed scheme coincided with the cutset outer bound and thus the capacity region is achieved. Also, using numerical examples, we showed that our 2CoMABC protocol performs superior to the wellknown HBC protocol (which has the same number of transmission phases). Although in general the comparison for few examples may not provide a general insight on which scheme outperforms the others, similar behavior has been observed by evaluating the achievable sum rates and the achievable rate regions in many other examples with different channel parameters.
Declarations
Acknowledgements
This work has been supported by the Iran NSF under Grant No. 93046836.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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