Energy-Efficient Relaying Strategy with Network Coding in Two-Way Parallel Channels

Abstract

We consider a two-way communication assisted by parallel regenerative decode-and-forward relays operating in orthogonal channels. In a system with limited channel state information at each source and relay node, an optimum distributed power allocation strategy is proposed to minimize the total transmit power, providing a target signal-to-noise ratio at each destination with a target outage probability. Moreover, combined with opportunistic relaying and network coding, a distributed decision mechanism is proposed for the relay node to decide whether to help the transmission or not. In this proposal, each source works out the transmit power and the decision threshold then broadcasts them. The selected relay compares the decision threshold with the channel gain of its weaker relay-to-destination link, then determines whether to forward the networkcoded data or not. Simulation results show the advantage of this strategy in terms of energy efficiency for a two-hop two-way communication scenario. The proposed strategy is very flexible as it can trade outage to power consumption and vice versa.

1. Introduction

Wireless relaying offers space diversity to extend the transmission range and to enhance end-to-end transmission performance such as outage probability and data rate. In conventional single link communication, the transmission between the source and the destination suffers from severe fading due to multipath fading effects and path loss, which results in unreliable communication. Fortunately, relays can provide cooperative diversity [1], and save power to improve the reliability without the need of physical antenna arrays. Recently, some works such as [2–4] introduced network coding [5] to the bidirectional cooperation. Especially, [2, 3] focus on the bit-level transmission, and working out the optimum system throughput provided that both the two source-to-relay channels are better than the direct channel between the two sources in the three-node model. As shown in Figure 1, two source nodes (𝑆₁ and 𝑆₂) exchange messages with the help of a relay (namely, 𝑅), using time-division scheme in two-way communications. The traditional method [2] needs four phases (Figure 1(a)) to complete the exchange of information. However, by using network coding (a bitwise XOR operation at the relay node), only three phases are needed (Figure 1(b)). In such a three-phase relaying scheme, the messages from 𝑆₁ and 𝑆₂ are first decoded at the relay node, network coded, re-encoded, and then sent as one message to both destinations simultaneously. Here in this paper, we extend the three-node case to the scenario in which there are multiple relay candidates that could assist the source nodes in the two-way communication as shown in Figure 2.

(a)

(b)

Figure 1 

(a) Traditional relaying; (b) three-phase relaying.

Figure 2 

Two-way communication with multiple candidates.

When there are multiple relay node candidates, the multiple relays could simultaneously assist the transmission [6–8], or the most suitable single relay could be selected for transmission according to the channel state information (CSI). This opportunistic idea is based on instantaneously selecting an “on-peak” receiver with the “good” channel condition to improve system performance [9–12]. These works focused on the multirelay transmission in one-way transmissions (i.e., the messages are sent from 𝑆₁ to 𝑆₂, but no messages come from 𝑆₂ to 𝑆₁). Meanwhile, they assume that either the destination cannot directly receive the signals from the source [9], or that the complete CSI of all communication channels can be available at the source node [10, 12].

In addition, from the control mechanism's point of view, the cooperative strategies can be categorized into two types: central control strategy and distributed strategy. In most cases of practical systems, the distributed strategy is more practical since it only needs local and partial CSI, which overcomes the obstacles of a centralized architecture such as the substantial feedback requirements, overhead and delay, and so forth. For example, in [13], users select cooperation partners based on a priority list in a distributed manner for multiuser networks with coded cooperation. Reference [14] provides a relay selection algorithm based on instantaneous channel measurements obtained by each relay node locally, which reduces the cooperative costs among relays work in but only the one-way transmission is considered in [8, 13, 14].

Especially in [8], several optimum distributed power allocation strategies minimize the total transmit power while providing a target signal-to-noise ratio at the destination with a target outage probability. Due to the one-way transmission, [8] only needs to consider the relay-to-destination link after the relays successfully decode the messages. When network coding is introduced to decrease the transmission phases in two-way communications, as shown in Figure 1(b), the outage probability should be investigated on both 𝑅↔𝑆₁ and 𝑅↔𝑆₂ links in the third transmission phase after the relay has correctly decoded the messages from the two source nodes.

In this paper, our design target is to maximize power efficiency in two-way relay systems. Due to the limited knowledge of CSI at both source nodes, our distributed strategy mainly focuses on the forwarding decision and distributed power allocation. Firstly, a single relay node will be chosen as the opportunistic relay. Then, the opportunistic relay makes a decision whether to forward the source data according to a proper forwarding threshold. Moreover, the distributed power allocation based on the decode-and-forward (DaF) scheme [15] works out the corresponding transmit powers (𝑃_𝑆1,𝑃_𝑆2) and the forwarding threshold (𝛼_𝑡) at the source nodes. The source nodes broadcast the optimum pair ((𝑃_𝑆1,𝛼_𝑡) and (𝑃_𝑆2,𝛼_𝑡)) to be used by the relay nodes.

The remainder of this paper is organized as follows. The next section describes the system model. The analysis of the distributed strategy is performed in Section 3. Simulation results can be found in Section 4. Finally, concluding remarks are given in Section 5.

2. System Model and Background

As shown in Figure 2, we consider a relay network consisting of an 𝑆₁↔𝑆₂ two-way pair and 𝑀 relay nodes employing DaF. For simplicity, we denote relay 𝑖 by 𝑅_𝑖. We assume that the relaying operates in orthogonal time-division channels. Let ℎ_1𝑖 and ℎ_2𝑖 denote the fading coefficient of the 𝑆₁↔𝑅_𝑖 and 𝑅_𝑖↔𝑆₂ channels for the 𝑖th relay node, respectively, 𝑖=1,…,𝑀. The fading coefficient of the 𝑆₁↔𝑆₂ link is denoted by ℎ₀. Assume that each channel is slowly varying flat fading, and {ℎ_1𝑖,ℎ_2𝑖,ℎ₀} are all independent realizations of zero-mean complex Gaussian random variables with variance 𝜎^21𝑖, 𝜎^22𝑖 and 𝜎²⁰, respectively. We define 𝛽_1𝑖=1/𝜎^21𝑖, 𝛽_2𝑖=1/𝜎^22𝑖, and 𝛽₀=1/𝜎²⁰.

Firstly, the relay selection is prior to the forwarding decision and the distributed power allocation. That is, in the opportunistic relaying, the “good” relay is chosen prior to the transmission among a collection of 𝑀 possible candidates and source nodes. As shown in Figure 1(b), in the first transmission phase, 𝑆₁ broadcasts 𝑋₁ with power 𝑃_𝑆1, then 𝑆₂ broadcasts 𝑋₂ with power 𝑃_𝑆2. The corresponding destinations (i.e., 𝑆₂ node and 𝑆₁ node) observe 𝑦₂ and 𝑦₁ as𝑦𝑦₂₁=𝑃_𝑆1ℎ₀𝑋₁+𝑍₂,=𝑃_𝑆2ℎ₀𝑋₂+𝑍₁,(1) and the opportunistic relay 𝑅_𝑟 observes 𝑦_1𝑟 from the 𝑆₁↔𝑅_𝑟 link and 𝑦_2𝑟 from the 𝑆₂↔𝑅_𝑟 as𝑦𝑦_1𝑟2𝑟=𝑃_𝑆1ℎ_1𝑟𝑋₁+𝑍_1𝑟,=𝑃_𝑆2ℎ_2𝑟𝑋₂+𝑍_2𝑟,(2) where 𝑍₂, 𝑍₁, 𝑍_1𝑟, and 𝑍_2𝑟 are additive white Gaussian noise (AWGN) terms at the corresponding destinations and the selected relay, respectively. Without loss of generality, they are of variance 𝑁₀.

The opportunistic relay is chosen based on the following criterion: ℎ_𝑖=min𝑟||ℎ=^1𝑖||²a,rg||ℎm^2𝑖a||²x_𝑖,𝑖=1,…,𝑀,ℎ_𝑖,(3) which is similar to that in [9, 16].

That is, a single “good” relay is selected based on the end-to-end instantaneous wireless channel conditions from the 𝑀 relay candidates to act as the cooperative partner. Among the 𝑀 relay candidates, the relay node that maximizes ℎ_𝑖 is defined as the “good” relay 𝑅_𝑟. We denote min{|ℎ_1𝑟|²,|ℎ_2𝑟|²} by 𝐴_𝑟. We use this relay selection criterion because the quality of the signals received by each destination depends on the quality of the weaker link [16]. Assume this opportunistic relay node can decode 𝑋₁ and 𝑋₂ correctly when its received signal-to-noise ratio (SNR) from 𝑆₁ and 𝑆₂ satisfiesΓ_1𝑟=𝑃^𝑆1||ℎ_1𝑟||²𝑁₀Γ_2𝑟≥𝛾_target,=𝑃^𝑆2||ℎ_2𝑟||²𝑁₀≥𝛾_target,(4) where 𝛾_target is the given SNR constraint for correctly decoding. By using network coding, a bitwise XOR operation at the selected relay node encodes the messages from both 𝑆₁ and 𝑆₂, then the encoded messages are sent to each destination node where 𝑆₁ and 𝑆₂ could get the desired messages by performing an XOR operation [5]. Thus, it only needs three phases (Figure 1(b)). The destination processes, the messages from the source, and the relay jointly when the decoder operates with log likelihood ratios [2] to achieve the performance gain of maximal ratio combining (MRC). Then, the received SNR at the destination 𝑆₂ is Γ₂=𝑃^𝑆1||ℎ₀||²+𝑃_𝑟||ℎ_2𝑟||²𝑁₀.(5) And the resulting SNR at the destination 𝑆₁ is Γ₁=𝑃^𝑆2||ℎ₀|²+𝑃_𝑟||ℎ_1𝑟|²𝑁₀.(6)

Assume each destination can correctly receive the source data whenever Γ₂≥𝛾_target and Γ₁≥𝛾_target. Based on the assumption above, the problem of power efficiency for DaF relay networks with parallel channels can be modeled as in Problem (𝑄₁): submjeinctto_{{PS1,PS2,Pr,𝛾}}𝑃𝑃_𝑆1𝑆1+𝑃_𝑆2+𝑃_𝑟||ℎ₀||²+𝑃_𝑟||ℎ_2𝑟||²𝑁₀≥𝛾_target𝑃_𝑆2||ℎ₀||²+𝑃_𝑟||ℎ_1𝑟||²𝑁₀≥𝛾_target𝑃_𝑆1||ℎ_1𝑟||²𝑁₀≥𝛾_target𝑃_𝑆2||ℎ_2𝑟||²𝑁₀≥𝛾_target.(𝑄¹)

Since the traditional two-way relaying is decomposed into two one-way transmissions (Figure 1(a)), the optimum power allocation strategy for one-way transmission in DaF relay networks [8] is briefly restated as follows. Assume the one-way transmission is from 𝑆₁ to 𝑆₂, then [8] works out the optimum power 𝑃^∗𝑆₁ and the forwarding threshold 𝛼_𝑡⋅𝑃^∗𝑆₁ can only be one of 𝑀+1 discrete values {𝛾_target𝑁₀/|ℎ₀|², 𝛾_target𝑁₀/|ℎ₁₁|², 𝛾_target𝑁₀/|ℎ₁₂|²,…,𝛾_target𝑁₀/|ℎ_1𝑀|²}. The parameters {𝑃^∗𝑆₁,𝛾} are obtained based on that the channel gain of relay-to-destination, |ℎ_2𝑖|² is exponentially distributed, which does not find out the “bottle-neck” of transmission (i.e., the weaker link of the relays). When relaying is selected (i.e., 𝑃^∗𝑆₁<𝛾_target𝑁₀/|ℎ₀|²), the source node broadcasts {𝑃^∗𝑆₁,𝛼_𝑡} to all the candidates. If the channel gain of relay-destination link is greater than the forwarding threshold, namely, |ℎ_2𝑖|²≥𝛼_𝑡, then the selected relays forward the source data. It implies that multiple relays could participate in the relaying simultaneously.

In the model with network coding and opportunistic relaying, as shown in Figure 1(b), when relaying is selected, 𝑃^∗𝑆₁=𝛾_target𝑁₀/|ℎ_1𝑟|², the target SNR can be guaranteed at the selected relay during the transmission from 𝑆₁ to the selected relay. And it is the same to 𝑆₂ when 𝑃^∗𝑆₂=𝛾_target𝑁₀/|ℎ_2𝑟|². But the outage event may occur in the third transmission phase because the instantaneous channel gain of forwarding links may be less than 𝛼_𝑡, which is calculated by the statistics of all the links. Thus, given the target SNR, 𝛾_target, at each destination with a target outage probability, 𝜌_target, in two-way communications, we rewrite Problem (𝑄₁) as Problem (𝑄₂). 𝐸[𝑃_𝑟] is the expected value of the transmit power of opportunistic relay. Problem (𝑄₂) is the problem of power efficiency in two-way relay networks.submjeinctto_{{PS1,PS2,Pr,𝛾}}𝑃𝑃_𝑆1rob+𝑃_𝑆2+𝐸𝑃_𝑟𝑃^𝑆1|ℎ₀|²+𝑃_𝑟||ℎ_2𝑟||²𝑁₀≤𝛾_target≤𝜌^target𝑃_rob𝑃^𝑆2||ℎ₀||²+𝑃_𝑟||ℎ_1𝑟||²𝑁₀≤𝛾_target≤𝜌^target𝑃_𝑆1||ℎ_1𝑟||²𝑁₀≥𝛾_target𝑃_𝑆2||ℎ_2𝑟||²𝑁₀≥𝛾_target.(𝑄²)

Obviously, the transmission through the relaying could be more power efficient than the direct transmission on the condition that each relaying link is better than the direct link (i.e., |ℎ_1𝑟|²≥|ℎ₀|² and |ℎ_2𝑟|²≥|ℎ₀|² ). In this paper, we will work out this problem.

3. Distributed Relaying Decision and Power Allocation Strategy

In this section, a distributed relaying decision mechanism followed by distributed power allocation strategy is proposed to crack the obstacles of centralized mechanism, using the limited CSI at each node.

When 𝑆₁ transmits the training messages, due to the broadcast nature of wireless medium, all relay nodes and 𝑆₂ can simultaneously estimate their 𝑆₁↔𝑅 and 𝑆₁↔𝑆₂ fading coefficients {ℎ_1𝑖,𝑖=1,…,𝑀},{ℎ₀}, respectively. Similarly, when the relay and 𝑆₂ transmit the training bits, {ℎ₀,ℎ₁₁,ℎ₁₂,…,ℎ_𝑖𝑀} can be estimated at 𝑆₁. However, {ℎ₂₁,ℎ₂₂,…,ℎ_2𝑀} may not be available at 𝑆₁. Thus, the distributed strategy is proposed with the realizations {ℎ₀,ℎ_1𝑖,ℎ₁₂,…,ℎ_1𝑀}, the statistics of all the links available at the source 𝑆₁, and the realizations {ℎ₀,ℎ₂₁,ℎ₂₂,…,ℎ_2𝑀}, the statistics of all the links available at 𝑆₂. The relay nodes are assumed to have their local CSI, that is, ℎ_1𝑖 and ℎ_2𝑖 for 𝑅_𝑖,𝑖=1,2,…,𝑀.

The nature of the distributed strategy requires that each relay should make its decision only on its local CSI. Since the opportunistic relay has been chosen in advance (as mentioned in Section 2), the selected relay only needs to decide whether to forward the source data or not. In this paper, the selected relay broadcasts the network-coded data when its minimal channel gain of relay-to-destination satisfies 𝐴_𝑟=min||ℎ_1𝑟||²,||ℎ_2𝑟||²≥𝛼^𝑡,(7) where 𝛼_𝑡 is the forwarding threshold value. One reason is because this network coding needs both ends to decode at the same rate, another reason is that the transmission data rate broadcast by the relay is limited by the weaker link. Thus, the opportunistic relay forwards the decoded signals with sufficient power 𝑃^∗𝑟𝑃^∗𝑟=𝛾^target𝑁₀𝐴_𝑟=max0,𝛾^target𝑁₀−𝑃^∗𝑆𝑘||ℎ_𝑘𝑟||²≤max0,𝛾_targetmin||ℎ_1𝑟||²,||ℎ_2𝑟||²(8)𝑁₀−𝑃^∗𝑆𝑘||ℎ_𝑘𝑟||²𝛼_𝑡,(9) where 𝛾^_target denotes the SNR contribution from the relay.

We note that such a distributed decision mechanism results in a nonzero probability that none of the relay nodes satisfies (7), and hence a nonzero outage probability. A large value of 𝛼_𝑡 means that the selected relay will transmit with less power and less often, which saves power but at the expense of a high outage probability. On the other hand, a small value for 𝛼_𝑡 means that the selected relay will transmit possibly higher power and more often, which results in a low outage probability but possibly high power. Hence, the source nodes should work out the pairs {𝑃^∗𝑆₁,𝛼_𝑡} and {𝑃^∗𝑆₂,𝛼_𝑡} to meet a system-given specification, that is, an outage probability requirement 𝜌_target.

From the source's point of view, the transmit power of the selected relay node is a random variable with the known statistics, since the realizations of the forwarding link are not available at the source node. Due to (3), we have the cumulative distribution function (CDF) of 𝐴_𝑟𝐹_𝐴𝑟(𝑥)=^𝑀_𝑖=11−𝑒^{−𝛽𝑖𝑥},𝑥≥0,(10) and its probability density function (pdf) is given by 𝑓_𝐴𝑟(𝑥)=^𝑀_𝑘=1𝛽_𝑘𝑒^{−𝛽𝑘𝑥𝑀}_𝑖=_{1,𝑖≠𝑘}1−𝑒^{−𝛽𝑖𝑥},𝑥≥0,(11) where 𝛽_𝑖=∑^2𝑘=1𝛽_𝑖𝑘, 𝑖=1,…,𝑀 (refer to the appendix).

Having the pdf of 𝐴_𝑟, the expected value of the transmit power of the selected relay 𝑅_𝑟 is obtained as 𝐸𝑃_𝑟=^∞𝛼_𝑡max0,𝛾_target𝑁₀−𝑃_𝑆𝑘||ℎ₀||²𝑥𝑓^𝐴𝑟(𝑥)𝑑𝑥,(12) where 𝑘=argmax_{𝑗}{|ℎ_𝑗𝑟|²}, 𝑗=1,2. The factor max{0,𝛾_target𝑁₀−𝑃_𝑆𝑘|ℎ₀|²} means that the relay should pay more power according to the channel gain of weaker link to guarantee the correct decoding at both ends in the third transmission phase.

From (3) and (7), the forwarding threshold 𝛼_𝑡 should be chosen as the value that satisfies the outage probability with equality, that is, 𝜌_target=Pr𝐴_𝑟<𝛼_𝑡=𝐹_𝐴𝑟𝛼_𝑡=^𝑀_𝑖=11−𝑒^{−𝛽𝑖𝛼∗𝑡}.(13) Exact expression for the threshold 𝛼_𝑡 is difficult to obtain but it is possible to derive bounds for it. For instance, by letting 𝛽_min=min{𝛽₁,𝛽₂,…,𝛽_𝑀} and 𝛽_max=max{𝛽₁,𝛽₂,…,𝛽_𝑀}, we can bound the outage probability as 1−𝑒^{−𝛽max𝛼∗𝑡}^𝑀≤^𝑀_𝑖=11−𝑒^{−𝛽𝑖𝛼∗𝑡}≤1−𝑒^{−𝛽min𝛼∗𝑡}^𝑀.(14) Therefore, 𝛼^∗𝑡 is bounded as −1𝛽_maxln1−𝜌^1/𝑀_target≤𝛼^∗𝑡≤−1𝛽_minln1−𝜌^1/𝑀_target.(15) The value of 𝛼^∗𝑡 can be obtained by a search in the given range in (15) numerically.

In addition, the transmit power of the source nodes can be calculated by Theorem 1, which provides the following optimal solution.

Theorem 1. The optimum transmit power of the source nodes, 𝑃^∗𝑆₁ and 𝑃^∗𝑆₂, can only be one of the discrete values in the following sets: 𝛾^target𝑁₀||ℎ_1𝑟||²,𝛾^target𝑁₀𝛾||ℎ₀||²,(16)_target𝑁₀||ℎ_2𝑟||²,𝛾^target𝑁₀||ℎ₀||².(17) respectively.
Each source node's transmit power is equal to 𝛾_target𝑁₀/|ℎ₀|² when the direct transmission is preferred. And the source nodes transmit power 𝛾_target𝑁₀/|ℎ_1𝑟|², 𝛾_target𝑁₀/|ℎ_2𝑟|², respectively, when the relaying is preferred. Thus, the power consumption of relaying is less than that of direct transmission, that is, 𝑃^∗𝑆₁+^∞𝛼_∗𝑡max0,𝛾_target𝑁₀−𝑃^∗𝑆₁||ℎ₀||²𝑥𝑓^𝐴𝑟(𝑥)𝑑𝑥≤𝛾^target𝑁₀||ℎ₀||²𝑃,(18)_∗𝑆2+^∞𝛼_∗𝑡max0,𝛾_target𝑁₀−𝑃^∗𝑆₂||ℎ₀||²𝑥𝑓^𝐴𝑟(𝑥)𝑑𝑥≤𝛾^target𝑁₀||ℎ₀||².(19)

Proof. Here, the distributed strategy is based on the opportunistic relaying and network coding. Due to the opportunistic relaying, the “good” relay is chosen prior to the transmission (3). And the source nodes can work out the forwarding threshold 𝛼_𝑡 with the known statistics of all the links (𝛼_𝑡 is the same to both 𝑆₁ node and 𝑆₂ node) by (15). Then, 𝑆₁ node can evaluate (18) to make a decision whether to perform relaying based on the knowledge of ℎ₀ and ℎ_1𝑟 (for 𝑃^∗𝑆₁). At the same time, 𝑆₂ node can also evaluate (19) based on the knowledge of ℎ₀ and ℎ_2𝑟 (for 𝑃^∗𝑆₂). Consequently, 𝑆₁ and 𝑆₂ nodes can exchange “1 bit” signaling information (here, this “1 bit” signaling information is little overhead and can be achieved by a one-time handshaking protocol between the two source nodes.) to finally decide whether to transmit the source data by the direct link or relaying link.
We consider the scenario in which the relaying is preferred. Clearly, the target SNR at the selected relay can be achieved without an outage event in each source-to-relay link with the source transmit power 𝛾_target/|ℎ_1𝑟|², 𝛾_target/|ℎ_2𝑟|², respectively.
Since each source node only knows the statistics of all the links without the knowledge of the realizations of channel gain in the relay-to-destination link, the outage event may occur in the third phase. Due to (13), the target outage probability can be satisfied when the forwarding threshold 𝛼^∗𝑡 is calculated from (15). In the following content, it can be shown that the target SNR at destination can be satisfied. Assume the 𝑅_𝑟↔𝑆₂ link is weaker than the 𝑅_𝑟↔𝑆₁ link, that is, |ℎ_2𝑟|²≤|ℎ_1𝑟|². From (8), we can get 𝑃^∗𝑟=(max{0,𝛾_target𝑁₀−𝑃^∗𝑆1|ℎ₀|²})/|ℎ_2𝑟|², 𝑃^∗𝑆₁=𝛾_target𝑁₀/|ℎ_1𝑟|²≤𝛾_target𝑁₀/|ℎ_2𝑟|² (i.e., the transmit power of 𝑆₁ is less than that of souce2, 𝑃^∗𝑆₁≤𝑃^∗𝑆₂).
At 𝑆₂, the received SNR is 𝑃^∗𝑆₁||ℎ₀||²+𝑃^∗𝑟||ℎ_2𝑟||²=𝑃^∗𝑆1||ℎ₀||²+max0,𝛾^target𝑁₀−𝑃^∗𝑆₁||ℎ₀||²||ℎ_2𝑟||²||ℎ_2𝑟||²≥𝛾_target.(20) At 𝑆₁, the received SNR is 𝑃^∗𝑆₂||ℎ₀||²+𝑃^∗𝑟||ℎ_1𝑟||²=𝑃^∗𝑆₂||ℎ₀||²+max0,𝛾^target𝑁₀−𝑃^∗𝑆₁||ℎ₀||²||ℎ_2𝑟||²||ℎ_1𝑟||²≥𝑃^∗𝑆₂||ℎ₀||²+max0,𝛾^target𝑁₀−𝑃^∗𝑆₁||ℎ₀||²||ℎ_2𝑟||²||ℎ_2𝑟||²≥𝛾_target.(21)
Using the expression in (12), the expected value of total transmit power is as follows: 𝐸𝑃_total==𝑃𝑃_𝑆1𝑆1+𝑃_𝑆2+𝐸𝑃_𝑟+𝑃_𝑆2+^∞𝛼_∗𝑡max0,𝛾_target𝑁₀−𝑃_𝑆1||ℎ₀||²𝑥𝑓^𝐴𝑟(𝑥)𝑑𝑥.(22)
Therefore, the target SNR and the target outage probability can be satisfied. Theorem 1 is proved.

The proposed “energy-efficient relaying strategy with network coding” in two-way parallel channels is summarized in Table 1. Here, the opportunistic relaying is applied to decrease the cooperative costs, to save power, and capture the nice link for the practical system. Moreover, the network coding is introduced to save power.

Table 1 

Energy-efficient relaying strategy with network coding in two-way parallel channels.

4. Simulation Results

In this section, we will consider the following three different cooperative schemes.(i)The proposed distributed strategy (for relaying decision and power allocation) combined with opportunistic relaying and network coding (DSON for short). (ii)The optimum distributed power allocation (ODPA) scheme [8]. (iii)Single relay model (SRM) [8].

The traditional two-way relaying is decomposed into two one-way transmissions (as shown in Figure 1(a)). Moreover, to the best of our knowledge, (ii) and (iii) outperform the existing distributed schemes in one-way relaying, so we focus on comparing the performance of (i) with that of (ii) and (iii). We consider the two-way transmission consisting of 𝑆₁ node and 𝑆₂ node 100 m apart, and 𝑀 relay nodes between the source nodes. The fading model is considered as [15], that is, the variance of the channel gain is proportional to the distance between nodes. Here, we have 𝜎^2𝑘𝑖=0.5𝐶𝑑^𝛼𝑆_𝑘𝑅𝑖𝜎,𝑘=1,2,₂₀=0.5𝐶𝑑^𝛼𝑆_1𝑆2,(23) where 𝑑_𝑗𝑖 is the distance between node 𝑗 and node 𝑖, the path-loss exponent is denoted by 𝛼, the factor 0.5 is due to the above variances defined by two dimension. 𝐶 is a constant that is expressed as 𝐶=𝐺_𝑡𝐺_𝑟𝜆²/𝐿(4𝜋)², where 𝐺_𝑡 is the transmitter antenna gain, 𝐺_𝑟 is the receiver antenna gain, 𝜆 is the wavelength, and 𝐿 is the system loss factor not related to propagation (𝐿≥1). The values 𝛼=3, 𝐺_𝑡=𝐺_𝑟=1, 𝜆=(1/3)𝑚, 𝐿=1, are used in the simulations. Assume the AWGN variances on all links to be 𝑁₀=10⁻¹⁰ and the target SNR (𝛾_target) to be 10 dB.

We first consider the case in which {𝑑_{𝑆1𝑅𝑖}}^𝑀𝑖=1={0.2,0.8}, {𝑑_{𝑅𝑖𝑆2}}^𝑀𝑖=1={0.8,0.2}, and 𝑀=2. Figure 3 illustrates the numerical results of the expected value of the total power 𝐸[𝑃_total] as a function of the target outage probability 𝜌_out. It is observed that the proposed strategy (DSON) outperforms the existing schemes at practical values of outage probability. For instance, for an outage 𝜌_out=0.05, approximately 22%, 37% is saved by the DSON scheme as compared to the ODPA and the SRM schemes, respectively. The performance of the DSON scheme is not better than that of the ODPA and the SRM at the higher outage probability regimes but we know, as fact, that the high outage probability is prohibited in practical systems.

Figure 3 

𝐸[𝑃total] versus the outage probability 𝜌out for the different distributed strategies with {𝑑𝑆1𝑅𝑖}𝑀𝑖=1={0.2,0.8}, {𝑑𝑅𝑖𝑆2}𝑀𝑖=1={0.8,0.2}, and 𝑀=2.

Generally, the relays are located in the midst of two source nodes. The following simulation results are based on the scenario in which all the relay nodes are in the midst of two source nodes. Namely, {𝑑_{𝑆1𝑅𝑖}}^𝑀𝑖=1={𝑑_{𝑅𝑖𝑆2}}^𝑀𝑖=1={0.5,…,0.5}.

Figure 4 illustrates the numerical results of the expected value of the total power 𝐸[𝑃_total] as a function of the target outage probability 𝜌_out for the case 𝑀=2. Similarly to the results of Figure 3, Figure 4 shows that the proposed scheme (DSON) outperforms both (ii) and (iii) at the lower 𝜌_out regime. It is also observed that the relative gain of the proposed scheme is larger in comparison to the case of Figure 3. For instance, at outage of 𝜌_out=0.05, approximately 30%, 31% is saved by the DSON with respect to the ODPA and the SRM, respectively. It is clear that, when the relays are in the middle between the two source nodes, they provide good help to both source nodes. With only two relays (𝑀=2), the opportunistic relaying has less spatial channels to choose from (less spacial diversity) and suffers from broadcasting the network-coded data according to the weaker link in the third transmission phase; this is the implication on the performance of the DSON scheme at higher outage probability regime.

Figure 4 

𝐸[𝑃total] versus the outage probability 𝜌out for the different distributed strategies with 𝑀=2.

Figure 5 illustrates the numerical results for the case when the number of relay nodes is 𝑀=8. Here, it is observed that the DSON strategy has improved considerably in comparison to the case of one- and two-relay nodes. The DSON strategy now enjoys more spatial diversity in the presence of more relay nodes as the candidates. Though the ODPA and the SRM also have more spatial diversities, both the ODPA and the SRM select relays mainly based on source-to-relay links, so their spatial dimension is half of that of the DSON strategy. Moreover, the DSON with network coding only has three transmission phases, this results in more saving in power consumption.

Figure 5 

𝐸[𝑃total] versus the outage probability 𝜌out for the different distributed strategies with 𝑀=8.

5. Conclusions

In this paper, an energy-efficient relaying strategy with network coding in two-way parallel channels is proposed. In the first stage of this proposal, based on the opportunistic relaying strategy, only one relay is selected to assist the transmission. It decreases the cooperative costs among relays, which is preferred in practical systems. Meanwhile, it guarantees that the DSON chooses the relay from both source-to-relay and relay-to-destination links. Moreover, network coding decreases the transmission phases from four to three. Combined with the opportunistic relaying and network coding, distributed relaying decision and power allocation strategy is applied to obtain the optimal power efficiency. Since only limited CSIs needed in this scheme, it is practical for real applications in two-way relaying networks.

Of course, the DSON suffers from broadcasting the network-coded data according to the weaker link in the third transmission phase for satisfying the requirements of target SNR and target outage probability. From a power saving point of view, the DSON outperforms the ODPA and the SRM in a two-way relaying communication link.

Appendix

For purposes of completeness, we briefly summarize a result from [16], Lemma A.

Lemma A (see [16]). Random variables 𝐴_𝑖,𝐴_𝑖1,…,𝐴_𝑖𝐾, (𝑖=1,…,𝑀), 𝐴_𝑖1,…,𝐴_𝑖𝐾 are independently exponential distributed with parameter 𝛽_𝑖1,…,𝛽_𝑖𝐾, respectively, 𝐴_𝑖=min{𝐴_𝑖1,…,𝐴_𝑖𝐾}. Let 𝑟=argmax_𝑖𝐴_𝑖, for 𝐹_𝐴𝑟(𝑥), the CDF of 𝐴_𝑟, there is 𝐹_𝐴𝑟(𝑥)=^𝑀_𝑖=11−𝑒^{−𝛽𝑖𝑥},(A.1) where 𝛽_𝑖=∑^𝐾𝑘=1𝛽_𝑖𝑘, 𝑖=1,…,𝑀.

Acknowledgments

This paper was supported by the International Science and Technology Cooperation Program (2008DFA12160), China's 863 Project no. 2009AA011501, National Natural Science Foundation of China no. 60832008, China's Major Project no. 2009ZX03003-009, Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT), National Science and Technology Pillar Program no. 2008BAH30B09, and National Basic Research Program of China no. 2007CB310608.