Multi-pair two-way relaying systems with physical layer network coding

Abstract

Physical layer network coding (PNC) significantly improves the throughput of the two-way relay channel (TWRC). However, most of current work on PNC just focuses on one pair users, and there are limit work on multi-pair PNC, in which the main bottleneck on system performance is the interference caused by the remaining communicating pairs. In this paper, we consider a scenario where multiple pairs communicate with each other through one relay node. For a purpose of comparison, we present the analytical capacities of the multi-pair version of the existing TWRC schemes in closed forms. Considering more practical issues such as the direction distribution of incident signals, we propose a spatial multi-pair physical layer network coding scheme. The novelty includes: in the multiple access phase, the log likelihood ratio combination scheme is proposed, which is robust in bad communication conditions; in the broadcast phase, two adaptive group broadcasting schemes are proposed, which can efficiently overcome the near interference problem. Simulation results validate the superiority of the proposed algorithms.

Appendix

Through comparing the value of SINR, we have $\gamma_{R,1}^{\text{SM2}} > \{ \gamma_{R,1,1}^{\text{SM1}} ,\gamma_{R,1,2}^{\text{SM1}} \}$ and we conclude the second mapping scheme has a better performance than the first one. Since $a_{1,1}$ and $a_{1,2}$ are independent BPSK symbols, we can choose uniform distribution, which would give the optimal channel capacity. Then, the distribution of the multiple access channel input, $a_{1,1} + a_{1,2}$, is

$$\begin{aligned} & p(a_{1,1} + a_{1,2} = 2) = p(a_{1,1} + a_{1,2} = - 2) = 1/4 \\ & p(a_{1,1} + a_{1,2} = 0) = 1/2 \\ \end{aligned}$$

(14)

After hard decision [5], we obtain the estimation of the input $a_{1,1} + a_{1,2}$, denoted by . Suppose $p_{2,1}$, $p_{2,2}$ and $p_{2,3}$ are the crossover probabilities from a symbol in $a_{1,1} + a_{1,2}$ to another symbol in such that

(15)

Then we have

$$\begin{aligned} & p_{2,1} =\, Q(\eta \sqrt {\gamma_{R,1}^{\text{SM2}} } ) \\ & p_{2,2} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {1 - Q((\eta - 2)\sqrt {\gamma_{R,1}^{\text{SM2}} } ) - Q((\eta + 2)\sqrt {\gamma_{R,1}^{\text{SM2}} } )} & {\eta > 2} \\ \end{array} } \\ {\begin{array}{*{20}c} {Q((2 - \eta )\sqrt {\gamma_{R,1}^{\text{SM2}} } ) - Q((\eta + 2)\sqrt {\gamma_{R,1}^{\text{SM2}} } )} & {1 < \eta \le 2} \\ \end{array} } \\ \end{array} } \right. \\ & p_{2,3} =\, Q((\eta + 2)\sqrt {\gamma_{R,1}^{\text{SM2}} } ) \\ \end{aligned}$$

(16)

where $\eta = 1 + \frac{1}{{2\gamma_{R,1}^{\text{SM2}} }}\ln (1 + \sqrt {1 - \exp ( - 4\gamma_{R,1}^{\text{SM2}} )} )$ is the optimum hard decision threshold obtained by applying the maximum posterior probability criterion [27]. The upper bound of the MAC capacity for the first pair users can be calculated as follows

(17)

The relay can at most obtain $\tilde{C}_{{{\text{MAC}},1}}^{\text{SM2}}$ bits of the first pair users’ summation information successfully. The relay will then MAC the received symbols to according to PNC mapping. And the distribution is

(18)

Note that there is some information loss due to this operation which corresponds to the mapping from both ‘2’ and ‘−2’ to the single bits ‘0’. The upper bound of the effective MAC capacity for the first pair users is therefore

(19)

Then the upper bound of the total effective MAC capacity is $\left( {C_{\text{MAC}}^{\text{SM2}} } \right)^{up} = \left( {\sum\limits_{k = 1}^{K} {\left( {C_{{{\text{MAC}},k}}^{\text{SM2}} } \right)^{up} } } \right)/K$. With reference to [28], the lower bound of SM-PNC2 MAC capacity for the first pair users can be denoted as

(20)

With the optimal decision threshold, we can obtain the conditional probability

(21)

(22)

(23)

(24)

Then, we can obtain

(25)

Then the lower bound of the total effective MAC capacity is $\left( {C_{\text{MAC}}^{\text{SM2}} } \right)^{low} = \left( {\sum\limits_{k = 1}^{K} {\left( {C_{{{\text{MAC}},1}}^{\text{SM2}} } \right)^{low} } } \right)/K.$