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A novel design of physical layer network coding in strong asymmetric twoway relay channels
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 166 (2013)
Abstract
The quality of twoway links is always asymmetric in practical twoway relay channels (TWRC) and therefore, the capacity of TWRC will be limited by the weaker link. An asymmetric modulation scheme, with physical layer network coding, was presented in order to improve the transmission reliability in TWRC. This makes full use of the stronger link to improve the overall transmission rate and also ensures the reliability of the weaker link. The simulation results show that the proposed asymmetric modulation scheme in the case of strongly asymmetric channels, compared to the symmetric transmission, enhances the system capacity significantly and also guarantees the system reliability.
1. Introduction
Network coding (NC) was originally proposed to improve the performance of multicast throughput in wired networks by Ahlswede et al. in 2000 [1]. Recently, the broadcast nature of wireless channels has attracted a lot of research activities on the application of NC in wireless networks.
The twoway relay channel (TWRC) is a typical scenario in wireless communications. Physical layer network coding (PLNC) was proposed to improve the throughput of TWRC [2], which maps the superimposition of the signals received simultaneously to a digital bit stream. PLNC improves the system performance by making use of the interference, instead of avoiding it. PLNC can further be classed into two categories  PNCF (PNC over finite field) and PNCI (PNC over infinite field)  according to whether the network coding field adopted is finite or infinite [3]. The capacity of TWRC with PLNC is higher than the traditional communication strategies [4, 5]. The design of modulation suited for TWRC with PLNC can be BPSK or QPSK [6], and an unconventional 5ary modulation which is optimized according to the channel condition [7]. In addition, PLNC can be combined with channel code to improve bit error rate (BER) performance of the system [8].
The research works reported above are all based on the same assumption that the transmission rate of two end nodes is symmetric. In practical TWRC, however, the quality of twoway links is always asymmetric. Therefore, the capacity of TWRC will be limited by the weaker link. Thus, in the symmetric rate transmission of TWRC, in order to ensure the reliability of the weaker link, the stronger link has to transmit and receive with loworder modulation as same as the weaker link. For this reason, the stronger link does not take advantage of its good channel conditions to improve the overall transmission rate, which lowers the validity.
Asymmetric modulation is a method to realize the asymmetric rate transmission. The power matching ratio of the two end nodes in TWRC is corresponding to the performance of symbol error rate in the multiple access phase [9, 10]. In the broadcast phase of TWRC, under the same BER constraint, the weaker link can decode at lower signal noise ratio (SNR) compared to the stronger link by exploiting a priori bit information in each transmit symbol [11].
This paper investigates the asymmetric rate transmission both in the multiple access phase and broadcast phase of the twophase TWRC by designing an asymmetric modulation scheme with PLNC. The simulation results show that the proposed scheme not only improves the transmission validity by increasing the system capacity but also guarantees the transmission reliability.
The rest of this paper is organized as follows. Section 2 introduces the system model. Section 3 presents the new scheme of asymmetric modulation with PLNC. Section 4 analyzes the performance of the asymmetric modulation scheme. Simulation experiments and performance comparisons between the symmetric transmission mode and the proposed scheme will be discussed in Section 5. Finally, Section 6 concludes this paper.
2. System model
We consider the twophase TWRC system as shown in Figure 1, in which two independent end nodes n_{1} and n_{2} exchange information with each other via the relay node n_{R}. As illustrated in Figure 1, B_{ i }, i∈ {1, 2, R} denotes the bit information of node n_{ i } which is modulated to symbol X_{ i }. The coefficient h_{ ij }, i,j∈ {1,2,R} is the complex pathloss coefficient for the channel from n_{ i } to n_{ j }. W_{ i } is the noise at n_{ i } to be zeromean complex Gaussian variable with variance σ_{ i }^{2}. P_{ i } is the transmitting power of n_{ i }. In this paper, the following assumptions are made: (1) all nodes operate in a halfduplex manner, i.e., a node cannot transmit and receive simultaneously; (2) symbollevel time synchronization is assumed; (3) there is no direct link between n_{1} and n_{2}; (4) channel coefficient is quasistatic and possess reciprocity, i.e., h_{1R} = h_{R1} = h_{1}, h_{2R} = h_{R2} = h_{2}; (5) nodes can perfectly estimate the channel state information (CSI) to realize the phase synchronization and amplitude preequalization; (6) the energy of symbol X_{ i } is normalized to unit 1; (7) power constrains P_{1} + P_{2} = 1, P_{R} = 1.
The process of information exchange comprises of two phases:

1.
Multiple access phase (MAC). First, n _{1} and n _{2} modulate B _{1} and B _{2} respectively to X _{1} and X _{2}. Second, n _{1} and n _{2} transmit X _{1} and X _{2} to the relay node n _{R} simultaneously. Then n _{R} demodulates the superimposition of the signal Y _{R} to B _{R} = B _{1}⊕B _{2}.

2.
Broadcast phase (BRC). The relay node n _{R} modulates B _{R} to X _{R} and broadcasts it to n _{1} and n _{2}, simultaneously. Then n _{1} demodulates Y _{1} to B _{R} and get the bit information of n _{2} by using exclusive or (XOR) operation B _{2} = B _{1}⊕B _{R}. Similarly, n _{2}demodulates Y _{2} to B _{R} and get the bit information of n _{1} by using XOR operationB _{1} = B _{2}⊕B _{R}.
So, n_{1} and n_{2} can finish the information exchange only by two phases. The importance of the information exchange is the design of modulation and demodulation at the relay node n_{R}. In multiple access phase, n_{R} is supposed to realize the demodulation of Y_{R} → B_{R} = B_{1}⊕B_{2}. In broadcast phase, n_{R} has to realize the modulation of B_{R} → X_{R}. The performance of the system depends on the design at n_{R}, which is a key point in this paper.
3. Asymmetric rate transmission
The design of the modulation and demodulation for n_{1}, n_{2}, and n_{R} will be presented in this section to realize the asymmetric rate transmission both in the multiple access phase and broadcast phase. For simplicity and without loss of generality, we assume ${\sigma}_{{h}_{1}}^{2}>{\sigma}_{{h}_{2}}^{2}$ (respectively denote the variances of h_{1} and h_{2}) which means that the stronger link C_{1} (the channel between n_{1} and n_{R}) has better quality than the weaker link C_{2} (the channel between n_{2} and n_{R}).
3.1 Design of asymmetric modulation
The asymmetric modulation is proposed by utilizing the asymmetric channel quality to make n_{1} and n_{2} transmit and receive at different rates. For simplicity and clarity, QPSK and 16QAM will be as examples to be depicted for the design of the asymmetric modulation, which are respectively 2 bit/symbol and 4 bit/symbol.
3.1.1 MAC phase
Before transmitting, n_{1} first inserts two dummy zeros ‘00’ for each two bits in B_{1} to obtain $\stackrel{\xb7}{{B}_{1}}$, making ‘00’,‘01’,‘10’, and ‘11’ become ‘$\stackrel{\xb7}{0}\stackrel{\xb7}{0}00$’,‘$\stackrel{\xb7}{0}\stackrel{\xb7}{0}01$’,‘$\stackrel{\xb7}{0}\stackrel{\xb7}{0}10$’, and‘$\stackrel{\xb7}{0}\stackrel{\xb7}{0}11$’ (these dummy zeros contain no information and their positions are fixed and known to both n_{1} and n_{2}); Then n_{1} modulates $\stackrel{\xb7}{{B}_{1}}$ to X_{1} by QPSK, and n_{2} modulates B_{2} to X_{2} by 16QAM (the constellation of X_{1} and X_{2} are shown in Figures 2 and 3. Finally, n_{1} and n_{2} transmit X_{1} and X_{2} to n_{R}, respectively. So in the same symbol period, n_{1} transmits twobit information while n_{2} transmits four bit information.
As illustrated in Figures 2 and 3, the stronger link C_{1} transmits signals by loworder modulation (QPSK), while the weaker link C_{2} transmits signals by highorder modulation (16QAM). This kind of design might be opposite to our straight thinking, but it lets the system make use of the stronger link C_{1} to improve the overall transmission rate and also guarantee the transmission reliability of the weaker link C_{2} in the BRC phase (the specific statement is next). When the transmitting power ratio is P_{QPSK} : P_{16QAM} = 4 : 5, the BER performance of the QPSK16QAM superimposition signals is optimal in the MAC phase, which is demonstrated in [9]. Note that there are 36 points in the QPSK16QAM constellation as shown in Figure 4.
3.1.2 BRC phase
In the BRC phase, n_{R} first modulates B_{ R } to X_{ R }. Then n_{R} broadcasts X_{R} to n_{1} and n_{2}. The constellation of X_{R} is shown in Figure 3, which is same as the constellation of X_{2} in the MAC phase.
Due to the fore 2 bit ‘$\stackrel{\xb7}{0}\stackrel{\xb7}{0}$’ in each 4 bit of $\stackrel{\xb7}{{B}_{1}}$ which is inserted by n_{1} as the dummy zeros, the fore 2 bits of B_{2} are kept unchanged in B_{R} after the XOR operation at n_{R}. So n_{2} can exploit these as a priori information and does not need to decode every bit in B_{ R }. Node n_{2} can discard the known bits and decode only the latter 2 bits of B_{ R } according to the subset of 16QAM constellation (QPSK), which is illustrated in Figure 5. Then n_{2} can obtain the bit information of n_{1} by executing the XOR operation B_{1} = B_{2}⊕B_{R}.
As illustrated in Figure 5, the constellation used to demodulate signals by n_{2} is QPSK, which is a subset of 16QAM constellation. In this way, the distance of the adjacent points is increased, so the BER performance is improved and the transmission reliability is guaranteed.
By using this scheme, in BRC phase, the signal modulated by highorder modulation (16QAM) is transmitted through the stronger link C_{1} to improve the transmission rate, while the signal modulated by loworder modulation (QPSK, which is the subset of 16QAM constellation) is transmitted through the weaker link C_{2} to ensure the transmission reliability.
3.2 Symmetric modulation for comparison
In order to show up the advantages of the proposed asymmetric modulation scheme, we use the symmetric modulation QPSKQPSK as examples for a simple comparison.
By using symmetric modulation, the stronger link C_{1} has to adopt the same loworder modulation as the weaker link C_{2} to transmit and receive signals. Both n_{1} and n_{2} adopt QPSK modulation to transmit and receive signals, and n_{R} adopt QPSK modulation to transmit the XOR signals. The constellations are shown in Figures 6 and 7.
4. Performance analysis
In this section, we will analyze the performance of the proposed asymmetric modulation and compare it to that of the symmetric modulation. As we known, C_{1} is better than C_{2}. Without loss of generality, we assume μ> 1. For simplicity, we assume that the noises at the three nodes have the same variance, ${\sigma}_{1}^{2}={\sigma}_{2}^{2}={\sigma}_{\mathrm{R}}^{2}={\sigma}^{2}$.
4.1 BER analysis
4.1.1 MAC phase
Here, we use d_{min} to denote the minimum distance between the adjacent points in the constellation of the superimposition of the signals received at n_{R} (as shown in Figures 4 and 7). Thus, we have
Then the approximation of the BER can be obtained according to [12, 13] as
Where γ_{R} is SNR of the signals received at n_{R}.
4.1.2 BRC phase
Also, we use d_{min} to denote the minimum distance between the adjacent points in the constellation of the signals received at n_{1} and n_{2} from n_{R} (as shown in Figures 3, 5, and 6). Then we have
Further, the approximation of the BER can be obtained according to [12, 13] as
where γ_{1} and γ_{2} denote SNR of the signals received at n_{1} and n_{2}, respectively.
4.3 Capacity analysis
According to [14, 15], the AWGN channel in MAC phase and BRC phase can be regarded as an equivalent virtual channel C_{V} shown in Figure 8.
The input of C_{V} is X and the output is Y = X + W, where W is a zeromean complex Gaussian random variable W ∼ CN(0, σ^{2}). C_{V} is a discreteinput and continuousoutput channel according to [15]. The input X comprises symbols selected from a finite and discrete input alphabet X = x_{ k } (k = 0, 1, …, q − 1), and the output is continuous Y = { − ∞, + ∞}. For a given X, it follows that Y is a complex Gaussian random variable with mean x_{ k } and variance σ^{2}, that is,
For any given input sequence X_{ i } (i = 1, 2,…, n), there is a corresponding output sequence of
The condition that the channel is memoryless can be expressed as
So, the capacity of C_{V} is defined as
The input X are equally probable symbols, and the probability p(x_{ i }) can be obtained according to the corresponding modulation constellation. Substituting p(x_{ i }) and Equation 7 into Equation 8, we can obtain the capacity of the equivalent virtual channel C_{V}.
5. Simulation results
In this section, simulation results are presented to demonstrate the performance of our proposed scheme and to verify the accuracy of our analytical analysis in Section 4. In the simulation, all the numerical results are calculated with averaging over 10,000,000 packets, and the number of bits contained by each packet is equal to the bits contained by each symbol in the corresponding constellation. For simplicity and without loss of generality, we consider two scenarios of μ = 4 and μ = 8. The SNR of the stronger link C_{1} is 6 and 9 dB higher than that of the weaker linkC_{2}.
5.1 BER performance
Here, the simulation results are presented to demonstrate the BER performance of the asymmetric modulation at n_{1} and n_{2}.
Figures 9 and 10 respectively show the BER performance of n_{1} and n_{2} when the asymmetric levels of the twoway links are 6 and 9 dB. As can be seen in Figures 9 and 10, when the asymmetric modulation QPSK16QAM is adopted, although the modulation is different (one is QPSK, another is 16QAM), the BER performance of n_{1} is close to n_{2}. Moreover, the closeness of the BER performance for the symmetric modulation QPSKQPSK is similar with the asymmetric modulation QPSK16QAM. When the asymmetric level of the twoway links increases (from 6 to 9 dB), the trend remains unchanged.
5.2 Capacity
Here, the simulation results are presented to demonstrate the capacity of the asymmetric modulation at the end nodes n_{1} and n_{2}. The sum of capacity is also illustrated here.
Figures 11 and 12 respectively show the capacity of n_{1} and n_{2}. The xaxis is the SNR of C_{2} in the two figures above, which needs to be paid attention to. As shown in Figure 11, when the SNR of the stronger link C_{1} is 6 dB higher than that of the weaker link C_{2} and the SNR is higher than 13 dB, the capacity of n_{1} by using the asymmetric modulation QPSK16QAM is higher than by using the symmetric modulation QPSKQPSK. When the SNR of the stronger link C_{1} is 9 dB higher than that of the weaker link C_{2}, the situation is similar when SNR is higher than 11 dB. As shown in Figure 12, when the channel quality is asymmetric, the capacity of n_{2} by using the asymmetric modulation QPSK16QAM is close to the capacity of n_{2} by using the symmetric modulation QPSKQPSK.
Figure 13 shows the total capacity of n_{1} and n_{2} when the asymmetric level of the twoway links is 6 and 9 dB. As shown in Figure 13, when the SNR of the stronger link C_{1} is 6 dB higher than that of the weaker link C_{2} and the SNR is higher than 15 dB, the capacity of n_{1} by using the asymmetric modulation QPSK16QAM is higher than by using the symmetric modulation QPSKQPSK. When the SNR of the stronger link C_{1} is 9 dB higher than that of the weaker link C_{2}, the situation is similar when SNR is higher than 13 dB. Overall, under higher SNR, compared with the symmetric modulation, the total capacity has been greatly improved by using the asymmetric modulation.
6. Conclusions
In this paper, an asymmetric modulation scheme with PLNC in TWRC is proposed, which aims to improve both the validity and reliability in twoway relay transmissions. The proposed asymmetric modulation scheme realized the asymmetric rate transmission both in MAC phase and BRC phase of TWRC. In MAC phase, the BER performance at the relay is improved. In BRC phase, the capacity is boosted by making full use of the stronger link, and the BER performance is guaranteed by exploiting a priori bit information to demodulate for the weaker link. We derived the approximated BER expressions for the scheme proposed, which were also demonstrated by simulation experiments. Through the comparisons of the symmetric modulation scheme, it is found that by using the proposed asymmetric modulation scheme, the total capacity is improved significantly under the asymmetric level of the twoway links.
In addition, as wellknown channel coding possesses the correcting ability and can improve the BER performance further, combining channel coding, network coding, and modulation for asymmetric transmissions in TWRC will be our future researches.
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Acknowledgement
This work was supported by the National Natural Science Foundation of China under No. 61271240.
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Wei, H., Zheng, B. & Ji, X. A novel design of physical layer network coding in strong asymmetric twoway relay channels. J Wireless Com Network 2013, 166 (2013). https://doi.org/10.1186/168714992013166
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Keywords
 Network coding
 Asymmetric modulation
 Twoway relay channel
 Subset constellation