## 4 Numerical Results
### Spectral Efficiency  
In the traditional denoising forwarding scheme, the relay receives signal \(y=x_{1}+x_{2}+n\). The constellation point of the in-phase branch of the transmitted signal after noise elimination mapping is \(\{-2L+2,-2L+4,\cdots,2L-4,2L-2\}\). Phase BC is approximately NQAM, i.e., \(N=\left(2L-1\right)^{2}\). If the modulation mode is 16 QAM, the signal received by the node is approximately 49 QAM. The signal received by the node in this scheme is still a 16 QAM. With the increase in the modulation order of the user, the modulation order of the signal received by the node increases exponentially. In the broadcast transmission phase, the distance between the constellation points of the modulation mapping scheme proposed in this paper is twice that of the denoising scheme. The scheme proposed in this paper solves the constellation ambiguity problem in the high order amplitude and phase modulation of bidirectional relay physical layer network coding. The num  
Figure 6: Performance comparison of signal space structure optimization. (**a**) Orthogonal alignment, (**b**) Alignment order optimization, (**c**) Alignment metric, (**d**) Scheme comparison. ber of signal constellation points is reduced by repositioning and merging the received signals at the relay node. The Euclidean distance between adjacent constellation points becomes larger, and the system bit error rate performance is improved. Figure 8 shows that the scheme in this paper can break through the limitations of BPSK modulation and adopt higher-order modulation when the relay encodes the aligned superimposed signal in the physical layer network. A greater sum rate can be obtained when 64 QAM is used. At the same time, since the distance between constellation points in the remapping scheme proposed in this paper is twice that of the denoising scheme, the channel transmission reliability is also better than the denoising scheme. The order for determining the orthogonal alignment direction is very important to further improve system performance. The basic idea is to determine the order of alignment directions according to the quality of the upstream channel from each node to the relay. That is, the first step is to solve and determine the direction corresponding to the channel with the worst quality. Finally, the alignment direction with optimal channel quality is determined. Taking a 3-user relay system as an example, the decoding accuracy of the received signals in the 3-aligned direction is compared through simu  
Figure 8: Sum rate comparison under different modulation orders. Figure 7: Comparison of BER among different schemes with [31; 32]. resents the amplitude gain in a certain orthogonal alignment direction. The amplitude gain of different orthogonal directions is not the same. The minimum Euclidean distance among different signal constellations is related to this. As shown in Figure 9, the first alignment direction can be selected from the whole matrix dimension, which has the optimal selection gain amplitude gains. Therefore, the best correct decoding performance can be obtained. The last alignment direction cannot be selected due to the limitation of spatial resource dimensions, which cause the worst performance. SSA randomly selects the alignment direction in the signal space, which may reduce the minimum Euclidean distance and cannot improve the decoding performance. Rayleigh fading channel is adopted, and two-way and multidirectional communication is assumed to be carried out by the relay. The transmitter knows the channel status information. Under the condition of 4 users, each user is equipped with 7 antennas and the relay is equipped with 13 antennas. The sum rate simulation results of the relay system are shown in Figure 10. Additionally, the rate optimization problem is also called the rate adaptive optimization problem. Specifically, rate optimization problems indirectly reflect the maximization of channel capacity. Resources are allocated adaptively according to time-varying CSI information and QoS parameters.