## 2 Two-Way Relaying System Model and Related Works
They analyze the performance of the considered systems through EXIT charts and verify it by BER estimation. Again, the authors apply a very idealized, unrealistic model with ideal symbol synchronization and AWGN as the only distortion. However, for multi-level modulations, allowing different attenuations and phase shifts of the signals received from both end nodes would result in much more complex schemes on the receiver side, so it can be the reason for justification of the applied idealized transmission model. In [27], [PERSON], [PERSON], and [PERSON] consider a PNC coding technique for a two-way relaying network that exploits the spatial modulation (SM) with convolutional codes at both source nodes and the relay node. It is assumed that all the nodes are equipped with multiple antennas. The relay node detects the signal by utilizing a maximum-likelihood detection technique based on a direct decoding or separate decoding algorithm. The authors claim that the SM-based PNC outperforms the conventional PNC technique. Simulation results achieved by the authors of [27] indicate that the direct decoding yields lower complexity than the separate decoding while achieving almost the same performance as the latter. It is worth noting that, in this case, the authors apply a more realistic model in which each constituent channel between any terminal antenna and any relay antenna is characterized by a complex channel coefficient. This leads to the channel representation in the form of complex channel matrices. To a certain extent, it is a necessity because using a fully ideal channel model in the MIMO configuration (with all channel coefficients equal to one) would not enable operation of the relay receiver. Application of a multi-antenna system in PNC two-way relaying is a subject of the paper [28]. The extension of the transmission range in an LTE-like system is considered using a relay station playing the role of a relay between two end nodes, namely, a base station (BS) and a mobile station (MS). It is shown that using more antennas in the relay enables detection of both MS and BS signals separately and, after turbo decoding of both streams, modulo-2 addition of the MS and BS packets can be performed, which results in a packet subsequently transmitted in the broadcast phase. BER estimation is selected as a performance measure. This topic is further investigated in [29]. The authors assume operation in the multiple access phase similar to that in [28], and compare possible arrangements of transmission in the broadcast phase. The first arrangement is a typical PNC system similar to that considered in [28], whereas, in the second one, precoding with the multi-user MIMO technique is applied. The authors prove that the multi-user MIMO method can obtain better performance than the network coding technique at the cost of higher computational requirements. Finally, a multiple antenna relay in two-way relaying transmission is also considered in [30]. The authors assume that both end nodes are equipped with single antennas. The subject of analysis is a posteriori probability (APP) relay detector. General formulas for log-likelihood ratios for received codeword bits, as well as mutual information characterizing each type of the considered APP detector, are given. The authors analyze the so-called separate decoding detector, in which the relay detects the codewords received from each node separately. They also consider two versions of a joint channel decoding and network coding detector (regular and generalized one), which determines the codewords being the sum of the codewords transmitted by both end nodes. The drawback of the two last detectors is the necessity of application of the same codes by both end nodes when transmitting the packets to the relay. Other questions are the computational complexity of the APP detectors despite their statistical optimality, and the sensitivity of the considered detectors to inaccuracies in the channel characteristics.