## 2 Two-Way Relaying System Model and Related Works
Channel estimation errors in PNC transmission over fading channels are the subject of the research reported in [31]. The authors model channel estimation errors as Gaussian distributed ones and formulate the network coding error as the distance between real and estimated points in the channel coefficients plane. Utilizing this model, the authors present a statistical lower bound on the variance of estimation error that can be toleratedby the relay terminal without imposing a network coding error on the system. In our own research, we also take into account non-ideal channel knowledge and estimate its influence on the system performance expressed by BER. Two-way relaying systems continue to be the subject of intensive investigation. In recent years, many new papers have been published in this area. Below, we consider only a few positions. In [32], a new approach for two-way relaying networks is proposed. The authors consider improving performance with a combination of successive cancellation, digital network coding, and opportunistic relay selection in a two-way cooperative scheme, in which two source nodes send their packets to each other via multiple relaying nodes. In [33], the authors extend the idea presented in [32] to the cognitive networks, where primary and secondary users operate concurrently. It is worth noting that the successive interference cancellation relay receiver considered in [32] and [33] uses a standard solution in which the weak signal is treated as noise in the detection of the stronger one. Then, after the stronger signal is detected and re-modulated, it is used in the detection of the weak one. Our experience, evidenced in [17], indicates that this approach has some serious limitations due to the necessity of careful selection of participating terminals. The reason for this is sensitivity to power differences in the contributing signals approaching the relay. This was the reason that we concentrated on improving the detection algorithm to make the selection of the terminals less sensitive to power differences between signals reaching the relay. Two-way relaying schemes are also considered in the application in a full-duplex cellular system [34]. The authors focus on the full-duplex wireless communication network, in which a full-duplex base station and half-duplex mobile station exchange their information in two phases via a half-duplex relay station. The amplify-and-forward and decode-and-forward relaying schemes are considered in this configuration. The relay station chooses the best scheme depending on the channel state information to achieve the maximum capacity. As the main contribution of our research is the investigation of the decision-making based on two received packets in the physical layer at the relay node, in our research we limit ourselves to relatively simple convolutional codes applied for orthogonal frequency division multiplexing (OFDM) transmission, which are typical for the WiFi IEEE 802.11 family of standards, including 802.11a/g/n/ac and ax. As we will see in the next sections of this paper, the investigated PNC algorithms operate on each subcarrier in parallel; therefore, in our opinion, the subcarrier spacing, OFDM symbol duration, and other parameters of WiFi transmission do not have influence on the operation of the improved SIC detector reported and analyzed in this paper, unless system configuration with more advanced than convolutional codes is applied. Therefore, we are convinced that our results achieved in the physical layer are valid both for traditional IEEE 802.11g and new IEEE 802.11 ax standards. The differences in performance of both detectors could be changed if--more powerful than convolutional codes--LDPC codes standardized in IEEE 802.11 ax were applied. These codes can diminish the difference in performance of both investigated PNC detectors applied in the relay. We will also report how channel estimation errors affect the performance of the whole two-way relaying system.