## 4 Simulation Results
### System Performance with Ideal Channel Coefficients  
To verify the detection quality of the proposed PNC algorithm applied in the relay in comparison to the reference one, first, ideal knowledge of channel coefficients is assumed. This enables evaluation of improvements in the detection of the algorithm itself. In the simulation experiments, the number of packets that were transmitted from each terminal was \(K=100\), \(1000\), \(5000\), or \(10000\), depending on the required accuracy and level of BER estimation. Every packet contained \(L=100\) payload OFDM symbols. The main aim of the simulation was to determine the estimated BER vs. signal-to-noise ratio (SNR) measurements for both users. In all our simulation runs we assumed the following definition of SNR:  
\[SNR=10\log_{10}\left(\frac{P_{Tx}(g_{1}^{2}+g_{2}^{2})}{N}\right) \tag{10}\]  
where \(P_{Tx}\) is the reference power transmitted by both terminals, \(N\) is the noise power, and the channel coefficients \(g_{1}\) and \(g_{2}\) model, as mentioned before, both propagation losses and the change of the power level with respect to the reference one. In all simulation results shown below, we set \(g_{1}=1\) without the loss of generality. A multipath channel model typical for WiFi was applied [35]. As already mentioned, the power delay profile was exponentially decaying, depending on the root mean square delay \(T_{rms}=50\) ns. We assumed the so-called block fading channel model, i.e., the time invariance of the channel impulse response within the transmission of a single packet. For each OFDM packet, new channel coefficients were randomly selected. We are aware of limitations in propagation conditions on channels in \(5\) GHz band; therefore, we assumed that the distance between end nodes and the relay was such that it ensures transmission with sufficient SNR level on the receiver side. For the performed link-level simulations reported in this paper, the assumed model is sufficient for estimation of BER against SNR. No shadowing or time evolution of channel parameters other than \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Bandwidth & 20 MHz \\ Cyclic prefix duration & 0.8 μs \\ Data duration & 3.2 μs \\ FFT size & 64 \\ No. of subcarriers & 52 \\ Operating frequency band & 5 GHz \\ Sampling rate & 40 MHz \\ Subcarrier spacing & 312.5 kHz \\ Throughput & 6 up 54 Mbps \\ Total symbol duration & 40 μs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of the simulated system. block fading was taken into account. The readers interested in more elaborate channel models and physical phenomena accompanying new propagation scenarios are directed to the papers [36; 37; 38]. Simulations were performed for several levels of a power imbalance between the signals reaching the relay from terminals A and B to show how robust both PNC algorithms are. As \(g_{1}=1\), we controlled the values of the coefficient \(g_{2}\). For \(g_{2}=1\), both signals from end terminals arrive at the relay with the same power, whereas in the case when \(g_{2}=0.2\), the difference in the power of strong and weak signals is 14 dB. We estimated the performance on the output of both terminal receivers (A and B) for \(g_{2}\) in the range between 0.2 and 0.9 (for the latter value, the difference in power is about 0.9 dB). As we can see in Figure 6, for relatively small power differences between the stronger and weaker signals, our proposed PNC algorithm applied in the relay results roughly a 4 dB improvement at BER equal to \(10^{-3}\) or lower. We can also notice that for a smaller power difference between the arriving signals (equivalent to a higher value of \(g_{2}\)), the performance in the form of BER deteriorates by about 1 dB as compared with two cases when \(g_{2}\) values are equal to 0.9 and 0.6. This results from the fact that when both signals reach the relay receiver at similar powers, the composite constellation can contain some constellation points which are very close to each other.