## 3 Considered Two-Way Relaying PNC Systems  
As OFDM transmission is applied in several radio communication systems, we consider a simple model of PNC with OFDM transmitters and a receiver, shown in the multiple access phase in Figure 3. Binary information data packets \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\) to be transmitted by terminal A and B, respectively, are encoded using a channel forward error correcting (FEC) code, resulting in blocks \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). These blocks are delivered to OFDM transmitters in both terminals and are subsequently transmitted over multipath channels featuring impulse responses \(\mathbf{h}_{1}\) and \(\mathbf{h}_{2}\), respectively.  
For the sake of simplicity, we assume that the channel impulse responses have unit energy, and we model the unequal power of the signals reaching the relay using the weighting coefficients \(g_{1}\) and \(g_{2}\). To model the propagation loss, we assign the power \(P_{l}\) (\(i=1\), \(2\)) of signals incoming to the receiver from each user by setting the values of the coefficients \(g_{i}=\sqrt{P_{l}}\). The received signal is disturbed by the additive white Gaussian noise \(\
u(t)\). In this paper, we assume that signals from both transmitters arrive at the receiver with such a propagation delay difference that the receiver can select a common orthogonality period in which OFDM signals from both transmitters are analyzed. Typically, e.g., in device-to-device (D2D) or WiFi communications, distances between transmitters and receivers are not large (e.g., up to a few hundred meters), so a coarse synchronization of both transmitters justifies such an assumption. The relay receives the noisy version of the summed signals and derives the sample block \(\mathbf{Y}\) which constitutes the subcarrier outputs of the OFDM correlator. Next, the PNC processing algorithm derives the data block which will be transmitted by the relay OFDM transmitter in the broadcast phase to both terminals A and B.  
If the OFDM systems between the relay and both terminals are sufficiently synchronized, as assumed above, we can consider transmission on each subcarrier as transmission over a flat fading channel. Consider transmission on the \(k\)-th subcarrier. Assume that the channels from terminal A and terminal B to the relay on the \(k\)-th subcarrier are characterized by the complex channel coefficients \(H_{1,k}\) and \(H_{2,k}\), respectively. On the output of the fast Fourier transform (FFT) correlator placed in the relay, the signal sample \(Y_{k}\) can be described by the equation (we omit the time index for simplicity):  
\[Y_{k}=H_{1,k}X_{1,k}+H_{2,k}X_{2,k}+N_{k}\;\;\text{for}\;\;k=0,1,\ldots,N-1 \tag{1}\]  
where \(N_{k}\) is the sample of the additive noise on the output of the correlator for the \(k\)-th subcarrier, and \(X_{1,k}\) and \(X_{2,k}\) are quadrature phase shift keying (QPSK) or quadrature amplitude modulation (QAM) data symbols transmitted in the multiple access phase on that subcarrier by terminal A and B, respectively. Based on the samples \(Y_{k}\) (\(k=0\), \(1\),..., \(N-1\)), the relay calculates the signal to be emitted in the broadcast phase.  
Several arrangements of the PNC processing algorithm are possible. We consider two of them. The first one [16] is treated as a reference to show the performance of the proposed second one.