## 2 Full Duplex System Model  
We consider \(N\) simultaneous, co-channel full duplex links each of which is located in a separate cell as shown in Figure 1. Focusing on a typical link, the receiver receives a desired signal \(x(t)\) from its associated transmitter, and some other interfering signals as shown in Figure 2. In fact, the desired signal \(x(t)\) will be initially distorted by the radio frequency (RF) analog circuits in the transmitter front end. After passing though the channel and corrupted by different interfering signals, the signal \(u(t)\) entering the receiver RF front end will be distorted again by the RF analog circuits. Figure 2 shows the discrete-time baseband-equivalent model for full duplex (FD) link, and captures the effects of analog circuits and the limited dynamic range in both sides of the link. An independent zero-mean Gaussian noise \(c(t)\) whose variance equals \(\kappa\) times the energy of the intended signal \(x(t)\) is injected in the model to capture the effect of the transmitter distortion created by the RF front end. This noise closely approximates the combined effects of additive power-amp harmonics, nonlinearities in the DAC and power-amp, as well as oscillator phase noise [15]. Therefore, the equivalent baseband signal in the air \(s(t)\) would be:  
Figure 1: A full duplex cellular network. Figure 2: A full duplex baseband-equivalent system for both uplink and downlink signals. \[s(t)=x(t)+c(t);s.t.\ \left\{ \begin{array}{c}c(t)\sim\mathcal{N}(0,\kappa|x(t)|^{2}),\\ c(t)\perp x(t),\\ c(t)\ \textit{is white}.\end{array} \right. \tag{1}\]  
Furthermore, an independent zero-mean Gaussian noise \(e(t)\) whose variance equals \(\beta\) times the energy collected at the receiving antenna is injected in the model to to capture the effects of the receiving RF front end and the DAC. Therefore, the received baseband signal after DAC would be:  
\[y(t)=u(t)+e(t);s.t.\ \left\{ \begin{array}{c}e(t)\sim\mathcal{N}(0,\beta|u(t)|^{2}),\\ e(t)\perp u(t),\\ e(t)\ \textit{is white}.\end{array} \right. \tag{2}\]  
Note that \(\kappa\ll 1\) and \(\beta\ll 1\) are system parameters. After careful manipulation presented in [22], the SINR at both ends can be written as:  
\[ \begin{split}\gamma_{n}^{d}&=\frac{p_{n}^{d}\varsigma_{nn}^{ d}}{(1+\beta)I_{n}^{d}+(P_{l}+\kappa+\beta)p_{n}^{u}g_{nn}^{du}+\sigma_{o}^{2}}, \\ \gamma_{n}^{u}&=\frac{p_{n}^{u}g_{nn}^{du}}{(1+\beta)I_{n}^{u }+(\kappa+\beta)p_{n}^{d}g_{nn}^{ud}+\sigma_{o}^{2}},\end{split}  \tag{3}\]  
where \(\gamma_{n}^{d}\) is the SINR of the downlink signal at the UE while \(\gamma_{n}^{u}\) is the SINR of the uplink signal at the BS (Throughout this paper, the superscript \(d\) and \(u\) indicate downlink and uplink, respectively). \(I_{n}^{d}\) and \(I_{n}^{u}\) are the total amount of intercell interference at UE and BS, respectively. Moreover, the \(p_{n}^{d}=|x_{n}^{d}|^{2}\) and \(p_{n}^{u}=|x_{n}^{u}|^{2}\) are the downlink and uplink transmit power, respectively. \(g_{nn}^{d}(t)\) and \(g_{nn}^{u}\) are the desired downlink and uplink channel gains. \(g_{nn}^{du}\) is the channel gain of the uplink to downlink self interference (i.e., self interference at UE), whereas \(g_{nn}^{ud}\) is the channel gain of the downlink to uplink self interference (i.e., self interference at BS). Finally, \(P_{l}\) is an indicator parameter that equals 0 in case of symmetric communication and 1 for asymmetric communication. \(\sigma_{o}^{2}\) is the power of the additive white Gaussian noise. In each cell \(n\), the required QoS (i.e., the minimum required data rate and the maximum bit error rate (BER) in both downlink and uplink) can be mapped into a pair of target SINRs \(\Gamma_{n}^{d}\) and \(\Gamma_{n}^{u}\). Therefore, the following two inequalities present two conditions that should be satisfied in every cell \(n\in\mathcal{N}\):  
\[