## 2 System Model  
In this section, the system model and the important underlying assumptions are presented in detail. The system model under consideration is shown in Figure 2. It consists of a primary base station (PBS) with \(P\)--PUs and a secondary base station (SBS) with \(S\)--SUs. SUs randomly access subcarriers of the primary network, spreading their interference among all the subcarriers (unused subcarriers and unused subcarriers by PUs). The channel between any \(i^{th}\) SU and SBS is denoted by \(\alpha_{i}\) and between any \(j^{th}\) PU and PBS by \(\gamma_{i}\). Interference channels between any \(j^{th}\) PU and SBS are denoted by \(\gamma_{js}\), while between \(i^{th}\) SU and PBS is denoted by \(\alpha_{ip}\), where \(i\in\{1,2,\ldots,S\}\) and \(j\in\{1,2,\ldots,P\}\).  
All these channels are assumed to be undergoing Rayleigh fading. The Rayleigh fading model is one of the most common and most used channel models for such theoretical studies, however, small-scale channel fading models such as Nakagami and Rician can also definitely be considered. Nonetheless, the insights and observations obtained from this study would remain the same. Therefore, the channel power distribution will follow an exponential distribution [21]. These exponential power distributions are characterized by their corresponding rate parameters, which depend on the mean value. Thus, a low mean channel power parameter would imply a larger distance between the PU-Rx and a SU-Tx than a high mean value. It is therefore important to highlight that given the immobile users in the network, the large-scale fading such as path-loss and shadowing will be constant [21; 22]. This is because the users are assumed to be immobile, however, different mobility models can be definitely considered for the future work.  
As mentioned in the earlier section, it is also assumed that the channel gains) information between PU and SBS (\(\gamma_{ps}\)) and between SU and SBS (\(\alpha_{s}\)) is available at the SU. Moreover, thermal additive white Gaussian noise (AWGN) in the network is assumed to have a circularly symmetric complex Gaussian distribution with zero mean and variance to \(\eta\), i.e., \(\mathcal{CN}(0,\eta)\). For further simplicity, the IT is  
Figure 2: A cognitive radio network (CRN) with a primary base station (PBS) and primary users (\(P\)-PUs), and secondary base station (SBS) with secondary users (\(S\)-SUs).  
kept constant for all subcarriers. Now, to maintain the IT for PU QoS in the CRN, the transmit power of the \(s^{th}\) SU for an arbitrary \(i^{th}\) subcarrier with the peak transmit power adaptation is given as  
\[P_{s}=\begin{cases}\bar{P}_{s},&q\geqslant\bar{P}_{s}\gamma_{ps},\\ \frac{q}{\gamma_{sp}},&q<P_{s}\gamma_{ps},\end{cases} \tag{1}\]  
\[=\min\left\{\bar{P}_{s},\frac{q}{\gamma_{ps}}\right\}, \tag{2}\]  
where \(\bar{P}_{s}\) is the peak power of SU-Tx, \(q\) is the IT and \(\gamma_{ps}\) is the channel power gain between PU and SU. As a first building block, a CRN consisting of a single primary user and single secondary user is considered in this letter. For the sake of readability, the system parameters used in this letter are given in Table 1.  
In the next section, we will look into the mathematical preliminaries, and we analyze the BER performance of such a system in the next section.