## 3 Mathematical Preliminaries  
In [9], it is shown that in a CRN with uniform allocation of subcarriers, the subcarrier collisions follow a hyper-geometric distribution, whose probability mass function (PMF) is given as  
\[\Pr(C_{ps})=\frac{\left(\begin{matrix}F_{p}\\ C_{ps}\end{matrix}\right)\left(\begin{matrix}F-F_{p}\\ F_{s}-C_{ps}\end{matrix}\right)}{\left(\begin{matrix}F\\ F_{s}\end{matrix}\right)} \tag{3}\]  
where the notation \((\cdot)\) stands for the binomial coefficient, \(F\) is the set of all available subcarriers, \(F_{s}\) is the set of subcarriers allocated to SUs, \(F_{p}\) is the set of subcarriers allocated to PUs, and \(C_{ps}\) is the number of collided subcarriers. From this expression, the average subcarrier collisions will then be given as  
\[\mathbb{E}[C_{ps}]=\frac{F_{s}F_{p}}{F}, \tag{4}\]  
and the average number of non-colliding subcarriers will be  
\[\mathbb{E}[C_{nc}]=\frac{F_{s}(F-F_{p})}{F}. \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol Used** & **Description** \\ \hline \(P\) & Total number of PUs present \\ \(S\) & Total number of SUs present \\ \(F\) & Total pool of subcarriers available \\ \(F_{p}\) & No. of subcarriers allocated to the PU \\ \(F_{s}\) & No. of subcarriers allocated to the SU \\ \(C_{ps}\) & No. of subcarrier collisions between SU and PU \\ \(F-F_{p}\) & No. of free subcarriers \\ \(C_{nc}\)\(or\)\((F_{s}-C_{ps})\) & No. of collision free subcarriers \\ \(P_{b,c}\) & Error probability with subcarrier collisions \\ \(P_{b,nc}\) & Error probability with no subcarrier collisions \\ \(P_{e}\) & Total bit error \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters.  
The instantaneous average BER of the \(s^{th}\) SU will also be the mean of all BER for each subcarriers. The total subcarriers for SU being \(F_{s}\), thus  
\[P_{e}=\frac{1}{F_{s}}\sum_{i=1}^{F_{s}}P_{e}(i)=\frac{1}{F_{s}}\Bigg{\{}\sum_{i=1 }^{C_{ps}}P_{b,c}+\sum_{j=1}^{C_{nc}}P_{b,nc}\Bigg{\}}, \tag{6}\]  
where \(P_{b,c}\) is the probability of error with subcarrier collisions, \(P_{b,nc}\) is the probability of error with no-collisions, \(C_{ps}\) is the number of subcarrier collision between PU and SU and \(C_{nc}\) is number of no-collision subcarriers. Therefore, the average probability of error of the SU can be given as  
\[\begin{split}\mathbb{E}\left[P_{e}\right]&= \mathbb{E}\left[\frac{1}{F_{s}}\sum_{i=1}^{C_{ps}}P_{b,c}+\frac{1}{F_{s}}\sum_{ i=1}^{C_{nc}}P_{b,nc}\right],\\ &=\frac{1}{F_{s}}\mathbb{E}\left[\sum_{i=1}^{C_{ps}}\mathbb{E} \left[P_{b,c}\right]\right]+\mathbb{E}\left[\sum_{i=1}^{C_{nc}}\mathbb{E} \left[P_{b,nc}\right]\right],\\ &=\frac{1}{F_{s}}\mathbb{E}\left[C_{ps}\mathbb{E}\left[P_{b,c} \right]\right]+\mathbb{E}\left[C_{nc}\mathbb{E}\left[P_{b,nc}\right]\right]. \end{split} \tag{7}\]  
Since \(C_{ps}\) and \(P_{b,c}\) are independent, \(C_{nc}\) and \(P_{b,nc}\) will be. Thus,  
\[\mathbb{E}[P_{e}]=\frac{1}{F_{s}}\big{\{}\mathbb{E}[C_{ps}]\mathbb{E}[P_{b,c}] +\mathbb{E}[C_{nc}]\mathbb{E}[P_{b,nc}]\big{\}}. \tag{8}\]  
Here, \(\mathbb{E}[C_{ps}]\) and \(\mathbb{E}[C_{nc}]\) are derived in Equations (4) and (5), respectively. The remaining average probability of subcarrier collisions (\(\mathbb{E}[P_{b,c}]\)) and average probability of no subcarrier collisions (\(\mathbb{E}[P_{b,nc}]\)) will be derived in the following subsections.