## 3 Spectrum Sensing Quality Analysis  
In this section, our objective is to analyze the spectrum sensing quality of each SSN via a local spectrum sensing model, aiming to provide the quantification result with the emphasis to evaluate the spectrum sensing performance of each SSN. More importantly, the analysis results will be employed to formulate the distributed power control framework for IT and MT SSNs. By revisiting Equations (9) and (10) in local spectrum sensing model, we can observe that the detection probability \(p_{d}(m,k)\) in \(\mathcal{P}_{d}\) and the false alarm probability \(p_{f}(m,k)\) in \(\mathcal{P}_{f}\) can be referred to the random variables for the \(m\)-th SSN under the \(k\)-th uplink channel due to the uncertainty of the presence and absence of the PU. It is worth noting that the errors in spectrum sensing for a SSN will be generally considered negligible due to imperfect spectrum sensing [27], e.g., misdetection and false alarm caused by hardware capability of SSN and practical time-varying channel conditions. So the errors in spectrum sensing for a SSN will further incur the fact that different uplink channels will hold different probabilities of detection and false alarm. In particular, the higher the detection probability in \(\mathcal{P}_{d}\), the better the PUs are protected; the lower the false alarm probability in \(\mathcal{P}_{f}\), the more efficiently the uplink channel can be reutilized by SUs [7]. Based on this observation, the higher the relative divergence between \(p_{d}(m,k)\) and \(p_{f}(m,k)\), the better the performance of local spectrum sensing. It has been revealed that the Kullback-Leibler divergence is an effective measure of how one probability diverges from a second probability [30]. Hence, the relative divergence between \(p_{d}(m,k)\) and \(p_{f}(m,k)\) for the \(m\)-th SSN under the \(k\)-th uplink channel can be defined as follows based on a Kullback-Leibler divergence framework:  
\[D\Big{(}p_{d}(m,k)\|p_{f}(m,k)\Big{)}\triangleq p_{d}(m,k)\log_{2}\frac{p_{d}( m,k)}{p_{f}(m,k)} \tag{11}\]  
With respect to the entire set of uplink channels, the relative divergence between \(\mathcal{P}_{d}\) and \(\mathcal{P}_{f}\) for the \(m\)-th SSN can be denoted as:  
\[D_{m}(\mathcal{P}_{d}\|\mathcal{P}_{f})=\sum_{k=1}^{N_{c}}p_{d}(m,k)\log_{2} \frac{p_{d}(m,k)}{p_{f}(m,k)} \tag{12}\]  
It is noticeable that the relative divergence between the detection probability and the false alarm probability just reflects the performance of local spectrum sensing by each SSN. Viewed from the SU perspective, we are also interested in the impact of the SOP usage probability on the spectrum sensingquality. To this end, we characterize the spectrum sensing quality factor which can be expressed by a function of two parameters including the relative divergence between \(\mathcal{P}_{d}\) and \(\mathcal{P}_{f}\) along with the SOP usage probability \(\delta_{b}\) for bottleneck SU \(b\). Specifically, the spectrum sensing quality factor \(\mathcal{F}_{m\to b}\) of the \(m\)-th SSN with respect to bottleneck SU \(b\) can be defined as:  
\[\mathcal{F}_{m\to b}\triangleq\delta_{b}\cdot D_{m}\Big{(}\mathcal{P}_{d}\| \mathcal{P}_{f}\Big{)},\quad b\in\mathcal{N}_{S} \tag{13}\]  
By using mathematical statistics theory, next we start by formulating the detection probability distribution \(\Upsilon_{d}(m)\) and the false alarm probability distribution \(\Upsilon_{f}(m)\), which have been derived from Algorithm 1. ```
1:Input: The detection probability set \(\mathcal{P}_{d}\) and the false alarm probability set \(\mathcal{P}_{f}\). 2:Output: The detection probability distribution \(\Upsilon_{d}(m)=\Big{\{}\Upsilon_{1}^{d}(m),\,\Upsilon_{2}^{d}(m),\cdots, \,\Upsilon_{\xi}^{d}(m)\Big{\}}\)

The false alarm probability distribution \(\Upsilon_{f}(m)=\Big{\{}\Upsilon_{1}^{f}(m),\,\Upsilon_{2}^{f}(m),\cdots, \,\Upsilon_{\xi}^{f}(m)\Big{\}}\).