# Rapid Estimation of TVWS: A Probabilistic Approach Based on Sensed Signal Parameters
## 5 Results and Discussion  
From each sensed signal, we calculated statistical parameters \(\overline{\mathbf{W}},\overline{\mathbf{X}},\sigma_{\mathbf{W}}\), and \(\sigma_{\mathbf{X}}\). We also calculated the optimal value for \(T\) minimizing \(P_{fa}\). With the ROC curves, we then identified \(P_{d}\) for each case. Additionally, we calculated the distance \(\tau\) of the statistical parameters to the new reference point of \(Z_{1}X\). The obtained information was next used to find relationships among those parameters in order to identify patterns to provide us with new information.  
Figure 15 illustrates the auxiliary parameters combined with the PDF curves for points P1 to P9. Here, we can visualize the variation of parameters \(\tau_{i}\) for each location. The numerical values calculated for \(\tau_{i}\) are represented in Table 4. When the obtained parameters are analyzed, we can identify two cases, as explained next:  
\[\overline{\mathbf{W}}\leq T\leq\overline{\mathbf{X}}\to Case\ I\]  
\[\overline{\mathbf{W}}\leq\overline{\mathbf{X}}\leq T\to Case\ II\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline _Point_ & \(\tau_{\overline{\mathbf{W}}}\) & \(\tau_{I}\) & \(\tau_{T}\) & \(\tau_{\overline{\mathbf{X}}}\) \\ \hline P1 & 13.226 & 10.631 & 9.698 & 11.279 \\ P2 & 16.859 & 13.774 & 12.802 & 13.962 \\ P3 & 19.577 & 16.302 & 15.548 & 16.040 \\ P4 & 19.296 & 15.989 & 15.387 & 17.005 \\ P5 & 19.546 & 16.407 & 15.779 & 15.884 \\ P6 & 23.139 & 20.110 & 19.912 & 18.668 \\ P7 & 13.598 & 11.009 & 9.814 & 10.704 \\ P8 & 22.532 & 19.064 & 18.480 & 18.227 \\ P9 & 20.394 & 17.173 & 16.571 & 16.537 \\ P10 & 19.134 & 15.891 & 15.135 & 15.847 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Auxiliary parameters used in the REM.  
Figure 14: In the plot that represents the PDF of the sensed signals, we draw a reference line named \(Z_{1}X\), which is defined by a Z-score that separates 99% vs. 1% of the AUC of \(X(p)\). Additionally, we observe its distance from \(T\), \(I\), \(\overline{\mathbf{X}}\), and \(\overline{\mathbf{W}}\), denoted as \(\tau_{T}\), \(\tau_{1}\)\(\tau_{\overline{\mathbf{X}}}\), and \(\tau_{\overline{\mathbf{W}}}\), respectively.  
where case I represents that \(W(p)\) and \(X(p)\) are more separated and there are more \(PU\) detected, whereas case II represents that \(W(p)\) and \(X(p)\) are closer and less PUs are using channels. By using \(\tau_{i}\) to find how overlapped the sensed signals are, two extra parameters must be obtained, and these are \(\alpha\) and \(\beta\) defined by  
\[\alpha = \tau_{\overline{W}}-\tau_{\overline{X}}\] \[\beta = \tau_{\overline{W}}-\tau_{T} \tag{13}\]  
accordingly, \(\alpha>\beta\) corresponds to case I, and \(\alpha<\beta\) corresponds to case II.  
When this parameter variation is identified in readings of point P1 that contain the lowest number of PUs detected and comparing it with readings taken on P6, which have the maximum number of PUs detected, we can identify possible patterns that provide us with the baseline for a method to decide the presence of TVWS in other locations. However, we need to find information related to those extreme cases, i.e., only PUs in all channels and no PU detected at all.