# Rapid Estimation of TVWS: A Probabilistic Approach Based on Sensed Signal Parameters
## 3 Analysis of the Collected Data
Signals below \(I\) are the noise components of \(X(p)\), and for that reason they are not considered as missdetected PU signals. To calculate \(P_{m}\), we can use  
Figure 4:0  
where \(I\) and \(T\) are the lateral limits to find the area under \(X(p)\). The value of \(T\) has to be calculated with the aim of reducing \(P_{fa}\) while maximizing \(P_{d}\), so that we use the statistical parameters of the noise signal \(W(p)\) to find its value. Hence, it can be defined by  
Figure 4:1  
where \(\sigma_{W}\) is the standard deviation of \(W(p)\) and \(z_{W}\) is the Z-score. Let us consider that \(T\) has to fulfill  
Figure 4:2  
Figure 4:3  
Figure 4:4  
where \(T_{min}\) is equal to \(I\) and \(T_{max}\) has to consider \(Z_{W(1\%)}=2.33\) that defines \(P_{fa}=1\%\) according to [20]. Based on this, a \(Z_{W(3\%)}=1.88\) to obtain a \(P_{fa}=3\%\) has been used to reach accurate results, as is explained in [6]. On the other hand, \(T\) referred to the composite signal \(X(p)\) defined by  
Figure 4:5  
where \(\sigma_{X}\) is the standard deviation of \(X(p)\) and \(z_{X}\) is the Z-score that defines the \(P_{d}\). By combining Equations (6) and (8), we find the value of \(Z_{X}\) as follows:  
Figure 4:6  
Figure 4:7  
Figure 4:8 Collected signal representation: (**a**) the Probability Density Function (PDF) of \(W(p)\) is represented in blue, the PDF of \(X_{o}(P)\) is plotted with a magenta dotted line, and the PDF of \(X(p)\) is plotted with red dotted line. Notice that \(X_{o}(P)\) and \(X(P)\) curves have the same means and standard deviations; (**b**) the normal PDF curves of \(W(p)\), \(X(p)\), and lines labelled as \(I\) and \(T\) delimit four areas under the curve (AUC). It is possible to identify \(P_{d}\) in green, \(P_{fa}\) in blue, \(P_{m}\) in orange, the \(P_{n}\) in gray. hence, after finding \(Z_{X}^{-1}\) we obtain the \(P_{d}\) for each point as observed in Figure 8a, which represents the ROC curve [21]. The outcome of this curve represents \(P_{d}\). By analyzing this curve and its variations as illustrated in Figure 8b, we can infer how overlapped the normal curves of \(W(p)\) and \(X(p)\) represented in Figure 7b are. This overlap depends on the number of detected PU. As we can observe, the positions and patterns of the ROC curves are variable; these changes depend on how wide and overlapped the PDF curves of the \(W(p)\) and \(X(p)\) signals are. These variations are seen in Figure 9, where panels (a) and (d) represent the sensed signals where we can identify the presence of PU above \(T\). In panels (b) and (e), the composite signal experiences an increment of its standard deviation (\(\sigma_{i}\)), which produces the expansion of the PDF curve, and as a logical consequence, the ROC curves shown in panels (c) and (f) modified their respective patterns. Figure 4:9 ROC curve where \(P_{fa}\) is the input and \(P_{d}\) is calculated: panel (**a**) shows a single ROC, and panel (**b**) shows the overlapped ROC curves from ten different points sensed. Figure 3:0 Two-point readings are plotted, for which panels (**a**,**d**) represent the \(X(p)\) signal sensed. In panels (**b**,**e**), the PDF representations for \(W(p)\) and \(X(p)\) are presented, and panels (**c**,**f**) illustrate their respective Receiver Operation Characteristics (ROC) curves, where we can observe different patterns. Accordingly, by understanding these variations, we can identify the behaviour of the sensed spectrum, can obtain related information, and can rapidly estimate the presence of TVWS in the analyzed TV-UHF band.