# Rapid Estimation of TVWS: A Probabilistic Approach Based on Sensed Signal Parameters
## 3 Analysis of the Collected Data
Figure 5: Different values for threshold are plotted to show that \(T_{high}\) produces missed detection of real PU signals and that \(T_{low}\) produces some noise signals which are be considered PUs. In both cases, there are errors in the obtained results. An adequate value for \(T\) is plotted, which maintains enough distance from the noise floor to avoid false alarms and missed detection. This equilibrium is obtained when \(T\) considers a \(P_{fa}\) of 3%. Our next step was to complement the information observed in Figure 4 with the statistic parameters obtained from the sensed signals, i.e., \(W(p)\) and \(X(p)\). From signals collected in parallel, i.e., noise \(W(p)\) and composite signal \(Xo(p)\), we obtained their statistical characteristics as means (\(\overline{\text{W}}\), \(\overline{\text{X}_{o}}\)) and standard deviations (\(\sigma_{W}\), \(\sigma_{X_{o}}\)). We represented the signals as Probability Density Function (PDF) curves for a better understanding of the statistical analysis. It is important to notice that the \(W(p)\) signal follows a normal distribution as is demonstrated in [6] by applying the Kolgomorov-Smirnov Test (KST) [19]; hence, its PDF can be represented, as Figure 7a illustrates with blue line. On the other hand, the composite signal-sensed \(X_{o}(p)\) does not follow a normal distribution, as demonstrated in [6] by applying the KST. This phenomena is due to the multimodal nature of the composite signal sensed, as observed in Figure 7a in magenta dotted line. If we try to calculate the intersection point of \(W(p)\) and \(X_{o}(p)\), it would be mathematically complicated due to the higher grade of the equation that represents the multimodal curve. This complexity is reduced if we replace \(X_{o}(p)\) with the normal curve \(X(p)\) by applying  
\[\overline{\text{X}_{o}} = \overline{\text{X}} \tag{3}\] \[\sigma_{X_{o}} = \sigma_{X} \tag{4}\]  
where \(\overline{\text{X}_{o}}\) and \(\overline{\text{X}}\) are the means, and \(\sigma_{X_{o}}\) and \(\sigma_{X}\) are the standard deviation for \(Xo(p)\) and \(X(p)\) signals respectively. Then, the assumed \(X(p)\) curve has the same mean and standard deviation values as \(X_{o}(p)\), and it is illustrated in Figure 7a in red dotted line. This assumption reduces the complexity of the mathematical calculations while maintains the accuracy in our results. Once the assumption that \(X_{o}(p)\) can be replaced with \(X(p)\) is accepted, in the rest of the manuscript, we will refer to the composite signal as \(X(p)\). The next step was to define a threshold \(T\) located close enough to the noise floor to prevent missed detection and far enough to avoid false alarms. The value of \(T\) may be calculated by applying Equations (1) and (2), yielding the reference level to decide if the channel is used or not. On the other hand, normal PDF curves \(W(p)\) and \(X(p)\), together with tentative \(T\), defined four areas under the curve (AUC). Each of these areas represents a specific probability value as explained next. The probability of detection (\(P_{d}\)) refers to the signals above \(T\) and represents real PU signals detected. The probability of false alarm (\(P_{fa}\)) accounts for noise peaks above \(T\) that may be misinterpreted as PU signals. The probability of noise (\(P_{n}\)) accounts for noise signals below \(T\) which are considered as noise. Finally, the probability of missed detection (\(P_{m}\)) accounts for undetected PU signals considered as noise. Both probabilities \(P_{m}\) and \(P_{fa}\) will account for errors in the obtained results. Figure 6: Channels detected as used are coloured in gray, whereas channels considered as TVWS are in white. Channel 37 is in yellow because it is assigned for other purposes. Panels **(a–c)** correspond to points 1–3, respectively. We need to consider an auxiliary line labelled as \(I\), which projects the intersection point of \(W(p)\cap X(p)\) on the power axis, as illustrated in Figure 7b. \(P_{m}\) is represented as the yellow AUC delimited by \(I\), \(X(p)\), and \(T\).