# An Accurate Probabilistic Model for TVWS Identification
## 4 Proposed Probabilistic Model
### Working with Normal Distributions  
In this research, both signals, i.e., the noise \(W(f)\) and the composite signal \(X(f)\), are collected simultaneously by two RFE devices connected in parallel. To verify that the sensed noise \(W(f)\) is a normally distributed signal, two complementary kinds of statistical methods were applied, namely, graphical and mathematical [30; 31]. The first one plots the samples of the noise signal, and its distribution is compared with a predefined pattern. Following this method, Figure 5(a) shows the histogram of the noise samples taken in a particular location and Figure 5(b) shows the probability distribution plot of the same samples. It is visually evident that the plotted samples correspond to a normal distribution. The second method is a statistical test; in this case, the Kolmogorov-Smirnov Test (KST) [31], which takes the signal samples and verifies if they follow specific parameters to be considered as normally distributed. This test with a 5% significance level was applied to the collected noise samples, and the corresponding results of the CDF are shown in Figure 5(c). These numbers suggest that the tested samples are normally distributed. On the other hand, as is shown in Figure 6d,e, the multi modal composite signal \(Xo(p)\) presents a distribution in which it is possible to identify the component of noise represented by a higher peak and the signal of PU represented by the multi modes on the right tail. When the KST [31] with a significance level of 5% is applied to the composite signal samples, the results suggest that it does not correspond to a normally distributed signal, as observed in Figure 6f. In our model, the composite signal with multi modal curve \(Xo(p)\) observed in Figure 7a can be represented as the normal distribution curve. Also, \(X(p)\) is shown in Figure 7b. By considering the statistical parameters of \(Xo(p)\), i.e., the mean and standard deviation, we generate the above mentioned normal curve that fits the sensed multi modal composite signal. In Figure 7c we observe the variations among the multi modal \(Xo(p)\) and the assumed \(X(p)\) curves. This assumption is necessary to reduce the complexity of the mathematical calculations. Figure 7 shows the pdf of different signals. The pdf of the noise \(W(p)\), the multi modal curve of the composite signal \(Xo(p)\), and the normal curve of the composite signal \(X(p)\) are plotted in blue, magenta, and red, respectively, for three different locations. Panel (a) shows the curves of the noise \(W(p)\), Panel (b) represents the intersection points of \(W(p)\cap Xo(p)\) named as \(A\) and its projection to the X-axis or \(A^{\prime}\), and Panel (c) shows the overlapping intersection points of \(W(p)\cap X(p)\) named as \(B\) and its projection to the X-axis or \(B^{\prime}\). In these plots, it is easy to observe that  
\[A^{\prime}-B^{\prime}\ll P(dBm) \tag{4}\]  
which means that \(A^{\prime}\) and \(B^{\prime}\) are very close together, and its difference is minimal compared with the power of the measured signals so that it can be neglected. This fact supports that the multi modal composite signal \(Xo(p)\) can be assumed as a normally distributed signal \(X(p)\) to reduce the mathematical calculation complexity without affecting the accuracy of the obtained results. Figure 6: Noise signal sensed \(W(p)\). (**a**) Histogram of the collected data. (**b**) Normal probability plot of the same data read. (**c**) Cumulative Density Function (CDF) of the Kolmogorov–Smirnov Test (KST) applied. Curve of the multi modal composite signal sensed \(Xo(p)\), (**d**) Histogram of the collected data, (**e**) Normal probability plot of the distribution for the same reading, and (**f**), CDF of the KST applied. This assumption is supported by the fact that the calculation of \(P_{fa}\) was done on the noise signal, i.e., \(W(p)\), which corresponds to a normal distribution. In addition, the following two points should also be noted.