# An Accurate Probabilistic Model for TVWS Identification
## 4 Proposed Probabilistic Model
### Sensing the TV Signal and the Noise  
The existence of the PU can be modeled by using the following hypotheses in the frequency domain:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Trajectory** & **Distance in Km** & **Readings** & **Noise Samples** & **Signal Samples** \\ \hline T1 & 16.05 & 5417 & 612,752 & 612,752 \\ T2 & 32.45 & 10,419 & 1,166,928 & 1,166,928 \\ T3 & 31.48 & 12,924 & 1,447,488 & 1,447,488 \\ T4 & 23.22 & 7362 & 824,544 & 824,544 \\ \hline
**Total** & **103.20** & **36,176** & **4,051,712** & **4,051,712** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Number of readings and samples collected with each RF Explorer (RFE).  
\[H_{0}:X(f)=W(f) \tag{1}\]  
\[H_{1}:X(f)=S(f)+W(f) \tag{2}\]  
where \(W(f)\) denotes Additive White Gaussian Noise (AWGN), \(S(f)\) denotes the received signal of the PU, \(X(f)\) denotes the sensed composite of signal and noise, and \(H_{0}\) represents the null hypothesis for the of PU absence in the channel, whereas \(H_{1}\) represents alternative hypothesis of the PU presence in the channel.  
By using the designed M-SSS, the information from the UHF-TV spectrum was collected at each instance of time \(t_{i}\) and plotted, as shown by the example in Figure 5. In both cases, for RFE and for SA, the signal presents a similar shape in general terms, with specific frequencies in which the power in dBm rises to its maximum values and other frequencies in which minimum values are reached. These panels show the power variations and differences between them in the time domain, which are due to the internal characteristics of each device.  
The information of \(X(f)\) alone is not enough to identify TVWS, so that is imperative to find more parameters allowing to identify the presence of PU. A crucial point to be considered is how the noise is measured and its mean \(\overline{W}(f)\) is calculated. In the reviewed literature, in some cases, the noise is sensed at the beginning of the measurement campaign and that value is used to be compared with all the readings, in other cases a theoretical value is considered, and a third approach is to consider a fixed threshold \(T\). In practice, it is very difficult to obtain accurate information about the noise power [24] due to its variability or unexpected noise sources along the measured pattern when the samples are taken.  
According to the IEEE 802.22 standard [5; 10], the channel in operation must be sensed several times, namely, during the transmission, inter-framing time intervals, and in-framing. it is important to obtain trustful and accurate channel noise and signal samples. In order to obtain accurate information regarding the noise, simultaneous sensing of the noise and the signal is here proposed. As it can be seen in Figure 4a, signal \(X(f)\) is received through the antenna and sensed with one DAI, and at the same time, another DAI with similar characteristics and a 50 \(\Omega\) matched impedance is used to sense the noise of the system. As a result, both the signals (i.e., \(X(f)\) and \(W(f)\)) are used in our proposed channel sensing mechanism.  
With the mentioned hardware configuration, the UHF-TV spectrum was sensed, and representative examples are shown in Figure 5a,c for the overlapped curves of \(W(f)\) in blue and \(X(f)\) in red, taken with the RFE and SA, respectively. The next logical step is to calculate the statistical mean of the noise \(\overline{W}(f)\) in black, taken with the same instruments at time instant \(t_{i}\). Note that this is the reference level for calculating \(T\).  
Figure 5: TV signals sensed with RFE (top) and SA (bottom) are compared. The \(X(f)\) in red, \(W(f)\) in blue, and \(\overline{W}(f)\) in black are observed in panels (**a**,**c**), \(\overline{W}(f)\) is the calculated noise level. In panels (**b**,**d**) are plotted \(X(f)\), \(\overline{W}(f)\), and the different values for \(M\) or \(T\).