# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 6 Results and Discussion  
In this section, the performance of the proposed SCAR-Phase of spectrum sensing and its simplified version are analyzed via Monte Carlo simulations. The number of Monte Carlo iterations is set to 5000. In the considered CR scenario, the PU transmits an ST-based signal given by (1). The ST sequence is designed using (8) with a TIR value \(\alpha=0.2\), a training period \(P=10\) and considering that \(\sigma_{d}^{2}+\sigma_{t}^{2}=1\). The received signal at the SU is modeled as in (2), with random channel gains and \(\tau\) selected from the interval \([0,P-1]\) in such a way that the time offset variate in each Monte Carlo simulation. Hence, \(\hat{\tau}\) is obtained in each trial using (7), thus the Monte Carlo simulations are carried out considering the corresponding synchronization errors for each SNR value. The noise variance is chosen to satisfy a determined value of instantaneous SNR (denoted by \(\gamma\)), hence \(\sigma^{2}=\theta^{2}(\sigma_{d}^{2}+\sigma_{t}^{2})/\gamma\). The region of interest is the low SNR between the PU and the SU, where the detection of the PU's signal can be tough. Given that, in such a scenario, the number of samples required by the energy detector dramatically increases at the time that the SNR decreases, the SUs' throughput is severely affected. Therefore, both proposed approaches are compared with the traditional energy detector, since their aim \begin{table}
\begin{tabular}{c c} \hline \hline
**Spectrum Sensing Method** & **Number of Real Operations** \\ \hline Energy detector & \(7N-1\)[19] \\ SCAR-Phase of spectrum sensing & \(216P^{3}+12P+17+7N+P_{H_{0}}(15N+5)\) \\ Simplified SCAR-Phase of spectrum sensing & \(216P^{3}+12P+15N+23\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexity comparison of the SCAR-Phase of spectrum sensing, the simplified SCAR-Phase of spectrum sensing and the energy detector in terms of the number of real operations. Figure 6: Computational complexity in terms of number of real operations against \(P_{H_{0}}\) for a fixed value of number of samples \(N\). is to reduce the number of samples used in the spectrum sensing phase to increase the maximum SUs' throughput when operating in the low SNR region. In order to carry out the Monte Carlo simulations, the values of \(\lambda_{st}^{\prime}\) and \(\lambda_{ed}=g(\lambda_{st}^{\prime})\) that maximize the overall probability of detection (\(P_{\text{D}}\)) for a target \(P_{\text{FA}}=\beta\) must be determined for the SCAR-Phase of spectrum sensing. For this purpose, \(P_{\text{D}}\) is first obtained as a function of \(\lambda_{st}^{\prime}\) using (39) and (41). Then, the value \(\lambda_{st}^{\prime}\) that maximizes \(P_{\text{D}}\) is found and used to obtain \(\lambda_{ed}=g(\lambda_{st}^{\prime})\). The detection performance of the proposed SCAR-Phase and simplified SCAR-Phase of spectrum sensing is shown in Figure 7, in terms of the attained overall probability of detection (satisfying a constraint on the overall probability of false alarm \(\beta=0.1\)) as a function of SNR. These results are obtained by finding the number of samples used for spectrum sensing, \(N\), that attains an objective probability of synchronization error, below which the theoretical expression of the overall probability of detection is accurate enough. For example, with \(N=6000\) the probability of synchronization error is \(2\times 10^{-2}\) at \(-20\) dB (see Figure 3 for reference). Therefore, for a set of collected samples of length \(N\geq 6000\), the probability of synchronization error will be less than \(2\times 10^{-2}\) in the low SNR region above \(-20\) dB.