# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 5 Simplified SCAR-Phase of Spectrum Sensing  
The simplified SCAR-Phase of spectrum sensing block diagram is shown in Figure 5. Different to the SCAR-Phase sensing, the simplified version consists of a sample collection period and only one sample processing period. This is due to the fact that the contribution of the energy detection metric in the enhanced ST-based detection (ST-Det) in (25) does not require the synchronization procedure. Moreover, the results of this metric are the same as in the first sample processing period of the SCAR-Phase sensing, thus this period can be omitted. Hence, after the SU collects the samples, the synchronization process with the transmitted PU's sequence is carried out at the same time that the energy of the received signal is calculated. Then, once \(\tau\) is estimated, the already synchronized ST sequence can be used in the ST-related metric of the ST-Det. If the enhanced ST-based detector decides that the PU's signal is present, the SU waits until the next spectrum sensing phase, or else, the SU can transmit data. Note that simplifying the sensing phase to one sample processing period reduces the computational complexity of the approach, which is characterized in terms of the total number of real operations needed to make a decision.  
After analyzing the mathematical structure of the proposed simplified SCAR-Phase of spectrum sensing, the computational complexity is found to be \(C=C_{sync}+C_{ed}+C_{t}\), where the number of real operations needed by the synchronization method is \(C_{sync}=216P^{3}+12P+18\), by the energy detection metric is \(C_{ed}=7N-1\). Therefore,  
\[C=216P^{3}+12P+15N+23. \tag{46}\]Next, the detection performance metrics of the simplified SCAR-Phase sensing are presented. The overall probability of false alarm is:  
\[P_{\text{FA}}=P_{\text{fa}}^{st}, \tag{47}\]  
where \(P_{\text{fa}}^{st}\) is given by (31).  
Additionally, the overall probability of detection, when \(\hat{\tau}=\tau\) modulo-\(P\), is:  
\[P_{\text{D}}=P_{\text{d}}^{st}, \tag{48}\]  
where \(P_{\text{d}}^{st}\) is given by (37).  
In Table 1 the computational complexity of the proposed approaches and the energy detector can be compared. Note that, for the same number of samples, the computational complexity of the energy detector is the lowest. Although the computational complexity of the energy detector is much simpler in contrast to the proposed approaches, the detection performance of the SCAR-Phase sensing and its simplified version is much higher in the very low SNR region, as it will be shown later in the results section, which makes it worth the increase in computational complexity. Additionally, since the number of real operations used by the SCAR-Phase of spectrum sensing depends on the value of \(P_{\mathcal{H}_{0}}\), its computational complexity will be greater than that of the simplified scheme as \(P_{\mathcal{H}_{0}}\) approaches 1. This is shown in Figure 6, where the number of real operations of the proposed schemes as a function of \(P_{\mathcal{H}_{0}}\) is shown for a fixed value of \(N\). Note in this figure that the computational complexity of the simplified SCAR-Phase of spectrum sensing remains constant, whereas for the SCAR-Phase of spectrum sensing increases according to \(P_{\mathcal{H}_{0}}\). It can be seen that for \(P_{\mathcal{H}_{0}}\) values greater than \(\approx\) 0.55 the computational complexity of the proposed simplified approach is lower than that of the SCAR-Phase of spectrum sensing. This makes sense given that the probability of executing the second processing period of the SCAR-Phase sensing is greater. In the next section, it will be shown that to attain a given detection performance in the very low SNR  
Figure 5: Block diagram of the simplified SCAR-Phase spectrum sensing.  
region, the computational complexity of the simplified version is lower than that of the SCAR-Phase of spectrum sensing.