# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 6 Results and Discussion
Hence, the theoretical results for the SCAR-Phase of spectrum sensing and the simplified version approximate the simulation results, as shown in Figure 7. Moreover, in order to provide a fair perspective on the performance of the proposed approaches, it is also shown in this figure the attained overall probability of detection against SNR with a number of samples that produce a higher probability of synchronization error. For example, with \(N=2250\) the probability of synchronization error is approximately \(0.28\) at \(-20\) dB (see Figure 3 for reference). Thus, it can be seen in Figure 7 that with \(N=2250\) the theoretical approximations (in both proposed approaches) do not perfectly match the simulation results as the SNR decreases. Indeed, in this case, the theoretical results overestimate the simulations in the very low SNR since more synchronization errors are made. It is worth mentioning that the results of both proposed approaches are similar, given the fact that in the low SNR the SCAR-Phase sensing results rely mostly on the ST-Det of the second sample processing period, whereas the simplified SCAR-Phase sensing results rely only on the ST-Det in the unique sample processing period. Additionally, for comparison purposes, Figure 7 also exhibits the theoretical detection performance of the energy detector with \(N=6000\), which is the number of samples used by the proposed approaches. For the results labeled as energy detector, the theoretical expressions in (10)-(12) are used. It can be noticed that the performance of the SCAR-Phase of spectrum sensing and the simplified version evince a similar operation in comparison to the energy detector in the SNR region above \(-13\) dB. However, for SNR values below \(-13\) dB, the second sample processing period of the SCAR-Phase sensing is enabled to exploit the ST sequence by means of the ST-Det. Therefore, in the low SNR, the performance of the SCAR-Phase sensing is significantly improved with respect to that of the energy detector. In consequence, for a target detection performance, the number of samples can be reduced in the low SNR when the proposed approaches are implemented. Finally, note in this figure that spectrum algorithms that are based on the sample covariance matrix, such as the Covariance Absolute Value and Covariance Frobenius Norm [7], have been included for comparison purposes under the signal model considered in (2). It can be seen that they exhibit a degradation in detecting uncorrelated signals and their performance drastically deteriorates since \(N\) is small [7], which is the scenario analyzed in this paper. Therefore, further analyses compare the proposed approaches only with the energy detector. Next, Figure 8 shows the required number of samples as a function of the SNR, for a given pair of target probabilities of detection and false alarm, \(\beta_{d}=0.9\) and \(\beta=0.1\), respectively. It quantifies the increasing difference in the required number of samples to achieve the target probabilities as the SNR decreases. For example, for an SNR of \(-16\) dB, the SCAR-Phase of spectrum sensing and the simplified SCAR-Phase of spectrum sensing need approximately \(2\times 10^{3}\) samples whilst the energy detector requires approximately \(1\times 10^{4}\) samples, which is around five times the number of samples required by the proposed approaches. Furthermore, when the SNR is equal to \(-20\) dB, the new schemes only require approximately \(4\times 10^{3}\), whereas the energy detector requires approximately \(6.6\times 10^{4}\) samples, which is 16.5 times the required number of samples by the proposed approaches. Moreover, Figure 8 evidences that as the SNR increases, the detection performance of the energy detector and the proposed approaches is similar. Note that, the SCAR-Phase of spectrum sensing and the simplified SCAR-Phase of spectrum sensing requires the same number of samples.