# On Optimal Cooperative Sensing with Energy Detection in Cognitive Radio
## 3 Cooperative Sensing with Optimal Thresholds
### Convex Optimization
Specifically, when compared to cooperative sensing with equal thresholds, under the target \(\overline{P^{f}}\) is 0.01, and \(P^{m}\) is reduced from 30% to 2% (and thus the corresponding \(P^{d}\) is improved from 70% to 98%). Next, to investigate the impact of \(N\) on optimal cooperative sensing performance, we compare the miss detection probabilities \(P^{m}\) of the equal threshold method and the proposed optimal threshold in Figure 7. As can be seen, the more sensing nodes participate in cooperation, the lower the achieved \(P^{m}\) is, in both cases. However, the slope of optimal threshold is much steeper than that of equal threshold. Furthermore, the miss detection probability is reduced by more than 100 times when \(N\) is 30 or more. Since the improvement is roughly proportional to \(N\), we conclude that using optimized thresholds is essential for a large scale cooperative sensing, e.g., without optimal thresholds, many cooperative sensing nodes possibly lead to stagnan

Figure 6: Receiver operating characteristic (ROC) curves for a single node sensing and cooperative sensing; different SNRs are considered in cooperation with both an equal threshold and optimal thresholds. In Figure 8, we investigate the impact of the mean of different average SNRs \(\mathbb{E}[\tilde{\gamma}_{i}]\) on \(P^{m}\). When it is relatively low (below \(-17\) dB), we do not observe a noticeable difference in \(P^{m}\) between equal and optimal thresholds. However, as it grows, optimal thresholds can achieve much lower \(P^{m}\) than that of equal thresholds. For example, a miss detection probability as low as \(10^{-5}\) times can be achieved when \(\mathbb{E}[\tilde{\gamma}_{i}]\) is \(-10\) dB. ## 5 Conclusions

In this paper we proposed an optimal cooperative sensing solution using energy detection when sensing nodes experience different PU SNRs. Specifically, the fusion center and its cooperative sensing nodes used the hard decision under the Neyman-Pearson criterion. Despite the non-convexity of the problem, we provided a sufficient optimality condition that interestingly holds in the practical region of interest, and thus a globally optimal solution was obtained. Furthermore, utilizing optimal thresholds provides a gain of several orders of magnitude of cooperation in terms of miss detection probability compared with the conventional equal threshold under various scenarios of the target probability of false alarm, the number of cooperative sensing nodes, and the different average SNRs. Some practical extensions of the proposed scheme may include topics such as imperfect reporting

Figure 8: The probability of miss detection for two different thresholds methods are shown when the mean of different average SNRs of cooperative sensing nodes increases; the target \(\overline{P^{f}}=1\%\) and \(N=7\). Figure 7: The probability of miss detection for two different thresholds methods are shown with the increasing number of cooperative sensing nodes; the target \(\overline{P^{f}}=1\%\). channels, reduced reporting overheads, and energy-efficient sensing techniques in more general hard and soft decisions. This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT under Grant (NRF-2017R1A1A1A05001377). **Author Contributions:** [PERSON] designed the algorithm, performed the simulations, and prepared the manuscript. [PERSON] contributed to the design of the system and supervised the activities. [PERSON] led the reasearch project and supervised the activities as the corresponding author. All authors discussed the results and approved the publication. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A

Proof of Proposition 1.: For notational simplicity we define \(a_{i}\triangleq\frac{1}{\sqrt{1+\gamma_{i}}}\) and \(\tilde{\gamma}_{i}\triangleq\frac{\gamma_{i}\sqrt{\tau B}}{2}\).