## 5 Energy Detection Process
### Detection Threshold Estimation  
The choice of detection threshold is the most important process that defines the performance of any sensing method, including the ED [8]. The detection threshold is a value that defines the detection efficiency, and optimal threshold selection represents the value needed to meet the detection performance requirements [3]. As previously mentioned, the threshold estimation techniques can be broadly classified as either fixed or dynamic. The fixed threshold can be calculated based on two principles: the constant false alarm rate (CFAR) and the constant detection rate (CDR). Noise power (\(\sigma_{n_{i}}^{2}\)) is needed to determine the threshold in both cases [3; 49]. If the required false alarm probability (\(P_{fa_{i}}\)) is predetermined, the false alarm threshold based on the CFAR principle (\(\lambda_{fai}\)) can be expressed according to relation (12) as [47; 49]:  
\[\lambda_{fai}=Q^{-1}\left(P_{fa_{i}}\right)\sigma_{n_{i}}^{2}\sqrt{\frac{2}{N }}+\sigma_{n_{i}}^{2} \tag{14}\]  
In the CFAR principle, the threshold is set to meet an aimed false alarm probability (\(P_{fa_{i}}\)) and this threshold is then used to calculate the corresponding detection probability \(P_{d_{i}}\). The CFAR principle is used in CR when it is required to guarantee the reuse probability of the unused spectrum that demands the setting of the probability of the false alarm \(P_{fa_{i}}\) to a small fixed value, while the detection probability \(P_{d_{i}}\) should be maximised. Similarly, in order to achieve a target detection probability \(P_{d_{i}}\) for the average received signal power \(P\) of a specific PU, the detection probability threshold \(\lambda_{d_{i}}\), in the case of the CDR principle, can be derived from relation (11) [47; 49]:  
\[\lambda_{di}=Q^{-1}\big{(}P_{d_{i}}\big{)}\Big{(}P+\sigma_{n_{i}}^{2}\Big{)} \sqrt{\frac{2}{N}}+\Big{(}P+\sigma_{n_{i}}^{2}\Big{)} \tag{15}\]  
The CDR principle is used when it is necessary to guarantee a non-interference probability of the incumbent systems. This requires setting the detection probability \(P_{d_{i}}\) to a high value and minimising the false alarm probability \(P_{fa_{i}}\). By comparing expressions (14) and (15), it can be noted that the CFAR approach does not need the average received signal power \(P\) of a PU to set the threshold \(\lambda_{fa_{i}}\). For this reason, the CFAR principle is more commonly used in practice. However, constantly setting the false alarm probability \(P_{fa_{i}}\) to low values means that the corresponding threshold (\(\lambda_{fai}\)) will be high. As a consequence, interference may occur since it is not easy to detect low-power signals due to the very demanding values of the CFAR threshold \(\lambda_{fa_{i}}\). Therefore, the CFAR approach based on a fixed threshold is not optimal. An optimal threshold setting can be achieved if each SU dynamically adjusts its threshold according to its channel state. In order to gain better system performance, selecting the appropriate value of the DT is a challenging task. However, performing analyses when DT variation is taken into account enables significantly more realistic analyses. Instead of remaining constant, the threshold can be scaled with DT factor \(\rho^{\prime}\) (\(\rho^{\prime}\geq 1\)) in such a way that the factor \(\rho^{\prime}\) sets the DT interval \(\lambda_{i}^{\prime DT}\epsilon\left[\frac{\lambda_{i}}{\rho^{\prime}},\ \rho^{\prime}\lambda_{i}\right]\).