## 2 System Model and GD Test Statistics
### The GD Optimal Threshold  
As a matter of fact, the ED and GD ignore the PU signal characteristics and rely only on the PU signal energy. Thus, the ED and GD optimal threshold should be proportional to the nominal noise power at the SU input. In practice, the noise power is unknown and should be estimated by the GD noise power estimator (NPE in Figure 2). As a result, both the ED and GD detection thresholds can be defined based on the total error rate minimization [34; 35; 36]. In the case of the additive white Gaussian noise (AWGN) channel, the GD optimal threshold can be defined using the minimal probability of error in the following form:  
\[{\mathit{THR}_{{}_{GD}}^{op}=\arg\min_{{\mathit{THR}_{{}_{GD}}}}P_{\sigma}^{{} _{\pi}}({\mathit{THR}_{{}_{GD}}})} \tag{54}\]  
where \(P_{\sigma}^{{}_{\pi}}\) is the probability of error given by  
As we can see from Figure 5, in the ideal case, _i.e._, the complete compensation of the noise component of the GD correlation channel \(2s_{i}^{\rm mod}[k]\zeta_{i}[k]\) and the random component of the GD ED channel \(2s_{i}[k]\xi_{i}[k]\), the GD presents the best sample complexity performance in comparison with the ED and MF under conditions of the noise power uncertainty. The GD overcomes a negative impact of the noise power uncertainty. Thus, the GD can detect the PU signal at any arbitrary low _SNR_ increasing the number \(N\) of samples. In other words, there is no _SNR_ wall. In the case of ED, when there is no noise power uncertainty, _i.e._, \(\rho=1\), there is no _SNR_ wall and the PU signal can be detected at any low _SNR_ by increasing the sensing time or the number \(N\) of samples. If there is the noise power uncertainty, there is the _SNR_ wall for the ED and its location depends on the value of \(\varepsilon\) and, consequently, the uncertainty parameter \(\rho\). For example, at \(\varepsilon=1\) dB, the \(SNR_{wall}^{ED}=-3\) dB, and at \(\varepsilon=0.1\) dB, the \(SNR_{wall}^{ED}=-13\) dB. Thus, the sample complexity tends to approach infinity as the _SNR_ decreases tending to approach the _SNR_ wall:  
\[\lim_{SNR^{ED}\to SNR_{wall}^{ED}}f(SNR^{ED},P_{FA},P_{miss},\rho)\ \to\infty \tag{58}\]  
Small values of \(\varepsilon\), the least uncertainty case, are preferred because, in this case, there is a decreasing in _SNR_ wall. The MF has the better sample complexity performance at the low _SNR_ in comparison with the ED. For example, at \(\varepsilon=0.1\) dB, \(SNR_{wall}^{ED}=-13\) dB and \(SNR_{wall}^{MF}=-23\) dB. The MF sample complexity depends on the channel coherence time \(K_{c}\). As we can see, at the same value of \(\varepsilon\), \(SNR_{wall}^{MF}\) is decreased if the channel coherence time \(K_{c}\) is increased, _i.e._, \(SNR_{wall}^{MF}=-23\) dB at \(K_{c}=100\) and \(SNR_{wall}^{MF}=-33\) dB at \(K_{c}=1000\). Figure 5: The sample complexity curves of ED, MF, and GD under the noise power uncertainty. The GD sample complexity under the non-ideal condition, the case in practice, \(s_{i}^{\text{mod}}[k]\
eq s_{i}[k]\) and \(\sigma_{\xi}^{2}\
eq\sigma_{\eta}^{2}\), is presented in Figure 6 at \(\varepsilon=1\) dB, \(M=2\) and several values of \(\mu\) and \(\beta\). As we can see from Figure 6, the best GD sample complexity efficiency is obtained at \(\mu=1\) or \(s_{i}^{\text{mod}}[k]=s_{i}[k]\) and \(\beta=1\) or \(\sigma_{\xi}^{2}=\sigma_{\eta}^{2}\). Additionally, we can see that there is no _SNR_ wall in the GD case, but the GD sample complexity efficiency decreases at \(\mu\
eq 1\) and \(\beta\
eq 1\). In this case, more samples are needed at the same _SNR_ value to achieve the required probability of false alarm. The complementary receiver operating characteristic (ROC) curves, which are widely used in practice, for example in [39, 40, 41], for the ED and GD are presented in Figure 7 with and without the noise power uncertainty at \(M=6\) and \(N=20\). In a general, for both detectors the noise power uncertainty leads to the complementary ROC curves shifting away from the (0,0) origin.