## 2 System Model and GD Test Statistics
### The GD Optimal Threshold
As shown in Figure 7, the GD demonstrates the better sensing performance in comparison with the ED and the sensing performance degradation rate of GD is less under the noise power uncertainty conditions. In the GD case, under the low _SNR_ or if the _SNR_ is above the \(SNR_{wall}^{ED}\), the sensing performance degradation caused by the noise power uncertainty can be compensated by increasing in the number of samples \(N\). However, if the _SNR_ is below the \(SNR_{wall}^{ED}\), the ED complementary ROC curve is over the dotted straight line corresponding to the random coin-tossing detector case in Figure 7. This situation is observed at \(\varepsilon=0.1\) dB, \(SNR=-15\) dB if \(SNR=-13\) dB (Figure 5) and \(\varepsilon=1\) dB, \(SNR=-5\) dB when \(SNR_{wall}^{ED}=-3\) dB (Figure 6). In Figure 8, a comparison between the ED and GD performance in terms of the probability of error \(P_{e}\) as a function of the sample number \(N\), the analogous performance is discussed in [36], is shown at \(\varepsilon=0.1\) dB, \(SNR=-10\) dB, and \(SNR=-13\) dB. The GD demonstrates the better sensing performance in comparison with the ED one. For example, at \(N=10^{2}\) the probability of error \(P_{e}\) is equal to 0.3126 in the GD case and 0.5346 in the ED case. At \(SNR=-13\) dB that corresponds to the \(SNR_{wall}^{ED}\) when \(\epsilon=0.1\) dB, we can see that the probability of error \(P_{e_{r}}\) in the ED case fails to be robust and is distinctly differed owing to the _SNR_ wall phenomenon. In this case, an increasing in the number of samples \(N\) is not effective to improve the probability of error \(P_{e_{r}}\) performance for ED. At the same time, the GD has the same normal behavior meaning that the probability of error \(P_{e_{r}}\) performance for GD is improved with increasing in the number of samples \(N\). Figure 8: Spectrum sensing performance for the ED and GD in terms of the probability of error \(P_{e_{r}}\) as a function of the sample number. Figure 7: The complementary ROC curves for ED and GD. Comparison between the probability of error \(P_{\sigma}\) for the ED and GD as a function of the normalized optimal detection threshold, where \(NM\) is the normalization factor, is presented in Figure 9. The probability of error \(P_{\sigma}\) is evaluated for both detectors in two cases: there is the noise power uncertainty and there is no noise power uncertainty at the \(SNR=-5\,\mathrm{dB},M=2\), \(N=100\), \(\epsilon=0.1\) and \(1\mathrm{dB}\). As shown in Figure 9, the GD can achieve the lower probability of error \(P_{\sigma}\) in comparison with the ED for both cases. For example, if there is no noise power uncertainty the minimal probability of error \(P_{\sigma}\) is equal to 0.13 in the GD case and 0.25 in the ED case. If there is the noise power uncertainty with \(\epsilon=0.1\) dB, the lowest probability of error \(P_{\sigma}\) for the GD is equal to 0.24 and 0.33 for the ED. In a general case, the noise power uncertainty affects negatively on the ED and GD probability of error \(P_{\sigma}\). Thus, we can make the following conclusion: increasing in the noise power uncertainty leads to increasing in the probability of error \(P_{\sigma}\). Figure 10 demonstrates the effect of inequality between the noise variances \(\sigma_{\xi}^{2}\
eq\sigma_{\eta}^{2}\) at the GD PF and AF outputs on the probability of error \(P_{\sigma}\) as a function of the normalized detection threshold given by Equation (56) at \(SNR=-5\,\mathrm{dB}\), \(M=2\), \(N=100\), \(\epsilon=0.1\,\mathrm{dB}\). We can notice that the \(\beta\) value effects GD performance. For example, at \(\beta=0.9\) the probability of error \(P_{\sigma}^{GD}\) is approximately equal to 0.26 and at \(\beta=0.5\) the probability of error \(P_{\sigma}^{GD}\) is equal to 0.32. As follows form the theoretical analysis and simulation results, the GD implementation allows us to improve the spectrum sensing accuracy that is defined by the probability of false alarm \(P_{FA}\) and the probability of detection \(P_{D}\).