## 5 Numerical Results  
For the analysis to be conducted herein, we consider a wireless network with a total of \(K=10\) multi-antenna SUs randomly deployed to detect a PU that appears at an unknown position. The spatial correlation between the antennas of a SU is modeled herein as \(c_{i,j}=\rho^{[i-j]}\), with \(\ \{i,j\}=1,\cdots,L\). Moreover, \(0<\rho<1\) with \(\rho=e^{-23\kappa^{2}(d\lambda_{c})^{2}}\), which is called correlation coefficient between two adjacent antennas, and it relies on the angular spread \(\kappa\), the wavelength \(\lambda_{c}\) and the distance \(d\) between two adjacent antennas [12]. Finally, the spatial correlation between SUs due to the correlated shadowing effects is modeled based on the model given in [26]. In order to analyze the performance of the proposed detectors, we use receiver operating characteristic curve (ROC) and area under the ROC curve (AUC), which varies between 0.5 (poor performance) and 1 (good performance). Moreover, we define the average signal-to-noise ratio (SNR) of all SUs as: \(\bar{\kappa}=\frac{1}{K}\sum_{i=1}^{K}\kappa_{i}\) and the SNR of \(i\)-th SU is \(\kappa_{i}=\zeta^{i-1}\kappa_{\text{min}}\) where \(\kappa_{\text{min}}\) is the minimum SNR among those of the SUs [13]. For a given average SNR \(\bar{\kappa}\) and SNR gap \(\zeta\), we can generate the random SNRs of the SUs. Similarly, in order to analyze the effect of noise power uncertainty we generate the noise power at \(k\)-th SU as \(\sigma_{w,k}^{2}\sim\mathcal{U}\left(\frac{\sigma_{n,k}^{2}}{\alpha_{nu}}, \alpha_{nu}\sigma_{n,k}^{2}\right)\), where \(\alpha_{nu}\geq 1\), and \(\alpha_{nu}=1\) means no noise uncertainty. In Figure 2 we plot ROC curves to compare the performance of our proposed spectrum sensing schemes with the optimal likelihood ratio test (LRT) and the energy detection schemes. For the the optimal LRT, to solve Equation (6) we assume that the covariance matrices under both hypothesis are known. On the other hand the energy detector is given as:  
\[\Lambda_{\text{ENG}}\left(\mathbf{X}_{N}\right)=\sum_{n=1}^{N}\sum_{k=1}^{K} \sum_{l=1}^{L}\left|x_{k,l}\left(n\right)\right|^{2}\gtrless_{\mathcal{H}_{ 0}}^{\mathcal{H}_{1}}\gamma. \tag{19}\]  
Note that dashed lines represent the plots for \(\alpha_{nu}=2\) and solid lines represent the case when \(\alpha_{nu}=1\). In Figure 3, we show the AUC plots to analyze the effects of the sample size \(N\) in the presence of noise power uncertainty. With these considerations, the results clearly show that \(\Lambda_{\text{MPKP}}\) and \(\Lambda_{\text{SPKP}}\) outperform the traditional GLRT\(\Lambda_{\text{T}}\) with a smaller sample support. Moreover, we can see that the energy detector performs better in the absence of the noise power uncertainty, however, it has poor performance in the presence of noise power uncertainty. Interestingly, we can also observe that in the presence of noise power uncertainty, the increase of sample size has very negligible impact on the performance of the energy based detection scheme \(\Lambda_{\text{ENG}}\) due to the phenomena of SNR-Wall [4]. In Figure 4, we show the AUC plots to further analyze the effects of noise power uncertainty. Once again, we can see that \(\Lambda_{\text{MPKP}}\) and \(\Lambda_{\text{SPKP}}\) have superior performance than \(\Lambda_{\text{T}}\) and \(\Lambda_{\text{ENG}}\). Moreover, we can observe the proposed schemes are robust against the noise power uncertainty compared to \(\Lambda_{\text{ENG}}\). It because the proposed schemes estimate the noise power in real time. In Figure 5, we show the AUC plots to analyze the effect of shadowing (_i.e._, \(\sigma_{\text{dB-Spread}}\)). From the result we can see that the effect of the shadowing is very small over the detection performance of the detection schemes as the presented spectrum sensing schemes are cooperative. However, in the case of \(\Lambda_{\text{ENG}}\), the detection performance slightly increases with the increase in the \(\sigma_{\text{dB-Spread}}\).