# Utilizing Multi-Dimensional MmWave MIMO Channel Features for Location Verification
## 4 Performance Analysis  
In this section, we focus on the modeling of false alarm rate (denoted by \(P_{f}\)) and detection rate (denoted by \(P_{d}\)). To facilitate the calculation of the two probabilities, we denote \(P_{\Theta_{\zeta},H_{i}}\) the probability of \((\Theta_{\zeta}>\eta_{\zeta})\) under \(H_{i}\) for \(i=1,2\) and \(\zeta\in\{\phi,\alpha,\theta\}\), and we consider \(\sigma_{\zeta_{i,LAB}}^{2}=\sigma_{\zeta_{LAB}}^{2}\), and \(\sigma_{\zeta_{i,LAB}}^{2}=\sigma_{\zeta_{EB}}^{2}\). Based on authentication decision in (45), we establish the following lemmas regarding the probability \(P_{\Theta_{\zeta},H_{i}}\). **Lemma 2**. : _Based on (44) and (45), \(P_{\Theta_{\phi},H_{0}}\) is expressed by_  
\[P_{\Theta_{\phi},H_{i}}=\exp\left(-\frac{\eta_{\phi}}{2\sigma_{\phi,H_{i}}^{2} }\right)\sum_{z=0}^{\frac{t}{2}-1}\frac{1}{z!}\left(\frac{\eta_{\phi}}{2\sigma _{\phi,H_{i}}^{2}}\right)^{z}, \tag{46}\]  
_where \(\sigma_{\phi,H_{0}}^{2}=2(1-\rho_{\phi})\sigma_{\phi,AS}^{2}+2\sigma_{\phi,A _{\phi}}^{2}\) and \(\sigma_{\phi,H_{1}}^{2}=\sigma_{\phi,EB}^{2}+\sigma_{\phi,AS}^{2}+2\sigma_{ \Delta_{\phi}}^{2}\)._Proof. : To calculate the probability distribution of \(\Theta_{\phi}\) in (44), we define \(\Lambda\phi\) as  
\[\Lambda\phi\triangleq\hat{\phi}_{l,XB}(t)-\hat{\phi}_{l,AB}(t-1). \tag{47}\]  
Under \(H_{0}\), by substituting (9) into (47), \(\Lambda\phi_{H_{0}}\) is obtained as  
\[\Lambda\phi_{H_{0}} =\hat{\phi}_{l,AB}(t)-\hat{\phi}_{l,AB}(t-1)\] \[=(\rho_{\phi}-1)\phi_{l,AB}(t-1)+\sqrt{1-\rho_{\phi}^{2}}u_{\phi,l,XB}(t)+\Delta\phi_{l}(t)-\Delta\phi_{l}(t-1). \tag{48}\]  
One can see that \(\Lambda\phi_{H_{0}}\) is a zero-mean Gaussian distribution with variance \(\sigma_{\phi,H_{0}}^{2}=2(1-\rho_{\phi})\sigma_{\phi_{AB}}^{2}+2\sigma_{\Delta _{\phi}}^{2}\). Therefore, \(\Theta_{\phi}\) under \(H_{0}\) is a central chi-square distribution random variable and the PDF of \(\Theta_{\phi}\) under \(H_{0}\) is then written as  
\[f_{\Theta_{\phi},H_{0}}(x)=\frac{\left(\frac{x}{2\sigma_{\phi,H_{0}}^{2}} \right)^{\frac{L}{2}-1}}{2\sigma_{\phi,H_{0}}^{2}\Gamma(L/2)}\exp(-\frac{x}{2 \sigma_{\phi,H_{0}}^{2}}),\;x\geq 0, \tag{49}\]  
where \(\Gamma(\cdot)\) is a Gamma function. The CDF of \(\Theta_{\phi}\) under \(H_{0}\) is written as  
\[F_{\Theta_{\phi},H_{0}}(x)=1-\exp(-\frac{x}{2\sigma_{\phi,H_{0}}^{2}})\sum_{z= 0}^{\frac{L}{2}-1}\frac{1}{z!}\Bigg{(}\frac{x}{2\sigma_{\phi,H_{0}}^{2}} \Bigg{)}^{z}. \tag{50}\]  
Accordingly, \(P_{\Theta_{\phi},H_{0}}\) is written as  
\[P_{\Theta_{\phi},H_{0}}=\Pr\{\Theta_{\phi}>\eta_{\phi}|H_{0}\}=\exp(-\frac{ \eta_{\phi}}{2\sigma_{\phi,H_{0}}^{2}})\sum_{z=0}^{\frac{L}{2}-1}\frac{1}{z!} \Bigg{(}\frac{\eta_{\phi}}{2\sigma_{\phi,H_{0}}^{2}}\Bigg{)}^{z}. \tag{51}\]  
Similarly, based on (47), we derive \(\Lambda\phi_{H_{1}}\) under hypothesis \(H_{1}\) as  
\[\Lambda\phi_{H_{1}}=\hat{\phi}_{l,EB}(t)-\hat{\phi}_{l,AB}(t-1)=\phi_{l,EB}(t) +\Delta\phi_{l}(t)-\phi_{l,AB}(t-1)-\Delta\phi_{l}(t-1). \tag{52}\]  
Because \(\phi_{l,EB}(t)\), \(\Delta\phi_{l}(t)\), \(\phi_{l,AB}(t-1)\) and \(\Delta\phi_{l}(t-1)\) are independent zero-mean Gaussian random variables with variances \(\sigma_{\phi_{l,EB}}^{2}\), \(\sigma_{\Delta_{\phi}}^{2}\), \(\sigma_{\phi_{l,AB}}^{2}\) and \(\sigma_{\Delta_{\phi}}^{2}\), respectively, and then the PDF of \(\Theta_{\phi}\) under \(H_{1}\) is written as  
\[f_{\Theta_{\phi},H_{1}}(x)=\frac{\left(\frac{x}{2\sigma_{\phi,H_{1}}^{2}} \right)^{\frac{L}{2}-1}}{2\sigma_{\phi,H_{1}}^{2}\Gamma(L/2)}\exp(-\frac{x}{2 \sigma_{\phi,H_{1}}^{2}}),\;x\geq 0, \tag{53}\]  
where \(\sigma_{\phi,H_{1}}^{2}=\sigma_{\phi_{EB}}^{2}+\sigma_{\phi_{AB}}^{2}+2\sigma _{\Delta_{\phi}}^{2}\).