# Utilizing Multi-Dimensional MmWave MIMO Channel Features for Location Verification
## 2 Problem Formulation and System Model
### Channel Model  
Bob has a \(N_{r}\)-antenna uniform planar array (UPA) where the numbers of antennas in the horizon and vertical directions are \(M\) and \(N\), respectively, where \(N_{r}=MN\). The number of radio frequency chains of Bob is \(M_{RF}\)[19]. Let \(d\) be the distance between any arbitrary two adjacent antennas, and \(\mathbf{G}_{RF}\) be the analog beamformer equipped at Bob. Considering the limited scattering characteristics of the mmWave communication environments, we here adopt the extended Saleh-Valenzuela (SV) model to characterize the mmWave MIMO channels [20; 21]. Denote \(\phi_{l}\) as the azimuth angle of arrival (AAoA) of the \(l\)-th propagation multipath of the transmitter and \(\theta_{l}\) as the elevation angle of arrival (EAoA) of that, for \(l=1,2,\cdots,L\), respectively. We assume \(\phi_{l}\sim\mathcal{N}(0,\sigma_{\phi_{l,XB}}^{2})\) and \(\theta_{l}\sim\mathcal{N}(0,\sigma_{\theta_{l,XB}}^{2})\). The steering matrix \(\mathbf{A}(\phi_{l},\theta_{l})\) is expressed by  
\[\mathbf{A}(\phi_{l},\theta_{l})=\frac{1}{\sqrt{MN}} \begin{bmatrix}1&\cdots&e^{ |2\pi d/\lambda(N-1)\sin\phi_{l}\cos\theta_{l}}\\ \vdots&\ddots&\vdots\\ e^{|2\pi d/\lambda(M-1)\cos\phi_{l}}&\cdots&e^{|2\pi d/\lambda((M-1)\cos\phi_{l }+(N-1)\sin\phi_{l}\cos\theta_{l})}\end{bmatrix} , \tag{1}\]  
where \(\lambda\) is the wavelength of the carrier signal [22]. For simplicity, let \(v_{1,l}=\frac{2\pi d}{\lambda}\cos\phi_{l}\) and \(v_{2,l}=\frac{2\pi d}{\lambda}\sin\phi_{l}\cos\theta_{l}\). The steering matrix \(\mathbf{A}(\phi_{l},\theta_{l})\) can be further written as  
\[\mathbf{A}(\phi_{l},\theta_{l})=\mathbf{A}(v_{1,l},v_{2,l})=\mathbf{a}(v_{1,l })\mathbf{a}^{T}(v_{2,l}), \tag{2}\]  
where \(\mathbf{a}(v_{1,l})=\frac{1}{\sqrt{M}}[1,\cdots,e^{i(M-1)v_{1,l}}]^{T}\) and \(\mathbf{a}(v_{2,l})=\frac{1}{\sqrt{N}}[1,\cdots,e^{i(N-1)v_{2,l}}]^{T}\). Similar to [23], the mmWave channel with \(L\) being the number of propagation paths is modeled as  
\[\mathbf{H}=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}\alpha_{l}\mathbf{A}(\phi_{l}, \theta_{l})=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}\alpha_{l}\mathbf{a}(v_{1,l}) \mathbf{a}^{T}(v_{2,l})=\frac{1}{\sqrt{L}}\mathbf{A}_{v_{1}}\mathbf{H}_{\alpha }\mathbf{A}_{v_{2}}^{T}, \tag{3}\]  
where \(\mathbf{A}_{v_{1}}=[\mathbf{a}  
where \(\rho_{\alpha}\) is the path gain correlation coefficient [28] and \(u_{\alpha_{l}}(t)\) is zero-mean Gaussian noise, i.e., \(u_{\alpha_{l}}(t)\sim\mathcal{CN}(0,\sigma_{\alpha_{l},X\beta}^{2})\). Similarly, we use first-order Gauss-Markov process to model the time-varying azimuth angle of arrival as  
\[\phi_{l}(t)=\rho_{\phi}\phi_{l}(t-1)+\sqrt{1-\rho_{\phi}^{2}}u_{\phi_{l}}(t), \tag{6}\]  
where \(\rho_{\phi}\) is the AAoA correlation coefficient [28] and \(u_{\phi_{l}}(t)\) is zero-mean Gaussian noise, i.e., \(u_{\phi_{l}}(t)\sim\mathcal{N}(0,\sigma_{\phi_{l},X\beta}^{2})\). We also use first-order Gauss-Markov process to model the time-varying elevation angle of arrival as  
\[\theta_{l}(t)=\rho_{\theta}\theta_{l}(t-1)+\sqrt{1-\rho_{\theta}^{2}}u_{\theta _{l}}(t), \tag{7}\]  
where \(\rho_{\theta}\) is the EAoA correlation coefficient [28] and \(u_{\theta_{l}}(t)\) is zero-mean Gaussian noise, i.e., \(u_{\theta_{l}}(t)\sim\mathcal{N}(0,\sigma_{\theta_{l},X\beta}^{2})\). Channel-based verification utilizes the fact that wireless channels decorrelate rapidly as the source (e.g., a transmitter or user using transmitter is transmitting signals to an intended destination/receiver) changes location in an environment with rich scatterers and reflectors. This indicates that wireless channel features are spatially decorrelated between different geographic locations. When the distance between a legitimate source and an adversary is greater than half of the wavelength, it is extremely difficult for the adversary to obtain the accurate wireless channel features.