# Utilizing Multi-Dimensional MmWave MIMO Channel Features for Location Verification
## 3 Proposed Location Verification Scheme  
### Estimation of MmWave Channels  
To exploit the fine-grained mmWave channel features, we estimate the channel parameters separately. We first estimate AAoA and EAoA, and then estimate the mmWave path gain. We can write the estimations of \(\phi_{l,XB}(t)\), \(\theta_{l,XB}(t)\) and \(\alpha_{l,XB}(t)\), respectively, as  
\[\hat{\phi}_{l,XB}(t)=\phi_{l,XB}(t)+\Delta\phi_{l}(t), \tag{9}\]  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]  
\[\hat{\alpha}_{l,XB}(t)=\alpha_{l,XB}(t)+\Delta\alpha_{l}(t), \tag{11}\]where \(\Delta\phi\sim\mathcal{N}(0,\sigma_{\Delta\phi}^{2})\), \(\Delta\theta\sim\mathcal{N}(0,\sigma_{\Delta\theta}^{2})\) and \(\Delta\alpha\sim\mathcal{CN}(0,\sigma_{\Delta\alpha}^{2})\) are the corresponding estimation errors, respectively. Next, we calculate the variances of the estimation errors, i.e., \(\sigma_{\Delta\phi}^{2}\), \(\sigma_{\Delta\theta}^{2}\) and \(\sigma_{\Delta\alpha}^{2}\). For notational simplicity, here we ignore the time index \(t\) and the path index \(l\). The signal received by [PERSON] at time \(t\) is given by  
\[\mathbf{r}=\mathbf{G}_{RF}^{H}\text{vec}(\mathbf{H})s+\mathbf{n}=\mathbf{G}_{ RF}^{H}\text{vec}(\mathbf{A})\alpha s+\mathbf{n}, \tag{12}\]  
where \(\mathbf{A}\) is the \(M\times N\) steering matrix with its \((m,n)\)-th element given by  
\[\mathbf{A}_{m,n}=\frac{1}{\sqrt{MN}}e^{j(mv_{1}+mv_{2})}. \tag{13}\]  
The conditional PDF of \(\mathbf{r}\) is expressed by  
\[f(\mathbf{r}|v_{1},v_{2},\alpha)=\frac{1}{(\pi\sigma_{n}^{2})M_{ RF}}\exp\Bigg{\{}-\frac{||\mathbf{r}-\mathbf{G}_{RF}^{H}\text{vec}(\mathbf{A}) \alpha s||^{2}}{\sigma_{n}^{2}}\Bigg{\}}. \tag{14}\]  
The maximum-likelihood (ML) method in [29] is used to estimate mmWave channel features \(v_{1}\), \(v_{2}\), and \(\alpha\). The estimation of these features can be expressed as  
\[[\hat{v_{1}},\hat{v_{2}},\hat{\alpha}]=\underset{v_{1},v_{2}, \alpha}{\arg\max}f(\mathbf{r}|v_{1},v_{2},\alpha). \tag{15}\]  
Based on (14) and (15), we can equivalently write (15) as  
\[[\hat{v_{1}},\hat{v_{2}},\hat{\alpha}]=\underset{v_{1},v_{2}, \alpha}{\arg\min}||\mathbf{r}-\mathbf{G}_{RF}^{H}\text{vec}(\mathbf{A})\alpha s ||^{2}. \tag{16}\]  
When giving \(v_{1}\) and \(v_{2}\), the estimation of \(\alpha\) is calculated as  
\[\hat{\alpha}=\text{vec}(\mathbf{A})^{H}(\mathbf{G}_{RF}^{H})^{ \dagger}s^{*}\mathbf{r}. \tag{17}\]  
Substituting (17) into (16), the estimations of \(v_{1}\) and \(v_{2}\) is written as  
\[[\hat{v_{1}},\hat{v_{2}}] =\underset{v_{1},v_{2}}{\arg\min}||\mathbf{r}-\mathbf{G}_{RF}^{H} \text{vec}(\mathbf{A})\text{vec}(\mathbf{A})^{H}(\mathbf{G}_{RF}^{H})^{ \dagger}s^{*}\mathbf{r}s||^{2}\] \[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]0  
where \(g(v_{1},v_{2})\) is the cost function related to \(v_{1}\) and \(v_{2}\), \(\sigma_{s}^{2}=s^{*}s\) is the signal power and \(\mathbf{P}_{b}=\text{vec}(\mathbf{A})\text{vec}(\mathbf{A})^{H}\). Next, we give Lemma 1 regarding the variance of the estimations for \(v_{1}\) and \(v_{2}\). Lemma 1: _The variance of the estimations for \(v_{1}\) and \(v_{2}\) in (18) are given by_  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]1  
_where \(\mathbf{P}_{b}^{\perp}=\mathbf{I}-\mathbf{P}_{b}\). \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) are the diagonal matrices given by_  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]2  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]3Proof. : The received signal without noise can be written as  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]4  
When considering the receiver noise, the received signal can be further written as \(\mathbf{r}=\mathbf{r}_{s}+\mathbf{n}\). By using Taylor series expansion, the first derivative of the cost function \(g(v_{1},v_{2})\) is given by  
\[\hat{\theta}_{l,XB}(t)=\theta_{l,XB}(t)+\Delta\theta_{l}(t), \tag{10}\]5  
where \(\Delta v_{k}=\hat{v}_{k}-v_{k}\) denotes perturbation on \(v_{k}\).