# Reconfigurable Intelligent Surface Assisted Target Three-Dimensional Localization with 2-D Radar
## 2 System Model
For the 2-D radar, the range and azimuth angle of the \(k\)-th target can be measured, which are assumed as \(r_{k}\) and \(\theta_{k}\),  
\[r_{k}=R_{T-R}^{k}+n_{r,k} \tag{3}\]  
\[\theta_{k}=\arctan\left(\frac{y_{k}-y_{R}}{x_{k}-x_{R}}\right)+n_{\theta,k} \tag{4}\]  
where  
\[R_{T-R}^{k}=\sqrt{\left(x_{k}-x_{R}\right)^{2}+\left(y_{k}-y_{R}\right)^{2}+ \left(z_{k}-z_{R}\right)^{2}} \tag{5}\]  
The measurement noise \(n_{r,k}\) and \(n_{\theta,k}\) obey the normal distribution with zero mean, i.e., \(n_{r,k}\sim N(0,\sigma_{r,k}^{2})\), \(n_{\theta,k}\sim N(0,\sigma_{\theta,k}^{2})\). \(\sigma_{r,k}\) and \(\sigma_{\theta,k}\) represent the range and azimuth measurement errors of the 2-D radar, which is determined by the Signal-to-Noise Ratio (SNR) of the \(k\)-th target in the target echo \(r_{k}(t)\),  
\[\sigma_{r,k}=\frac{1.4\text{c}}{B\sqrt{SNR_{k}}} \tag{6}\]  
\[\sigma_{\theta,k}=\frac{\theta_{3 dB}}{1.6\sqrt{2 SNR_{k}}} \tag{7}\]  
where \(B\) is the bandwidth, and \(\theta_{3 dB}\) is the 3 dB beam width in the azimuth dimension. In the \(m\)-th auxiliary channel, the target echo \(r_{m}(t)\) is  
\[r_{m}(t)=\sum_{k=1}^{K}r_{m}^{k}(t)+w_{m}(t) \tag{8}\]  
\(w_{m}(t)\) is the Gaussian noise, which is independent from \(w(t)\). \(r_{m}^{k}(t)\) is the echo of the \(k\)-th target is received in the \(m\)-th auxiliary channel,  
\[r_{m}^{k}(t)=\alpha_{m}^{k}a_{t}\mathbf{\Lambda}_{m}\mathbf{a}_{r}^{\mathsf{T}}s(t-t_{ m}^{k}) \tag{9}\]  
where \(\alpha_{m}^{k}\) is the amplitude of the \(k\)-th target of the \(m\)-th RIS. \(\mathbf{a}_{t}=\left[\text{e}^{\text{i}\psi_{t,1}},\text{e}^{\text{i}\psi_{t,2}},\cdots,\text{e}^{\text{i}\psi_{t,L}}\right]^{\mathsf{T}}\) and \(\mathbf{a}_{r}=\left[\text{e}^{\text{i}\psi_{r,1}},\text{e}^{\text{i}\psi_{r,2}},\cdots,\text{e}^{\text{i}\psi_{t,L}}\right]^{\mathsf{T}}\) are the target phase vectors of the target-RIS and the RIS-radar channels, respectively. \(L\) is the number of sub-\(\lambda\)-sized surface element of the RIS. \(\mathbf{\Lambda}_{m}=\text{diag}(\left[\text{e}^{\phi_{1}^{m}},\text{e}^{\phi_{2} ^{m}},\cdots,\text{e}^{\phi_{L}^{m}}\right])\) is the adjustable phase matrix of the \(m\)-th RIS. \(t_{k}^{m}\) refers to the time delay of the \(k\)-th target in the \(m\)-th RIS-assisted auxiliary channel. \(\left[\cdot\right]^{\mathsf{T}}\) stands for the matrix transpose. \(\text{diag}(\cdot)\) represents a diagonal matrix with the elements of the vector as its diagonal elements. Based on the maximum SNR criterion, the phase matrix \(\mathbf{\Lambda}_{m}\) for the under-test resolution cell should be set as  
\[r_{k}(t)=\sum_{k=1}^{K}\alpha_{k}s(t-t_{k})+w(t) \tag{2}\]0  
Clearly, \(\mathbf{\Lambda}_{m}=\text{diag}(\left[\text{e}^{-\psi_{t,1}-\psi_{t,1}},\text{e} ^{-\psi_{t,2}-\psi_{t,2}},\cdots,\text{e}^{-\psi_{t,L}-\psi_{t,L}}\right])\), and the phase of the \(l\)-th element in the \(m\)-th RIS is \(\phi_{l}^{m}=-\psi_{t,l}-\psi_{r,l}\), \(l=1,2,\ldots,L\). Since the distances between the radar and the RISs are known, the measured range sum \(d_{m}^{k}\) in the \(m\)-th auxiliary channel represents the total range from the radar to the \(k\)-th target and then to the \(m\)-th RIS, which can be written as  
\[r_{k}(t)=\sum_{k=1}^{K}\alpha_{k}s(t-t_{k})+w(t) \tag{2}\]1where \(R_{T\to RIS,m}^{k}\) represents the distance from the \(k\)-th target to the \(m\)-th RIS,  
\[r_{k}(t)=\sum_{k=1}^{K}\alpha_{k}s(t-t_{k})+w(t) \tag{2}\]2  
The measurement noise \(n_{k,m}\sim N(0,\sigma_{k,m}^{2})\), and  
\[r_{k}(t)=\sum_{k=1}^{K}\alpha_{k}s(t-t_{k})+w(t) \tag{2}\]3  
where \(SNR_{k,m}\) represents the SNR of the \(k\)-th target in the \(m\)-th auxiliary channel. According to the relative position relationship between the RIS and the 2-D radar, \(SNR_{k,m}\) can be expressed with the SNR of the direct channel \(SNR_{k}\), \(SNR_{k,m}=SNR_{k}+20\lg\left(R_{T-R}^{k}/\left(R_{T-RIS,m}^{k}+R_{RIS,m-R}^{k} \right)\right)\). \(R_{RIS,m-R}^{k}\) represents the distance from the \(m\)-th RIS to radar.