# Reconfigurable Intelligent Surface Assisted Target Three-Dimensional Localization with 2-D Radar
## 3 Three-Dimensional Target Localization  
### Localization Method  
In this section, the measurements \(r_{k}\) and \(\theta_{k}\) in the direct channel and \(d_{m}^{k}\), \(m=1\), \(2\), \(\ldots\), \(M\) in the auxiliary channels are combined to estimate the three-dimensional location of the \(k\)-th target. It is assumed that the estimated location of the \(k\)-th target is represented as \(\hat{\mathbf{X}}_{k}=[\hat{x}_{k},\hat{y}_{k},\hat{z}_{k}]^{\mathrm{T}}\). According to (3) and (11), the range measurements \(r_{k}\) in the direct channel and \(d_{m}^{k}\) in the auxiliary channels are the sum of their true values and measurement errors. Based on the criterion of Minimum Mean Square Error (MMSE), the estimation of the target location \(\hat{\mathbf{X}}_{k}=[\hat{x}_{k},\hat{y}_{k},\hat{z}_{k}]^{\mathrm{T}}\) can be established as a least squares problem,  
\[\hat{\mathbf{X}}_{k}=\underset{\hat{x}_{k},\hat{y}_{k},\hat{z}_{k}}{\mathrm{argmin }}F(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k}) \tag{14}\]  
where  
\[F(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})=\left(r_{k}-R_{T\to R}^{k} \right)^{2}+\sum_{m=1}^{M}\left(d_{m}^{k}-R_{T\to RIS,m}^{k}-R_{T\to R}^{k} \right)^{2} \tag{15}\]  
The relationships between the distance \(R_{T\to R}^{k}\), \(R_{T\to RIS,m}^{k}\) and the target location \([x_{k},y_{k},z_{k}]\) are given in (5) and (12). Considering the azimuth \(\theta_{k}\) of the \(k\)-th target in the direct channel, the following constraints can be obtained,  
\[\theta_{k}-\Delta\theta_{k}<\arctan\left(\frac{\hat{y}_{k}-y_{R}}{\hat{x}_{k} -x_{R}}\right)<\theta_{k}+\Delta\theta_{k} \tag{16}\]  
where \(\Delta\theta_{k}\) is determined by the azimuth error \(\sigma_{\theta,k}\) in (7). With the confidence interval set as 99.7%, \(\Delta\theta_{k}=3\sigma_{\theta,k}\). Although the target pitch is not known in 2-D radar, its sine and cosine values must be less than 1. Therefore, the upper limit of the X-axis and Y-axis coordinates of the target can be obtained as shown in (17) and (18),  
\[\hat{x}_{k}<(r_{k}+\Delta r_{k})\cos(\theta_{k}-\Delta\theta_{k}) \tag{17}\]\[\hat{y}_{k}<(r_{k}+\Delta r_{k})\sin(\theta_{k}+\Delta\theta_{k}) \tag{18}\]  
where \(\Delta r_{k}\) is also determined by the range error of 2-D radar. Herein, \(\Delta r_{k}=3\sigma_{r,k}\). The \(\Delta r_{k}\) and \(\Delta\theta_{k}\) in (17) and (18) are considered to make the upper limit as large as possible, and the probability of the X-axis and Y-axis coordinates of the \(k\)-th target exceeding the set upper limit is extremely small. There is certain restriction on the flying height of aircrafts; it is reasonably assumed that the height of aircraft \(\hat{z}_{k}\) is usually less than 20 km,  
\[\hat{z}_{k}<20\text{ km} \tag{19}\]  
Therefore, the target three-dimensional localization can be formulated as the following optimization problem,  
\[ \begin{array}{ll}\hat{\mathbf{X}}_{k}=\underset{\hat{x}_{k}\in\Omega}{\arg\! \min}F(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})\\ s.t.&\theta_{k}-\Delta\theta_{k}<\arctan\!\left(\frac{\hat{y}_{k}-y_{k}}{\hat {x}_{k}-x_{k}}\right)<\theta_{k}+\Delta\theta_{k}\\ &\hat{x}_{k}<(r_{k}+\Delta r_{k})\cos(\theta_{k}-\Delta\theta_{k})\\ &\hat{y}_{k}<(r_{k}+\Delta r_{k})\sin(\theta_{k}+\Delta\theta_{k})\\ &\hat{z}_{k}<20\text{km}\end{array}  \tag{20}\]  
where \(\Omega\) represents the three-dimensional detection space where the target may exist. The optimization problem in (20) is convex, which will be proved, and then the convex optimization method is used to find its minimum value. Firstly, it is proved that the optimization function is convex. Since \(R^{k}_{T\to R}\) and \(R^{k}_{T\to RIS,m}\) are both square roots of Euclidean distance, they are both convex functions about \((\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})\). As shown in (15), \(F(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})\) is the sum of several square terms, and each square term is a convex function. Therefore, \(F(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})\) is also a convex function. Then, it is proved that the constraint set is a convex set.