# Reconfigurable Intelligent Surface Assisted Target Three-Dimensional Localization with 2-D Radar
## 3 Three-Dimensional Target Localization
### Crlb  
The CRLB is widely used in signal detection and estimation to provide a lower bound for the variance of any unbiased parameter estimator. In this section, the CRLB is derived to measure the target localization accuracy for the proposed localization algorithm. Its Mean Square Error (MSE) or Root Mean Square Error (RMSE) can be compared with the CRLB, and the performance gap between the algorithm and the best performance will be known. It is assumed that \(M\) RISs are used to assist the 2-D radar. Let the position of the \(k\)-th target be \(\mathbf{X}_{k}=\left[x_{k},y_{k},z_{k}\right]^{\mathrm{T}}\). The measurement in the direct channel and the auxiliary channels for the \(k\)-th target can be written as  
\[\mathbf{Z}_{k}=\left[r_{k},\theta_{k},d_{1}^{k},\cdots,d_{M}^{k}\right]^{\mathrm{T}} \tag{21}\]  
The \(k\)-th target measurement equation can be expressed as  
\[\mathbf{Z}_{k}= \begin{bmatrix}r_{k}\\ \theta_{k}\\ d_{1}^{k}\\ \vdots\\ d_{M}^{k}\end{bmatrix} = \begin{bmatrix}h_{1}(\mathbf{X}_{k})\\ h_{2}(\mathbf{X}_{k})\\ h_{3}(\mathbf{X}_{k})\\ \vdots\\ h_{M+2}(\mathbf{X}_{k})\end{bmatrix} +\mathbf{v},\ \mathbf{v}\sim N(0,\mathbf{R}) \tag{22}\]  
where  
\[h_{1}(\mathbf{X}_{k})=\sqrt{\left(x_{k}-x_{R}\right)^{2}+\left(y_{k}-y_{R}\right)^ {2}+\left(z_{k}-z_{R}\right)^{2}} \tag{23}\]  
\[h_{2}(\mathbf{X}_{k})=\arctan\frac{y_{k}-y_{R}}{x_{k}-x_{R}} \tag{24}\]  
\[h_{i}(\mathbf{X}_{k})=\sqrt{\left(x_{k}-x_{r,i-2}\right)^{2}+\left(y_{k}-y_{r,i-2 }\right)^{2}+\left(z_{k}-z_{r,i-2}\right)^{2}}+h_{1}(\mathbf{X}_{k}),\ 3\leq i\leq M+2 \tag{25}\]  
Let  
\[\mathbf{H}_{k}=\left[h_{1}(\mathbf{X}_{k}),h_{2}(\mathbf{X}_{k}),\ldots,h_{M+2}(\mathbf{X}_{k })\right]^{\mathrm{T}} \tag{26}\]  
\[ \begin{array}{l}\mathbf{R}=\mathrm{diag}[\sigma_{r,k^{\prime}}^{2},\sigma_{0,k^ {\prime}}^{2},\sigma_{k,1}^{2},\ldots,\sigma_{k,M}^{2}]\\ \triangleq\mathrm{diag}[\sigma_{1}^{2},\sigma_{2}^{2},\sigma_{3}^{2},\ldots,\sigma_{M+2}^{2}]\end{array}  \tag{27}\]  
Thus, the likelihood function of vector \(\mathbf{H}_{k}\) can be expressed as  
\[p(\mathbf{Z}_{k};\mathbf{H}_{k})=\frac{\exp\Bigl{\{}-\frac{1}{2}(\mathbf{Z}_{k}-\mathbf{H}_{k} )^{\mathrm{T}}\mathbf{R}^{-1}(\mathbf{Z}_{k}-\mathbf{H}_{k})\Bigr{\}}}{\left(2\pi\right)^{ \frac{M+2}{2}}|\mathbf{R}|^{\frac{1}{2}}} \tag{28}\]  
Let  
\[L_{k}=\ln p(\mathbf{Z}_{k};\mathbf{H}_{k})=-\frac{1}{2}\sum_{i=1}^{M+2}\frac{1}{\sigma _{i}^{2}}(\mathbf{Z}_{k}(i)-\mathbf{H}_{k}(i))^{2} \tag{29}\]  
where \(\mathbf{Z}_{k}(i)\) represents the \(i\)-th element in \(\mathbf{Z}_{k}\), and \(\mathbf{H}_{k}(i)\) represents the \(i\)-th element in \(\mathbf{H}_{k}\). The likelihood function satisfies the regularity condition for each argument, i.e.,  
\[ \begin{array}{l}E\Big{(}\frac{\partial L_{k}}{\partial\mathbf{H}_{k}}\Big{)}\\ =\Big{[}E\Big{(}\frac{\partial L_{k}}{\partial h_{1}(\mathbf{X}_{k})}\Big{)},\dots,E \Big{(}\frac{\partial L_{k}}{\partial h_{M+2}(\mathbf{X}_{k})}\Big{)}\Big{]}^{ \mathsf{T}}\\ =[0,\dots,0]^{\mathsf{T}}=\mathbf{0}_{(M+2)\times 1}\end{array}  \tag{30}\]  
For each unbiased estimate of the variance of \(\hat{\mathbf{X}}_{k}\), there exists a lower bound, which can be expressed as  
\[\operatorname{var}(\hat{\mathbf{X}}_{k})\geq\Big{[}\mathbf{I}_{k}^{-1}(\mathbf{X}_{k}) \Big{]}_{ll},\ l=1,2,3 \tag{31}\]  
This lower bound is referred to as the CRLB lower bound of the unbiased estimator, where \(\mathbf{I}_{k}(\mathbf{X}_{k})\) is the Fisher information matrix with the dimension as \(3\times 3\), and each element of the matrix is defined,  
\[\left[\mathbf{I}_{k}(\mathbf{X}_{k})\right]_{ab}=-E\Big{(}\frac{\partial^{2}L_{k}}{ \partial x_{a}\partial x_{b}}\Big{)},\ a,b=1,2,3 \tag{32}\]  
where \(x_{1}=x_{k}\), \(x_{2}=y_{k}\), \(x_{3}=z_{k}\).