## 4 Proposed Localization Method
### Algorithm Development
#### 4.1.1 Calibration Stage
Thus according to the Bayesian Gauss-Markov theorem [33], the linear minimum mean square error (LMMSE) estimate of \(\mathbf{\Delta}\mathbf{s}\) can be obtained from (45) as  
\[ \begin{split}\mathbf{\Delta}\mathbf{\hat{s}}&= \mathrm{E}(\mathbf{\Delta}\mathbf{s})+\left(\mathrm{cov}(\mathbf{\Delta} \mathbf{s})^{-1}+\mathbf{G}_{0}^{\mathrm{T}}\mathrm{cov}(\mathbf{\Delta} \mathbf{h}_{0})^{-1}\mathbf{G}_{\mathrm{s}}\right)^{-1}\mathbf{G}_{0}^{ \mathrm{T}}\mathrm{cov}(\mathbf{\Delta}\mathbf{h}_{0})^{-1}(\mathbf{h}_{0}- \mathbf{G}_{0}\mathrm{E}(\mathbf{\Delta}\mathbf{s}))\\ &=\left(\mathbf{Q}_{\mathbf{s}}^{-1}+\mathbf{G}_{0}^{\mathrm{T} }(\mathbf{G}_{\mathrm{c}}\mathbf{Q}_{\mathrm{c}}\mathbf{G}_{\mathrm{c}}^{ \mathrm{T}}+\mathbf{Q}_{\mathrm{rc}})^{-1}\mathbf{G}_{0}\right)^{-1}\!\! \mathbf{G}_{0}^{\mathrm{T}}(\mathbf{G}_{\mathrm{c}}\mathbf{Q}_{\mathrm{c}} \mathbf{G}_{\mathrm{c}}^{\mathrm{T}}+\mathbf{Q}_{\mathrm{rc}})^{-1}\!\! \mathbf{h}_{0}\end{split}  \tag{51}\]  
Under the assumption that the noise in \(\mathbf{G}_{\mathrm{c}}\) and \(\mathbf{G}_{0}\) is sufficiently small to be ignored, the covariance matrix of \(\mathbf{\Delta}\mathbf{\hat{s}}\) can be given as  
\[\mathrm{cov}(\mathbf{\Delta}\mathbf{s}-\mathbf{\Delta}\mathbf{\hat{s}})= \left(\mathbf{Q}_{\mathbf{s}}^{-1}+\mathbf{G}_{0}^{\mathrm{T}}(\mathbf{G}_{ \mathrm{c}}\mathbf{Q}_{\mathrm{c}}\mathbf{G}_{\mathrm{c}}^{\mathrm{T}}+ \mathbf{Q}_{\mathrm{rc}})^{-1}\mathbf{G}_{0}\right)^{-1} \tag{52}\]  
Using the estimate of transmitter and receiver position error in (51), we can refine the transmitter and receiver positions as  
\[\mathbf{\hat{s}}=\mathbf{s}-\mathbf{\Delta}\mathbf{\hat{s}} \tag{53}\]  
Utilizing the fact \(\mathbf{s}=\mathbf{s}^{\mathrm{o}}+\mathbf{\Delta}\mathbf{s}\), we can rewrite \(\mathbf{\hat{s}}\) in (53) as \(\mathbf{\hat{s}}=\mathbf{s}^{\mathrm{o}}+\mathbf{\Delta}\mathbf{s}-\mathbf{ \Delta}\mathbf{\hat{s}}\). Hence, the refined estimate of transmitter/receiver positions \(\mathbf{\hat{s}}\) has a covariance matrix identical with (52). Forming the inverse of \(\mathrm{cov}(\mathbf{\Delta}\mathbf{s}-\mathbf{\Delta}\mathbf{\hat{s}})\) and then comparing it to \(\mathbf{Q}_{\mathbf{s}}^{-1}\) results in \(\mathrm{cov}(\mathbf{\Delta}\mathbf{s}-\mathbf{\Delta}\mathbf{\hat{s}})^{-1}- \mathbf{Q}_{\mathbf{s}}^{-1}=\mathbf{G}_{0}^{\mathrm{T}}(\mathbf{G}_{\mathrm{ c}}\mathbf{Q}_{\mathrm{c}}\mathbf{G}_{\mathrm{c}}^{\mathrm{T}}+ \mathbf{Q}_{\mathrm{rc}}  
\[\mathbf{D}_{1,r,m}=-2\left[ \begin{array}{cccc}(\mathbf{u}^{\mathrm{o}}-\hat{ \mathbf{s}}_{r,1}-R_{r,1}^{\mathrm{o}}\rho_{\mathrm{t},m,r,1})^{\mathrm{T}}&0_{3 \times 1}^{\mathrm{T}}&\cdots&0_{3\times 1}^{\mathrm{T}}\\ 0_{3\times 1}^{\mathrm{T}}&(\mathbf{u}^{\mathrm{o}}-\hat{\mathbf{s}}_{r,2}-R_{ r,2}^{\mathrm{o}}\rho_{\mathrm{t},m,r,2})^{\mathrm{T}}&\cdots&0_{3\times 1}^{\mathrm{T}}\\ \vdots&\vdots&\ddots&\vdots\\ 0_{3\times 1}^{\mathrm{T}}&0_{3\times 1}^{\mathrm{T}}&\cdots&(\mathbf{u}^{ \mathrm{o}}-\hat{\mathbf{s}}_{r,N}-R_{r,N}^{\mathrm{o}}\rho_{\mathrm{t},m,r,N })^{\mathrm{T}}\end{array} \right] \tag{66}\]  
From the set of linear equations in (56), the WLS estimate of \(\theta^{\mathrm{o}}\), denoted by \(\theta\), which minimizes \(\mathbf{\Delta h}_{1}^{\mathrm{T}}\mathbf{W}_{1}\mathbf{\Delta h}_{1}\) can be produced as  
\[\theta=\left(\mathbf{G}_{1}^{\mathrm{T}}\mathbf{W}_{1}\mathbf{G}_{1}\right)^{ -1}\mathbf{G}_{1}^{\mathrm{T}}\mathbf{W}_{1}\mathbf{h}_{1} \tag{67}\]  
where \(\mathbf{W}_{1}\) represents the weighting matrix and it can be computed by  
\[ \begin{split}\mathbf{W}_{1}&=\left[\mathrm{E}(\Delta \mathbf{h}_{1}\mathbf{\Delta h}_{1}^{\mathrm{T}})\right]^{-1}\\ &=\left[\mathbf{B}_{1}\mathbf{Q}_{\alpha}\mathbf{B}_{1}^{\mathrm{T }}+\mathbf{D}_{1}\mathrm{cov}(\mathbf{\Delta s}-\mathbf{\Delta\hat{s}}) \mathbf{D}_{1}^{\mathrm{T}}\right]^{-1}\end{split}  \tag{68}\]  
However, to compute \(\mathbf{W}_{1}\), the unknown target position has to be acquired in advance.