## 4 Proposed Localization Method
### Algorithm Development
#### 4.1.1 Calibration Stage  
To make use of the BR measurements from the calibration targets, the calibration stage begins by reorganizing (13) as  
\[r_{\rm c,}k,m,n}-\|\mathbf{c}_{k}^{\rm o}-\mathbf{s}_{t,m}^{\rm o}\|-\| \mathbf{c}_{k}^{\rm o}-\mathbf{s}_{r,n}^{\rm o}\|+\|\mathbf{s}_{t,m}^{\rm o} -\mathbf{s}_{r,n}^{\rm o}\|=\mathbf{\Delta}\mathbf{r}_{\rm c,}k,m,n \tag{40}\]  
Since only the erroneous versions of \(\mathbf{c}_{k}^{\rm o},\mathbf{s}_{t,m}^{\rm o}\) and \(\mathbf{s}_{r,n}^{\rm o}\) are available, we put \(\mathbf{c}_{k}^{\rm o}=\mathbf{c}_{k}-\mathbf{\Delta}\mathbf{c}_{k},\mathbf{s}_{t,m}^{\rm o}=\mathbf{s}_{t,m}-\mathbf{\Delta}\mathbf{s}_{t,m}\) and \(\mathbf{s}_{r,n}^{\rm o}=\mathbf{s}_{r,n}-\mathbf{\Delta}\mathbf{s}_{r,n}\) into (40), and then expand it around erroneous values \(\mathbf{c}_{k}\), \(\mathbf{s}_{t,m}\) and \(\mathbf{s}_{r,n}\) to the linear error terms as  
\[r_{\rm c,}k,m,n}-\|\mathbf{c}_{k}-\mathbf{s}_{t,m}\|-\|\mathbf{c}_{k}- \mathbf{s}_{r,n}\|+\|\mathbf{s}_{t,m}-\mathbf{s}_{r,m}\|-(\rho_{\rm c,}^{\rm T }\mathbf{,}k,m}+\rho_{\rm t,m,r,n}^{\rm T})\mathbf{\Delta}\mathbf{s}_{t,m}-(\rho _{\rm c,}^{\rm T}\mathbf{,}k,r,n}-\rho_{\rm t,m,r,n}^{\rm T})\mathbf{\Delta} \mathbf{s}_{t,n} \tag{41}\] \[=-(\rho_{\rm c,}k,m}+\rho_{\rm c,}k,r,n}^{\rm T}\mathbf{\Delta} \mathbf{c}_{k}+\mathbf{\Delta}r_{\rm c,}k,m,n\]  
where  
\[\rho_{\rm c,}k,m}=\frac{\mathbf{c}_{k}-\mathbf{s}_{t,m}}{\|\mathbf{c}_{k}- \mathbf{s}_{t,m}\|} \tag{42}\]  
\[\rho_{\rm c,}k,r,n}=\frac{\mathbf{c}_{k}-\mathbf{s}_{r,n}}{\|\mathbf{c}_{k}- \mathbf{s}_{r,n}\|} \tag{43}\]  
\[\rho_{\rm t,m,r,n}=\frac{\mathbf{s}_{t,m}-\mathbf{s}_{r,n}}{\|\mathbf{s}_{t,m}-\mathbf{s}_{r,n}\|} \tag{44}\]  
Stacking (41) for all the \(k\), \(m\) and \(n\), we can formulate them in matrix form as  
\[\mathbf{h}_{0}-\mathbf{G}_{0}\mathbf{\Delta}\mathbf{s}=\mathbf{\Delta}\mathbf{h}_{0} \tag{45}\]  
The elements of \(\mathbf{h}_{0}\), \(\mathbf{G}_{0}\) and \(\mathbf{\Delta}\mathbf{h}_{0}\) are given by  
\[\mathbf{h}_{0}(i_{k,m,n},1)=r_{\rm c,}k,m,n}-\|\mathbf{c}_{k}-\mathbf{s}_{t, m}\|-\|\mathbf{c}_{k}-\mathbf{s}_{r,n}\|+\|\mathbf{s}_{t,m}-\mathbf{s}_{r,n}\| \tag{46}\]\[\mathbf{G}_{0}=\left[ \begin{array}{cc}\mathbf{G}_{0,t}&\mathbf{G}_{0,t}\\ \end{array} \right]\mathbf{G}_{0,t}(i_{k,m,n},3m-2:3m)=\rho_{c,k,t,m}^{\mathrm{T} }+\rho_{t,m,r,u}^{\mathrm{T}},\mathbf{G}_{0,r}(i_{k,m,n},3n-2:3n)=\rho_{c,k,t,m}^ {\mathrm{T}}-\rho_{t,m,r,n}^{\mathrm{T}} \tag{47}\]  
\[\mathbf{G}_{\mathrm{c}}(i_{k,m,n},3k-2:3k)=-(\rho_{c,k,t,m}+\rho_{c,k,r,n})^{ \mathrm{T}} \tag{48}\]  
\[\mathbf{\Delta}\mathbf{h}_{0}(i_{k,m,n},1)=-(\rho_{c,k,t,m}+\rho_{c,k,r,n})^{ \mathrm{T}}\mathbf{\Delta}\mathbf{c}_{k}+\Delta r_{c,k,m,n} \tag{49}\]  
for \(i_{k,m,n}=(k-1)MN+(m-1)N+n\), \(k=0,1,\ldots,K-1\), \(m=0,1,\ldots,M-1\), \(n=0,1,\ldots,N-1\), and zeros elsewhere. Furthermore, the error vector \(\mathbf{\Delta}\mathbf{h}_{0}\) can be recast using a compact representation as follows  
\[\mathbf{\Delta}\mathbf{h}_{0}=\mathbf{G}_{\mathrm{c}}\mathbf{\Delta} \mathbf{c}+\mathbf{\Delta}\mathbf{r}_{\mathrm{c}} \tag{50}\]  
from which we have the mean \(\mathrm{E}(\mathbf{\Delta}\mathbf{h}_{0})=0_{KMN\times 1}\) and the covariance \(\mathrm{cov}(\mathbf{\Delta}\mathbf{h}_{0})=\mathbf{G}_{\mathrm{c}}\mathbf{Q }_{\mathrm{c}}\mathbf{G}_{\mathrm{c}}^{\mathrm{T}}+\mathbf{Q}_{\mathrm{rc}}\). In (45), \(\mathbf{\Delta}\mathbf{s}\) represents the difference between the true and the nominal transmitter/receiver positions. In order to refine the transmitter and receiver positions, \(\mathbf{\Delta}\mathbf{s}\) shall be estimated as accurately as possible. Recall that \(\mathbf{\Delta}\mathbf{s}\) is a Gaussian distributed random vector with mean \(\mathrm{E}(\mathbf{\Delta}\mathbf{s})=0_{3(M+N)\times 1}\) and covariance matrix \(\mathrm{cov}(\mathbf{\Delta}\mathbf{s})=\mathbf{Q}_{\mathbf{s}}\), and it is independent of the error vector \(\mathbf{\Delta}\mathbf{h}_{0}\).