## 4 Proposed Localization Method
### Performance Analysis  
As mentioned above, the CRLB traces out a lower bound for minimum possible variance that an unbiased estimator can achieve. Next, we will analyze the efficiency of the proposed solution by comparing its covariance matrix with the benchmark, i.e., CRLB. For derivation simplicity, we would compare their inverse, rather than directly compare the two separately. The CRLB has been presentedin (33). By invoking the matrix inversion lemma [32] to (33) and using the definitions of \(\mathbf{X}\) and \(\mathbf{Y}\), we have after mathematical simplifications,  
\[\mathbf{CRLB}_{\mathrm{c}}(\mathbf{u}^{\mathrm{o}})^{-1}=\left(\frac{\partial \mathbf{r}^{\mathrm{o}}}{\partial\mathbf{u}^{\mathrm{o}}}\right)^{\mathrm{T}} \mathbf{Q}_{\mathrm{r}}^{-1}\!\!\left(\frac{\partial\mathbf{r}^{\mathrm{o}}}{ \partial\mathbf{u}^{\mathrm{o}}}\right)-\left(\frac{\partial\mathbf{r}^{ \mathrm{o}}}{\partial\mathbf{u}^{\mathrm{o}}}\right)^{\mathrm{T}}\!\!\mathbf{Q}_ {\mathrm{r}}^{-1}\!\!\left(\frac{\partial\mathbf{r}^{\mathrm{o}}}{\partial \mathbf{s}^{\mathrm{o}}}\right)^{\mathrm{T}}\!\!\mathbf{Q}_{\mathrm{r}}^{-1} \!\!\left(\frac{\partial\mathbf{r}^{\mathrm{o}}}{\partial\mathbf{u}^{\mathrm{o }}}\right) \tag{83}\]  
where the expression of \(\widetilde{\mathbf{Z}}\) has been given in (32). On the other hand, using (82), (81), (69), (68) and (52) successively, we can reformulate the inverse of \(\mathrm{cov}(\mathbf{u})\) as  
\[\mathrm{cov}(\mathbf{u})^{-1}=\mathbf{G}_{3}^{\mathrm{T}}\mathbf{Q}_{\mathrm{ r}}^{-1}\mathbf{G}_{3}-\mathbf{G}_{3}^{\mathrm{T}}\mathbf{Q}_{\mathrm{r}}^{-1} \mathbf{G}_{4}\widetilde{\mathbf{Z}}^{-1}\mathbf{G}_{4}^{\mathrm{T}}\mathbf{Q }_{\mathrm{r}}^{-1}\mathbf{G}_{3} \tag{84}\]  
where \(\mathbf{G}_{3}=\mathbf{B}_{1}^{-1}\mathbf{G}_{1}\mathbf{B}_{2}^{-1}\mathbf{G} _{2}\), \(\mathbf{G}_{4}=\mathbf{B}_{1}^{-1}\mathbf{D}_{1}\), and \(\widetilde{\mathbf{Z}}=\mathbf{Q}_{\mathbf{s}}^{-1}+\mathbf{G}_{4}^{\mathrm{T} }\mathbf{Q}_{\mathrm{r}}^{-1}\mathbf{G}_{4}+\mathbf{G}_{0}^{\mathrm{T}}( \mathbf{G}_{\mathrm{c}}\mathbf{Q}_{\mathrm{c}}\mathbf{G}_{\mathrm{c}}^{\mathrm{ T}}+\mathbf{Q}_{\mathrm{rc}})^{-1}\mathbf{G}_{0}\). Comparing (83) with (84), we observe that \(\mathbf{CRLB}_{\mathrm{c}}(\mathbf{u}^{\mathrm{o}})^{-1}\) and \(\mathrm{cov}(\mathbf{u})^{-1}\) are identical in structure. Next, we proceed to prove their equivalency under the following conditions:  
(C1) \(\|\Delta\mathbf{s}_{n,m}\|\ll\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{n,m} \|\)\(\|\Delta\mathbf{s}_{n,m}\|\ll\|\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}-\mathbf{s }_{n,m}^{\mathrm{o}}\|\)\(\|\Delta\mathbf{s}_{n,m}\|\ll\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s }_{n,m}^{\mathrm{o}}\|\), \(\|\Delta\mathbf{s}_{n,m}\|\ll\|\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}-\mathbf{ s}_{n,m}^{\mathrm{o}}\|\), and \(\|\Delta\mathbf{c}_{k}\|\ll\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{n,m}^{ \mathrm{o}}\|\), \(\|\Delta\mathbf{c}_{k}\|\ll\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{n,m}^{ \mathrm{o}}\|\), for \(k=1,2,\ldots,K\), \(m=1,2,\ldots,M\) and \(n=1,2,\ldots,N\);  
(C2) \(\|\Delta r_{m,m}\|\ll\|\mathbf{u}^{\mathrm{o}}-\mathbf{s}_{n,m}^{\mathrm{o}}\|\), \(\|\Delta r_{m,m}\|\ll\|\mathbf{u}^{\mathrm{o}}-\mathbf{s}_{n,m}^{\mathrm{o}}\|\), \(\|\Delta r_{m,m}\|\ll\|\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}-\mathbf{s}_{n,m }^{\mathrm{o}}\|\), and \(\|\Delta\mathbf{s}_{n,m}-\mathbf{s}_{n,m}^{\mathrm{o}}\|\ll\|\mathbf{u}^{ \mathrm{o}}-\mathbf{s}_{n,m}^{\mathrm{o}}\|\) for \(m=1,2,\ldots,M\) and \(n=1,2,\ldots,N\);  
The condition C1 implies the transmitter/receiver position error and the calibration target position error are negligibly small compared with the range between the calibration target and the transmitter/receiver. The condition C2 implies the BR measurement noise and the error in the refined transmitter/receiver position are negligibly small compared to the range between the calibration target and the transmitter/receiver.