## 3 Evaluation of the CRLB with Calibration Targets
Invoking the partitioned matrix inversion formula as well as the matrix inversion lemma [32] twice on (16), leads to the CRLB of \(\mathbf{u}^{\mathbf{o}}\) as  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}^{\text{o}}}{\partial\mathbf{u}^{ \text{o}}}\right)^{\text{T}}\!\mathbf{Q}_{\mathbf{r}}^{-1}\!\left(\frac{ \partial\mathbf{r}^{\text{o}}}{\partial\mathbf{s}^{\text{o}}}\right) \tag{18}\]3  
For comparison purposes, the CRLB of \(\mathbf{u}^{\mathbf{o}}\) with transmitter/receiver position error but without calibration derived in [22], denoted by \(\mathbf{CRLB}_{\mathbf{s}}(\mathbf{u}^{\mathbf{o}})\), is also given below  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}^{\text{o}}}{\partial\mathbf{u}^{ \text{o}}}\right)^{\text{T}}\!\mathbf{Q}_{\mathbf{r}}^{-1}\!\left(\frac{ \partial\mathbf{r}^{\text{o}}}{\partial\mathbf{s}^{\text{o}}}\right) \tag{18}\]4  
where \(\widetilde{\mathbf{Z}}=\mathbf{Q}_{\mathbf{s}}^{-1}+(\partial\mathbf{r}^{ \mathbf{o}}/\partial\mathbf{s}^{\mathbf{o}})^{\mathrm{T}}\mathbf{Q}_{\mathbf{r} }^{-1}(\partial\mathbf{r}^{\mathbf{o}}/\partial\mathbf{s}^{\mathbf{o}})\). For the sake of comparison, we proceed to construct an equivalent form of \(\mathbf{CRLB}_{\mathbf{c}}(\mathbf{u}^{\mathbf{o}})\) by denoting \(\widetilde{\mathbf{Z}}\) as \(\widetilde{Through the comparison of (31) and (33), it is readily to observe that the two CRLBs are identical in structure, except that \(\widetilde{\mathbf{Z}}\) is substituted by \(\widetilde{\mathbf{Z}}\). More specifically, the use of calibration targets introduces an additional component into the bracketed matrix expression to be inverted as  
\[ \begin{array}{l}\widetilde{\mathbf{Z}}=\widetilde{\mathbf{Z}}-\widetilde{ \mathbf{Z}}\\ =(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/\partial\mathbf{s}^{\mathrm{o}} )^{\mathrm{T}}\mathbf{Q}_{\mathrm{rc}}^{-1}(\partial\mathbf{r}_{\mathrm{c}}^{ \mathrm{o}}/\partial\mathbf{s}^{\mathrm{o}})-\mathbf{R}^{\mathrm{T}}\mathbf{P} ^{-1}\mathbf{R}\\ =(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/\partial\mathbf{s}^{\mathrm{o}} )^{\mathrm{T}}\big{(}\mathbf{Q}_{\mathrm{rc}}+(\partial\mathbf{r}_{\mathrm{c}} ^{\mathrm{o}}/\partial\mathbf{c}^{\mathrm{o}})\mathbf{Q}_{\mathrm{c}}( \partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/\partial\mathbf{c}^{\mathrm{o}} )^{\mathrm{T}}\big{)}^{-1}(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/ \partial\mathbf{s}^{\mathrm{o}})\end{array}  \tag{34}\]  
Using (34), we can rewrite \((\widetilde{\mathbf{Z}}-\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1}\mathbf{Y})^{-1}\) in (33) as \(((\widetilde{\mathbf{Z}}-\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1}\mathbf{Y})+ \widetilde{\mathbf{Z}})^{-1}\). Invoking the matrix inversion lemma [32] to the term \(((\widetilde{\mathbf{Z}}-\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1}\mathbf{Y})+ \widetilde{\mathbf{Z}})^{-1}\) in (33), we obtain after some algebraic manipulations,  
\[\mathbf{CRLB}_{\mathrm{c}}(\mathbf{u}^{\mathrm{o}})-\mathbf{CRLB}_{\mathrm{s}} (\mathbf{u}^{\mathrm{o}})=\mathbf{X}^{-1}\mathbf{Y}\mathbf{Y}^{\mathrm{T}} \mathbf{X}^{-1} \tag{35}\]  
where  
\[\mathbf{\Gamma}=\mathbf{H}^{-1}\mathbf{Y}(\mathbf{I}+\mathbf{Y}^{\mathrm{T}} \mathbf{H}^{-1}\mathbf{Y})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{H}^{-1} \tag{36}\]  
\[\mathbf{H}=(\widetilde{\mathbf{Z}}-\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1} \mathbf{Y}) \tag{37}\]  
\[\mathbf{Y}=\left(\frac{\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}}{\partial \mathbf{s}^{\mathrm{o}}}\right)^{\mathrm{T}}\mathbf{L}_{\mathrm{rc}} \tag{38}\]  
and \(\mathbf{L}_{\mathrm{rc}}\) is the Cholesky decomposition of \((\mathbf{Q}_{\mathrm{rc}}+(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/ \partial\mathbf{c}^{\mathrm{o}})\mathbf{Q}_{\mathrm{c}}(\partial\mathbf{r}_{ \mathrm{c}}^{\mathrm{o}}/\partial\mathbf{c}^{\mathrm{o}})^{\mathrm{T}})^{-1}\), i.e., \(\mathbf{L}_{\mathrm{rc}}\mathbf{L}_{\mathrm{rc}}^{\mathrm{T}}=(\mathbf{Q}_{ \mathrm{rc}}+(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/\partial\mathbf{c}^ {\mathrm{o}})\mathbf{Q}_{\mathrm{c}}(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{ o}}/\partial\mathbf{c}^{\mathrm{o}})^{\mathrm{T}})^{-1}\).