## 3 Evaluation of the CRLB with Calibration Targets  
The CRLB does not address the specific estimators employed, but simply reflects minimum possible variance that an unbiased estimator can achieve with existing observations. In this section, in order to justify the necessity of refining the inaccurate transmitter and receiver positions using calibration targets, we shall first set up the CRLB for the target localization problem described above. Besides the BR measurement noise, the position errors of transmitter, receivers and calibration targets are also included. From the localization scenario presented in Section 2, the deterministic but unknown parameters for the CRLB evaluation, collected into a \(3(M+N+K+1)\)-by-\(1\) vector \(\boldsymbol{\varphi}=\left[\left(\mathbf{u}^{\text{o}}\right)^{\text{T}}, \left(\mathbf{s}^{\text{o}}\right)^{\text{T}},\left(\mathbf{c}^{\text{o}} \right)^{\text{T}}\right]^{\text{T}}\), include the target position vector \(\mathbf{u}^{\text{o}}\), the transmitter and receiver position vector \(\mathbf{s}^{\text{o}}\), and the calibration target position vector \(\mathbf{c}^{\text{o}}\); the observations, collected into a \((MN+KMN+3M+3N+3K)\)-by-\(1\) vector \(\mathbf{z}=\left[\mathbf{r}^{\text{T}},\mathbf{r}_{\text{c}}^{\text{T}}, \mathbf{s}^{\text{T}},\mathbf{c}^{\text{T}}\right]^{\text{T}}\), include the BR measurement vector \(\mathbf{r}\) from the unknown target, the BR measurement vector \(\mathbf{r}_{\text{c}}\) from the calibration targets, the inaccurate measured transmitter and receiver position vector \(\mathbf{s}\), and the nominal calibration target position vector \(\mathbf{c}\), which are Gaussian distributed and independent with one another. Based on this, the joint probability density function (pdf) of the observations parameterized by the unknown parameter vector is readily shown to be  
\[ \begin{array}{c}p(\mathbf{z}\big{|}\boldsymbol{\varphi})=p(\mathbf{r}\big{|} \mathbf{u}^{\text{o}},\mathbf{s}^{\text{o}})\cdot p(\mathbf{r}_{\text{c}} \big{|}\mathbf{s}^{\text{o}},\mathbf{c}^{\text{o}})\cdot p(\mathbf{s}\big{|} \mathbf{s}^{\text{o}})\cdot p(\mathbf{c}\big{|}\mathbf{c}^{\text{o}})\\ \hskip 14.226378 pt=\mathbf{\kappa}\cdot\exp\!\left[-\frac{1}{2}(\mathbf{r}- \mathbf{r}^{\text{o}})^{\text{T}}\mathbf{Q}_{\mathbf{r}}^{-1}(\mathbf{r}- \mathbf{r}^{\text{o}})-\frac{1}{2}(\mathbf{r}_{\text{c}}-\mathbf{r}_{\text{c} }^{\text{o}})^{\text{T}}\mathbf{Q}_{\mathbf{r}}^{-1}(\mathbf{r}_{\text{c}}- \mathbf{r}_{\text{c}}^{\text{o}})\right.\\ \hskip 14.226378 pt\left.-\frac{1}{2}(\mathbf{s}-\mathbf{s}^{\text{o}})^{\text{T }}\mathbf{Q}_{\mathbf{s}}^{-1}(\mathbf{s}-\mathbf{s}^{\text{o}})-\frac{1}{2}( \mathbf{c}-\mathbf{c}^{\text{o}})^{\text{T}}\mathbf{Q}_{\mathbf{c}}^{-1}( \mathbf{c}-\mathbf{c}^{\text{o}})\right]\end{array}  \tag{15}\]  
where \(\mathbf{\kappa}\) is a constant with respect to the unknown parameters. By taking the logarithm of (15), partial derivatives with respect to the unknown parameters twice, and then expectation, the Fisher information matrix (FIM) can be calculated as  
\[ \begin{array}{c}FIM(\boldsymbol{\varphi})=\mathbb{E}\!\left[\frac{\partial \ln p(\mathbf{z}\big{|}\boldsymbol{\varphi})}{\partial\boldsymbol{\varphi}} \left(\frac{\partial\ln p(\mathbf{z}\big{|}\boldsymbol{\varphi})}{\partial \boldsymbol{\varphi}}\right)\right]\\ =\left[\begin{array}{ccc}\mathbf{X}&\mathbf{Y}&\mathbf{O}_{3\times 3}\\ \mathbf{Y}^{\text{T}}&\mathbf{Z}&\mathbf{R}^{\text{T}}\\ \mathbf{O}_{3\times 3}&\mathbf{R}&\mathbf{P}\end{array}\right]\end{array}