## 3 Evaluation of the CRLB with Calibration Targets
In form, the right side of (35) is just the performance enhancement because of the use of calibration targets. It is positive semi-definite (PSD) since it has a symmetric structure and \(\mathbf{Y}^{\mathrm{T}}\) is not full column rank. Even if the nominal positions of calibration targets and the corresponding BR measurements are very noisy, (35) can still remain PSD. In theory, only in the edge case when \((\mathbf{Q}_{\mathrm{rc}}+(\partial\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}/ \partial\mathbf{c}^{\mathrm{o}})\mathbf{Q}_{\mathrm{c}}^{-1}(\partial\mathbf{r }_{\mathrm{c}}^{\mathrm{o}}/\partial\mathbf{c}^{\mathrm{o}})^{\mathrm{T}})^{-1}\) tends to zero and then \(\mathbf{L}_{\mathrm{rc}}\rightarrow\mathbf{O}\) and \(\mathbf{Y}\rightarrow\mathbf{O}\), the performance enhancement in (34) would tend to zero. However, this edge case hardly exists in reality. Thus, mathematically, we can arrive at  
\[\mathbf{CRLB}_{\mathrm{s}}(\mathbf{u}^{\mathrm{o}})\geq\mathbf{CRLB}_{\mathrm{ c}}(\mathbf{u}^{\mathrm{o}}) \tag{39}\]  
The matrix inequality \(\mathbf{A}\geq\mathbf{B}\) means that \(\mathbf{A}-\mathbf{B}\) is PSD. It can be further deduced from (39) that \(\mathrm{tr}(\mathbf{CRLB}_{\mathrm{s}}(\mathbf{u}^{\mathrm{o}}))\geq\mathrm{ tr}(\mathbf{CRLB}_{\mathrm{c}}(\mathbf{u}^{\mathrm{o}}))\). The trace of \(\mathbf{CRLB}_{\mathrm{c}}(\mathbf{u}^{\mathrm{o}})\) and \(\mathbf{CRLB}_{\mathrm{s}}(\mathbf{u}^{\mathrm{o}})\) respectively represents minimum possible variance of target position estimation with and without using calibration targets. Therefore, we can conclude that using calibration targets brings potential enhancement to the target localization accuracy, at least at the CRLB level. _Example 1_. : To substantiate the evaluation on the CRLB presented above, a numerical example using a typical multi-static passive radar localization scenario was conducted, as presented in Figure 2. There are \(M=3\) transmitters, \(N=4\) receivers and \(K=3\) calibration targets in the scenario, and their true positions are listed in Table 1. The noise covariance matrix of the BR measurements from the unknown target are given by \(\mathbf{Q}_{\mathrm{r}}=\sigma_{\mathrm{r}}^{2}\mathbf{V}_{\mathrm{r}}\), where \(\sigma_{\mathrm{r}}\) reflects BR measurement noise level and \(\mathbf{V}_{\mathrm{r}}\) is set to 1 in the diagonal elements and 0.5 elsewhere. The covariance matrix of the transmitter/receiver position error is given as \(\mathbf{Q}_{\mathrm{s}}=\sigma_{\mathrm{s}}^{2}\mathbf{V}_{\mathrm{s}}\) where \(\sigma_{s}\) reflects the transmitter/receiver position error level and \(\mathbf{V}_{\mathrm{s}}=\mathrm{diag}(5\mathbf{I}_{3M\times 3M},\mathbf{I}_{3N\times 3N})\). The covariance matrix of calibration target position error is \(\mathbf{Q}_{\mathrm{c}}=\sigma_{\mathrm{c}}^{2}\mathbf{V}_{\mathrm{c}}\) where \(\sigma_{\mathrm{c}}\) reflects the calibration target position error level and \(\mathbf{V}_{\mathrm{c}}=\mathbf{I}_{3K\times 3K}\), and the covariance matrix of the corresponding calibration BR measurement noise is \(\mathbf{Q}_{\mathrm{rc}}=\sigma_{\mathrm{rc}}^{2}\mathbf{V}_{\mathrm{rc}}\) where \(\sigma_{\mathrm{rc}}=\sigma_{\mathrm{r}}\) reflects calibration BR measurement noise level and \(\mathbf{V}_{\mathrm{rc}}\) is set to 1 in the diagonal elements and 0.5 elsewhere. The target of interest is located at position \(\mathbf{u}^{\mathrm{o}}=[50000,15000,5000]^{\mathrm{T}}\mathbf{m}\). The effect of the calibration targets on the target localization accuracy, in the sense of CRLB, is presented in Figure 3.  
Figure 3a compares the CRLB curves with and without using calibration targets when the BR measurement noise level \(\sigma_{\rm r}\) is varied from \(10^{-2}\) m to \(10^{3}\) m while the transmitter/receiver position error level and calibration target position error level are fixed at \(\sigma_{\rm s}=20\) m and \(\sigma_{\rm c}=10\) m respectively. It can be observed from Figure 3a that the CRLB with calibration targets is generally below the one without, this coincides with the analytical conclusion given in (39).