**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 ϕ = [(**u**o) T,(**s**o) T,(**c**o) T] T , include the target position vector **u**o, the transmitter and receiver position vector **s**o, and the calibration target position vector **c**o; the observations, collected into a (*MN* + *KMN* + 3*M* + 3*N* + 3*K*)-by-1 vector **z** = [**r**T,**r**<sup>T</sup> <sup>c</sup> , **s**T, **c**T] T , include the BR measurement vector **r** from the unknown target, the BR measurement vector **r**c from the calibration targets, the inaccurate measured transmitter and receiver position vector **s**, and the nominal calibration target position vector **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

$$\begin{split} p(\mathbf{z}|\boldsymbol{\uprho}) &= p(\mathbf{r}|\mathbf{u}^{\alpha}, \mathbf{s}^{\alpha}) \cdot p(\mathbf{r}\_{\mathbf{c}}|\mathbf{s}^{\alpha}, \mathbf{c}^{\alpha}) \cdot p(\mathbf{s}|\mathbf{s}^{\alpha}) \cdot p(\mathbf{c}|\mathbf{c}^{\alpha}) \\ &= \boldsymbol{\upkappa} \cdot \exp\Big[ -\frac{1}{2} (\mathbf{r} - \mathbf{r}^{\alpha})^{\mathsf{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} (\mathbf{r} - \mathbf{r}^{\alpha}) - \frac{1}{2} (\mathbf{r}\_{\mathbf{c}} - \mathbf{r}\_{\mathbf{c}}^{\alpha})^{\mathsf{T}} \mathbf{Q}\_{\mathbf{r}\mathbf{c}}^{-1} (\mathbf{r}\_{\mathbf{c}} - \mathbf{r}\_{\mathbf{c}}^{\alpha}) \\ & \quad - \frac{1}{2} (\mathbf{s} - \mathbf{s}^{\alpha})^{\mathsf{T}} \mathbf{Q}\_{\mathbf{s}}^{-1} (\mathbf{s} - \mathbf{s}^{\alpha}) - \frac{1}{2} (\mathbf{c} - \mathbf{c}^{\alpha})^{\mathsf{T}} \mathbf{Q}\_{\mathbf{c}}^{-1} (\mathbf{c} - \mathbf{c}^{\alpha}) \Big] \end{split} \tag{15}$$

where κ 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

$$\begin{aligned} FM(\boldsymbol{\upmu}) &= \operatorname{E} \begin{bmatrix} \frac{\partial \ln p(\mathbf{z}|\boldsymbol{\upmu})}{\partial \boldsymbol{\upmu}} \left( \frac{\partial \ln p(\mathbf{z}|\boldsymbol{\upmu})}{\partial \boldsymbol{\upmu}} \right) \end{bmatrix} \\ &= \begin{bmatrix} \mathbf{X} & \mathbf{Y} & \mathbf{O}\_{3 \times 3} \\ \mathbf{Y}^{\mathrm{T}} & \mathbf{Z} & \mathbf{R}^{\mathrm{T}} \\ \mathbf{O}\_{3 \times 3} & \mathbf{R} & \mathbf{P} \end{bmatrix} \end{aligned} \tag{16}$$

where the blocks **X**, **Y**, **Z**, **R** and **P** are respectively given by

$$\mathbf{X} = \left(\frac{\partial \mathbf{r}^o}{\partial \mathbf{u}^o}\right)^T \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^o}{\partial \mathbf{u}^o}\right) \tag{17}$$

$$\mathbf{Y} = \left(\frac{\partial \mathbf{r}^{\diamond}}{\partial \mathbf{u}^{\diamond}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{\diamond}}{\partial \mathbf{s}^{\diamond}}\right) \tag{18}$$

$$\mathbf{Z} = \mathbf{Q}\_{\mathbf{s}}^{-1} + \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right) + \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{s}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}\mathbf{c}}^{-1} \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{s}^{o}}\right) \tag{19}$$

$$\mathbf{R} = \left(\frac{\partial \mathbf{r}\_{\mathbf{c}}^{\alpha}}{\partial \mathbf{c}^{\alpha}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathrm{nc}}^{-1} \left(\frac{\partial \mathbf{r}\_{\mathbf{c}}^{\alpha}}{\partial \mathbf{s}^{\alpha}}\right) \tag{20}$$

*Sensors* **2019**, *19*, 3365

$$\mathbf{P} = \mathbf{Q}\_{\mathbf{c}}^{-1} + \left(\frac{\partial \mathbf{r}\_{\mathbf{c}}^{o}}{\partial \mathbf{c}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}\mathbf{c}}^{-1} \left(\frac{\partial \mathbf{r}\_{\mathbf{c}}^{o}}{\partial \mathbf{c}^{o}}\right) \tag{21}$$

Denote *im*,*<sup>n</sup>* = (*m* − 1)*N* + *n*, and *ik*,*m*,*<sup>n</sup>* = (*k* − 1)*MN* + (*m* − 1)*N* + *n*. From the formulations of (7) and (13), the elements of the partial derivatives ∂**r**o/∂**u**o, ∂**r**o/∂**s**o, ∂**r**<sup>o</sup> c/∂**c**<sup>o</sup> and ∂**r**<sup>o</sup> c/∂**s**<sup>o</sup> in (17)–(21), can be determined as

$$\frac{\partial \mathbf{r}^{\rm o}}{\partial \mathbf{u}^{\rm o}}(i\_{\mathcal{W},\rm{\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm}}(i\_{\mathcal{W},\rm{\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\}}{R\_{\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm}}+\frac{\left(\mathbf{u}^{\rmo}-\mathbf{s}\_{\rm\rm t,\rm\rm m}^{\rm\rm\rm\rm}\right)^{\rm\rm T}}{R\_{\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm\rm}}}\tag{22}$$

$$\frac{\partial \mathbf{\hat{r}}^{o}}{\partial \mathbf{\hat{s}}^{o}} = \begin{bmatrix} \frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}\_{t}^{o}} & \frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}\_{t}^{o}} \end{bmatrix} \tag{23}$$

$$\frac{\partial \mathbf{r}^{\rm{o}}}{\partial \mathbf{s}\_{\rm{t}}^{\rm{o}}} (i\_{m,n} \, 3m - 2: 3m) = \frac{\left(\mathbf{s}\_{\rm{t}, \rm{n}}^{\rm{o}} - \mathbf{u}^{\rm{o}}\right)^{\rm{T}}}{R\_{\rm{t}, \rm{n}}^{\rm{o}}} - \frac{\left(\mathbf{s}\_{\rm{t}, \rm{n}}^{\rm{o}} - \mathbf{s}\_{\rm{t}, \rm{n}}^{\rm{o}}\right)^{\rm{T}}}{R\_{\rm{t}, m, r, n}^{\rm{o}}} \tag{24}$$

$$\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}\_{\mathrm{r}}^{o}} (i\_{m,\mathrm{u}}, 3\mathrm{u} - 2:3\mathrm{u}) = \frac{\left(\mathbf{s}\_{\mathrm{r},\mathrm{u}}^{o} - \mathbf{u}^{o}\right)^{\mathrm{T}}}{R\_{\mathrm{r},\mathrm{u}}^{o}} - \frac{\left(\mathbf{s}\_{\mathrm{r},\mathrm{u}}^{o} - \mathbf{s}\_{\mathrm{t},\mathrm{u}}^{o}\right)^{\mathrm{T}}}{R\_{\mathrm{t},\mathrm{m},\mathrm{r},\mathrm{u}}^{o}} \tag{25}$$

$$\frac{\partial \mathbf{r}\_{\text{c}}^{\text{o}}}{\partial \mathbf{c}^{\text{o}}} (i\_{k, \text{m}, \text{n}}, 3k - 2 : 3k) = \frac{\left(\mathbf{c}\_{k}^{\text{o}} - \mathbf{s}\_{\text{t}, \text{m}}^{\text{o}}\right)^{\text{T}}}{R\_{\text{c}, k, \text{t}, \text{m}}^{\text{o}}} + \frac{\left(\mathbf{c}\_{k}^{\text{o}} - \mathbf{s}\_{\text{r}, \text{n}}^{\text{o}}\right)^{\text{T}}}{R\_{\text{c}, k, \text{r}, \text{n}}^{\text{o}}} \tag{26}$$

$$\frac{\partial \mathbf{r}\_{\mathbf{c}}^{o}}{\partial \mathbf{s}^{o}} = \begin{bmatrix} \frac{\partial \mathbf{r}\_{\mathbf{c}}^{o}}{\partial \mathbf{s}\_{\mathbf{t}}^{o}} & \frac{\partial \mathbf{r}\_{\mathbf{c}}^{o}}{\partial \mathbf{s}\_{\mathbf{r}}^{o}} \end{bmatrix} \tag{27}$$

$$\frac{\partial \mathbf{r}\_{\text{c}}^{\text{o}}}{\partial \mathbf{s}\_{\text{t}}^{\text{o}}} (i\_{k,m,n}, 3m - 2: 3m) = \frac{\left(\mathbf{s}\_{\text{t},n}^{\text{o}} - \mathbf{c}\_{\text{k}}^{\text{o}}\right)^{\text{T}}}{R\_{\text{c},k,\text{tr}}^{\text{o}}} - \frac{\left(\mathbf{s}\_{\text{t},n}^{\text{o}} - \mathbf{s}\_{\text{r},n}^{\text{o}}\right)^{\text{T}}}{R\_{\text{t},m,\text{r},n}^{\text{o}}} \tag{28}$$

$$\frac{\partial \mathbf{r}\_{\mathbf{c}}^{\rm o}}{\partial \mathbf{s}\_{\mathbf{r}}^{\rm o}} (i\_{k,m,n}, 3n - 2: 3n) = \frac{\left(\mathbf{s}\_{\mathbf{r},n}^{\rm o} - \mathbf{c}\_{k}^{\rm o}\right)^{\rm T}}{R\_{\mathbf{c},k,x,n}^{\rm o}} - \frac{\left(\mathbf{s}\_{\mathbf{r},n}^{\rm o} - \mathbf{s}\_{\mathbf{t},m}^{\rm o}\right)^{\rm T}}{R\_{\mathbf{t},m,x,n}^{\rm o}} \tag{29}$$

for *k* = 1, 2, ... ,*K*, *m* = 1, 2, ... , *M* and *n* = 1, 2, ... , *N*, and zeros elsewhere.

By definition, the CRLB of ϕ, denoted by **CRLB**c(ϕ), is given as FIM(ϕ) −1 , where only the upper left 3-by-3 block is for the target position **u**o. Invoking the partitioned matrix inversion formula as well as the matrix inversion lemma [32] twice on (16), leads to the CRLB of **u**<sup>o</sup> as

$$\mathbf{CRLB}\_{\mathbf{c}}(\mathbf{u}^{o}) = \mathbf{X}^{-1} + \mathbf{X}^{-1}\mathbf{Y}(\mathbf{Z} - \mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1}\mathbf{Y} - \mathbf{R}^{\mathrm{T}}\mathbf{P}^{-1}\mathbf{R})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1} \tag{30}$$

For comparison purposes, the CRLB of **u**<sup>o</sup> with transmitter/receiver position error but without calibration derived in [22], denoted by **CRLB**s(**u**o), is also given below

$$\mathbf{CRLB}\_{\mathbf{s}}(\mathbf{u}^{\rm o}) = \mathbf{X}^{-1} + \mathbf{X}^{-1}\mathbf{Y}(\widehat{\mathbf{Z}} - \mathbf{Y}^{\rm T}\mathbf{X}^{-1}\mathbf{Y})^{-1}\mathbf{Y}^{\rm T}\mathbf{X}^{-1} \tag{31}$$

where **Z** = **Q**−<sup>1</sup> **<sup>s</sup>** + (∂**r**o/∂**s**o) <sup>T</sup>**Q**−<sup>1</sup> **<sup>r</sup>** (∂**r**o/∂**s**o). For the sake of comparison, we proceed to construct an equivalent form of **CRLB**c(**u**o) by denoting **<sup>Z</sup>** as **<sup>Z</sup>** = **<sup>Z</sup>** <sup>−</sup> **<sup>R</sup>**T**P**−1**R**. After invoking the matrix inversion lemma [32] to (∂**r**<sup>o</sup> c/∂**s**o) <sup>T</sup>**Q**−<sup>1</sup> **<sup>r</sup>**<sup>c</sup> (∂**r**<sup>o</sup> c/∂**s**o) <sup>−</sup> **<sup>R</sup>**T**P**−1**<sup>R</sup>** and some algebraic manipulations, we further represent **Z** as

$$\widetilde{\mathbf{Z}} = \mathbf{Q}\_{\mathbf{s}}^{-1} + \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right) + \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{s}^{o}}\right)^{\mathrm{T}} \left(\mathbf{Q}\_{\mathbf{r}\mathbf{c}} + \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{c}^{o}}\right) \mathbf{Q}\_{\mathbf{c}} \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{c}^{o}}\right)^{\mathrm{T}}\right)^{-1} \left(\frac{\partial \mathbf{r}\_{\mathrm{c}}^{o}}{\partial \mathbf{s}^{o}}\right) \tag{32}$$

Using (32), we obtain an equivalent expression of **CRLB**c(**u**o) as

$$\mathbf{CRLB}\_{\mathbb{C}}(\mathbf{u}^{\bullet}) = \mathbf{X}^{-1} + \mathbf{X}^{-1}\mathbf{Y}(\breve{\mathbf{Z}} - \mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1}\mathbf{Y})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{X}^{-1} \tag{33}$$

*Sensors* **2019**, *19*, 3365

Through the comparison of (31) and (33), it is readily to observe that the two CRLBs are identical in structure, except that **<sup>Z</sup>** is substituted by **Z**. More specifically, the use of calibration targets introduces an additional component into the bracketed matrix expression to be inverted as

$$\begin{split} \mathbf{\tilde{Z}} &= \mathbf{\tilde{Z}} - \mathbf{\hat{Z}} \\ &= \left( \partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{s}^{o} \right)^{\mathrm{T}} \mathbf{Q}\_{\mathrm{rc}}^{-1} (\partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{s}^{o}) - \mathbf{R}^{\mathrm{T}} \mathbf{P}^{-1} \mathbf{R} \\ &= \left( \partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{s}^{o} \right)^{\mathrm{T}} \Big( \mathbf{Q}\_{\mathrm{rc}} + \left( \partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{c}^{o} \right) \mathbf{Q}\_{\mathrm{c}} (\partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{c}^{o})^{\mathrm{T}} \Big)^{-1} (\partial \mathbf{r}\_{\mathrm{c}}^{o} / \partial \mathbf{s}^{o}) \end{split} \tag{34}$$

Using (34), we can rewrite ( **<sup>Z</sup>** <sup>−</sup> **<sup>Y</sup>**T**X**−1**Y**) −1 in (33) as (( **<sup>Z</sup>** <sup>−</sup> **<sup>Y</sup>**T**X**−1**Y**) + **<sup>~</sup> Z**) −1 . Invoking the matrix inversion lemma [32] to the term (( **<sup>Z</sup>** <sup>−</sup> **<sup>Y</sup>**T**X**−1**Y**) + **<sup>~</sup> Z**) −1 in (33), we obtain after some algebraic manipulations,

$$\mathbf{CRLB\_c(u^o) - CRLB\_s(u^o) = X^{-1}YY^TX^{-1}}\tag{35}$$

where

$$\mathbf{T} = \mathbf{H}^{-1}\mathbf{Y}(\mathbf{I} + \mathbf{Y}^{\mathrm{T}}\mathbf{H}^{-1}\mathbf{Y})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{H}^{-1} \tag{36}$$

$$\mathbf{H} = (\tilde{\mathbf{Z}} - \mathbf{Y}^{\mathrm{T}} \mathbf{X}^{-1} \mathbf{Y}) \tag{37}$$

$$\mathbf{Y} = \left(\frac{\partial \mathbf{r}\_c^o}{\partial \mathbf{s}^o}\right)^T \mathbf{L}\_{\mathbf{rc}} \tag{38}$$

and **Lr**<sup>c</sup> is the Cholesky decomposition of (**Qr**<sup>c</sup> + (∂**r**<sup>o</sup> c/∂**c**o)**Qc**(∂**r**<sup>o</sup> c/∂**c**o) T) −1 , i.e., **Lr**c**L**<sup>T</sup> **<sup>r</sup>**<sup>c</sup> = (**Qr**<sup>c</sup> + (∂**r**<sup>o</sup> c/∂**c**o)**Qc**(∂**r**<sup>o</sup> c/∂**c**o) T) −1 . 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 **Υ**<sup>T</sup> 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 (**Qr**<sup>c</sup> + (∂**r**<sup>o</sup> c/∂**c**o)**Q**−<sup>1</sup> **<sup>c</sup>** (∂**r**<sup>o</sup> c/∂**c**o) T) −1 tends to zero and then **Lr**<sup>c</sup> → **O** and **Υ** → **O**, the performance enhancement in (34) would tend to zero. However, this edge case hardly exists in reality. Thus, mathematically, we can arrive at

$$\mathbf{CRLB}\_{\mathbf{s}}(\mathbf{u}^{\diamond}) \ge \mathbf{CRLB}\_{\mathbf{c}}(\mathbf{u}^{\diamond}) \tag{39}$$

The matrix inequality **A** ≥ **B** means that **A** − **B** is PSD. It can be further deduced from (39) that tr(**CRLB**s(**u**o)) <sup>≥</sup> tr(**CRLB**c(**u**o)). The trace of **CRLB**c(**u**o) and **CRLB**s(**u**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 **Qr** = σ<sup>2</sup> <sup>r</sup>**Vr**, where σ<sup>r</sup> reflects BR measurement noise level and **Vr** is set to 1 in the diagonal elements and 0.5 elsewhere. The covariance matrix of the transmitter/receiver position error is given as **Qs** = σ<sup>2</sup> <sup>s</sup>**Vs** where σ*<sup>s</sup>* reflects the transmitter/receiver position error level and **<sup>V</sup>**<sup>s</sup> = diag(5**I**3*M*×3*M*,**I**3*N*×3*N*). The covariance matrix of calibration target position error is **Qc** = <sup>σ</sup><sup>2</sup> <sup>c</sup>**Vc** where σ<sup>c</sup> reflects the calibration target position error level and **Vc** = **I**3*K*×3*K*, and the covariance matrix of the corresponding calibration BR measurement noise is **Qr**<sup>c</sup> = σ<sup>2</sup> rc**Vr**<sup>c</sup> where σrc = σ<sup>r</sup> reflects calibration BR measurement noise level and **Vr**<sup>c</sup> is set to 1 in the diagonal elements and 0.5 elsewhere. The target of interest is located at position **u**<sup>o</sup> = [50000, 15000, 5000] Tm. The effect of the calibration targets on the target localization accuracy, in the sense of CRLB, is presented in Figure 3.

ϒ

**Figure 2.** Localization scenario geometry for simulation.

**Table 1.** Positions (in meters) of the transmitters, receivers and calibration targets.


**Figure 3.** Comparison of the CRLBs with and without using calibration targets: (**a**) for different BR measurement noise level σr; (**b**) for different transmitter/receiver position error level σs; (**c**) for different calibration target position error level σc.

Figure 3a compares the CRLB curves with and without using calibration targets when the BR measurement noise level σ<sup>r</sup> is varied from 10−<sup>2</sup> m to 103 m while the transmitter/receiver position error level and calibration target position error level are fixed at σ<sup>s</sup> = 20 m and σ<sup>c</sup> = 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). However, in the edge case where the BR measurement noise is very large, two CRLBs would tend to be the same. This is because in this case, the BR measurement noise dominates and effect of transmitter/receiver position error on the localization accuracy is relatively small. The CRLB curves versus the transmitter/receiver position error level σ<sup>s</sup> are plotted in Figure 3b where the BR measurement noise level and calibration target position error level are fixed at σ<sup>r</sup> = 10 m and σ<sup>c</sup> = 10 m respectively. A similar trend, i.e., two CRLBs would tend to be the same, appears in Figure 3b, when the transmitter/receiver position error is sufficiently small. A reasonable explanation is that, in this case, the transmitter/receiver positions are

known very accurately and their influence on the localization accuracy can be ignored compared to the BR measurement noise. The CRLB comparison versus calibration target position error level σ<sup>c</sup> is provided in Figure 3c where σ<sup>r</sup> = 10 m and σ<sup>s</sup> = 20 m. Interestingly, the trend of CRLB curves implies that, even when the calibration target position error is extremely large, the CRLB with utilization of calibration targets are still remarkably below the one without. This justifies again the analysis under (35), and similar results have also been presented in previous studies [23–30] on source localization and sensor network localization issues. Generally, from Figure 3, the use of calibration targets brings a significant improvement in the localization accuracy in the normal case, at least at the CRLB level.
