*4.2. 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 presented

in (33). By invoking the matrix inversion lemma [32] to (33) and using the definitions of **X** and **Y**, we have after mathematical simplifications,

$$\mathbf{CRLB}\_{\mathbf{c}} \left(\mathbf{u}^{o}\right)^{-1} = \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{u}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{u}^{o}}\right) - \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{u}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right) \widetilde{\mathbf{Z}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{s}^{o}}\right)^{\mathrm{T}} \mathbf{Q}\_{\mathbf{r}}^{-1} \left(\frac{\partial \mathbf{r}^{o}}{\partial \mathbf{u}^{o}}\right) \tag{83}$$

where the expression of **Z** has been given in (32).

On the other hand, using (82), (81), (69), (68) and (52) successively, we can reformulate the inverse of cov(**u**) as

$$\text{cov}(\mathbf{u})^{-1} = \mathbf{G}\_3^T \mathbf{Q}\_\mathbf{r}^{-1} \mathbf{G}\_3 - \mathbf{G}\_3^T \mathbf{Q}\_\mathbf{r}^{-1} \mathbf{G}\_4 \mathbf{Z}^{-1} \mathbf{G}\_4^T \mathbf{Q}\_\mathbf{r}^{-1} \mathbf{G}\_3 \tag{84}$$

where **G**<sup>3</sup> = **B**−<sup>1</sup> <sup>1</sup> **<sup>G</sup>**1**B**−<sup>1</sup> <sup>2</sup> **<sup>G</sup>**2, **<sup>G</sup>**<sup>4</sup> <sup>=</sup> **<sup>B</sup>**−<sup>1</sup> <sup>1</sup> **<sup>D</sup>**1, and **¯ Z** = **Q**−<sup>1</sup> **<sup>s</sup>** + **G**<sup>T</sup> 4**Q**−<sup>1</sup> **<sup>r</sup> G**<sup>4</sup> + **G**<sup>T</sup> <sup>0</sup> (**GcQcG**<sup>T</sup> **<sup>c</sup>** + **Qr**c) −1 **G**0.

Comparing (83) with (84), we observe that **CRLB**c(**u**o) <sup>−</sup><sup>1</sup> and cov(**u**) <sup>−</sup><sup>1</sup> are identical in structure. Next, we proceed to prove their equivalency under the following conditions:

(C1) **Δs**t,*m***c**<sup>o</sup> *<sup>k</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> t,*m*, **Δs**t,*m***s**<sup>o</sup> t,*<sup>m</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, **Δs**r,*n***c**<sup>o</sup> *<sup>k</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, **Δs**r,*n***s**<sup>o</sup> t,*<sup>m</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, and **Δc***k***c**<sup>o</sup> *<sup>k</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> t,*m*, **Δc***k***c**<sup>o</sup> *<sup>k</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, for *k* = 1, 2, ... ,*K*, *m* = 1, 2, ... , *M* and *n* = 1, 2, ... , *N*;

(C2) **Δ***rm*,*n***u**<sup>o</sup> <sup>−</sup> **<sup>s</sup>**<sup>o</sup> t,*m*, **Δ***rm*,*n***u**<sup>o</sup> <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, **Δ***rm*,*n***s**<sup>o</sup> t,*<sup>m</sup>* <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n*, and **Δs**t,*<sup>m</sup>* − **Δs**ˆt,*m* **u**<sup>o</sup> <sup>−</sup> **<sup>s</sup>**<sup>o</sup> t,*m*, **Δs**r,*<sup>n</sup>* <sup>−</sup> **<sup>Δ</sup>s**ˆr,*n***u**<sup>o</sup> <sup>−</sup> **<sup>s</sup>**<sup>o</sup> r,*n* for *m* = 1, 2, ... , *M* and *n* = 1, 2, ... , *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. Using the conditions C1 and C2, we obtain, after some involved algebraic manipulations, that

$$\mathbf{G}\_3 = \frac{\partial \mathbf{r}^0}{\partial \mathbf{u}^0}, \mathbf{G}\_4 = -\frac{\partial \mathbf{r}^0}{\partial \mathbf{s}^0}, \mathbf{G}\_0 = -\left(\frac{\partial \mathbf{r}\_c^0}{\partial \mathbf{s}^0}\right), \mathbf{G}\_\mathbf{c} = -\left(\frac{\partial \mathbf{r}\_c^0}{\partial \mathbf{c}^0}\right) \tag{85}$$

By this point, we can draw the conclusion that

$$\text{cov}(\mathbf{u})^{-1} = \mathbf{CRLB}\_{\mathbf{c}}(\mathbf{u}^{\diamond})^{-1} \tag{86}$$

That is, the proposed solution accomplishes the CRLB accuracy if the two conditions C1 and C2 are satisfied. In reality, localization scenarios, which satisfy the conditions C1 and C2, are not rare. These two conditions can be satisfied if the unknown target and the calibration targets are far from the transmitters and receivers, if not these conditions can still be satisfied if the BR measurement noise and the transmitter/receiver/calibration target position errors are sufficiently small.
