## 3 Localization Problem  
The proposed passive target localization problem uses TDOA signal parameter to estimate the passive target's unknown position under the conditions of LOS environment. Figure 1 illustrates the considered localization system, which consists of a set of transmitters \(\mathbf{x}_{i}^{t}= \begin{bmatrix}x_{i}^{t}&y_{i}^{t}\end{bmatrix} ^{T}\in \mathbb{R}^{2}\), \(\forall i\in\{1,2,...,N\}\), whose positions are previously known and the passive target \(\mathbf{x}= \begin{bmatrix}x&y\end{bmatrix} ^{T}\in\mathbb{R}^{2}\) at the unknown coordinates \(x\) and \(y\). The target is illuminated by the signals coming from the transmitters, which is then reflected from the target and captured at the receiver, which is, without the loss of generality, placed in the origin \(\mathbf{x}_{r}= \begin{bmatrix}0&0\end{bmatrix} ^{T}\). Here, we model TDOA measurement error using the Gaussian distribution, which in a LOS context, is a justified proposition [2]. Assuming the time synchronisation of \(\mathbf{x}_{i}^{t},\forall i\in\{1,2,...,N\}\) and \(\mathbf{x}_{r}\) is obtained and that the passive target simply reflects signals back and forth between \(\mathbf{x}_{i}^{t}\) and \(\mathbf{x}_{r}\)[48], the noisy TDOA measurements are obtained as the signal time difference  
\[t_{i}=t_{r,i}-t_{d,i},\quad i\in\{1,2,...,N\} \tag{1}\]  
where \(t_{r,i}\) denotes the TOA of a signal which is reflected from the target coming from the \(i\)th transmitter and captured at the receiver, and \(t_{d,i}\) is the TOA of a signal which is directly coming from the \(i\)th transmitter to a receiver. Then, by substituting the coordinates of receiver, transmitters and target the TDOA measurement is represented as  
\[t_{i}=\frac{1}{c}\big{(}\big{\|}\mathbf{x}_{i}^{t}-\mathbf{x}\big{\|}_{2}+ \big{\|}\mathbf{x}_{r}-\mathbf{x}\big{\|}_{2}-\big{\|}\mathbf{x}_{i}^{t}- \mathbf{x}_{r}\big{\|}_{2}\big{)}+\bar{n}_{i},\forall i\in\{1,2,...,N\}, \tag{2}\]  
where \(\big{\|}\cdot\big{\|}_{2}\) denotes the two-dimensional Euclidean distance, \(c\) denotes propagation speed of light, and \(\bar{n}_{i}\) is the Gaussian modelled measurement noise. Eliminating the term \(c\), the range measurement of the signal coming from \(i\)th transmitter is obtained  
\[r_{i}=c\cdot t_{i} \tag{3}\] \[=\big{(}\big{\|}\mathbf{x}_{i}^{t}-\mathbf{x}\big{\|}_{2}+\big{\|} \mathbf{x}_{r}-\mathbf{x}\big{\|}_{2}-\big{\|}\mathbf{x}_{i}^{t}-\mathbf{x}_{ r}\big{\|}_{2}\big{)}+n_{i},\forall i\in\{1,2,...,N\},\]  
where \(n_{i}=c\bar{n}_{i}\) is Gaussian zero-mean distributed variable. Since the distance \(\big{\|}\mathbf{x}_{i}^{t}-\mathbf{x}_{r}\big{\|}_{2}\) between each of the transmitters and receiver is previously known, we introduce a new variable to rearrange a range measurement as follows  
\[\tilde{r}_{i}=r_{i}+\big{\|}\mathbf{x}_{i}^{t}-\mathbf{x}_{r}\big{\|}_{2} \tag{4}\]  
Therefore, the Equation (3) can be rewritten as  
\[\tilde{r}_{i}=\big{\|}\mathbf{x}_{r}-\mathbf{x}\big{\|}_{2}+\big{\|}\mathbf{ x}_{i}^{t}-\mathbf{x}\big{\|}_{2}+n_{i},\forall i\in\{1,2,...,N\}. \tag{5}\]  
Figure 1: Illustration of the placement of transmitters, receiver and a target and a distribution of signals in a considered localization problem. Which is represented in the corresponding vector form  
\[\bar{\mathbf{r}}=\mathbf{d}(\mathbf{x})+\mathbf{n}, \tag{6}\]  
in which  
\[\mathbf{d}(\mathbf{x})= \begin{bmatrix}\left\|\mathbf{x}_{1}^{t}-\mathbf{x} \right\|_{2}+\left\|\mathbf{x}_{r}-x\right\|_{2}\\ \vdots\\ \left\|\mathbf{x}_{N}^{t}-\mathbf{x}\right\|_{2}+\left\|\mathbf{x}_{r}-x \right\|_{2}\end{bmatrix} , \tag{7}\]  
and \(\mathbf{n}=[n_{1},n_{2},\ldots,n_{N}]^{T}\)is the measurement noise vector, under assumption that each component of a vector \(\mathbf{n}\) is independent and an identically distributed zero-mean Gaussian variable. The \(\mathbf{C}\) denotes the appropriate matrix of covariance, whose elements can be calculated using the expectation operator \(E[\cdot]\) as \(\mathbf{C}=E\left[\mathbf{n}\mathbf{n}^{T}\right]\).