# Target Localization via Integrated and Segregated Ranging Based on RSS and TOA Measurements
## 2 Problem Formulation  
Let us consider a \(p\)-dimensional (\(p=2\) or \(3\)) sensor network comprising \(N\) reference sensors with known locations (called anchors) and a sensor whose location we wish to determine (called target). It is assumed that the target emits a signal to anchors, which are suitably equipped to withdraw RSS and TOA information from the received signal.  
According to [2; 3; 13; 18; 21; 23; 25; 26], RSS and TOA in NLOS conditions can be modeled as  
\[P_{i}=P_{0}-b_{i}-10\gamma\log_{10}\frac{\|\mathbf{x}-\mathbf{a}_{i}\|}{d_{0}}+n_{i}, \tag{1a}\] \[d_{i}=\|\mathbf{x}-\mathbf{a}_{i}\|+\beta_{i}+m_{i}, \tag{1b}\]respectively, where \(P_{0}\) (dBm) is the target's transmit power, \(b_{i}\) (dB) and \(\beta_{i}\) (m) are the (positive) NLOS biases (\(b_{i}=\beta_{i}=0\) if \(i\in\mathcal{L}_{I}\), and \(b_{i},\beta_{i}>0\) if \(i\in\mathcal{L}_{n}\), with \(\mathcal{L}_{I}\), \(\mathcal{L}_{n}\) representing the set of all LOS and NLOS links, respectively), \(\gamma\) is the path loss exponent, \(\mathbf{x},\mathbf{a}_{i}\) represent the true target's and \(i\)-th anchor's location (\(i=1,...,N\)), respectively, \(d_{0}\) is a reference distance (\(\|\mathbf{x}-\mathbf{a}_{i}\|\geq d_{0}\)), \(n_{i}\) is the log-normal shadowing term (dB) modeled as a zero-mean Gaussian random variable with variance \(\sigma_{n_{i}}^{2}\), i.e., \(n_{i}\sim\mathcal{N}(0,\sigma_{n_{i}}^{2})\), and \(m_{i}\) is the measurement noise (m) modeled as \(m_{i}\sim\mathcal{N}(0,\sigma_{m_{i}}^{2})\). Moreover, many authors assume that the magnitudes of the NLOS biases are bounded by a known constant [5, 7, 17], i.e., \(0\leq b_{i}\leq b_{\max}\) and \(0\leq\beta_{i}\leq\beta_{\max}\).  
If all RSS and TOA measurements are stacked in a single vector, i.e., \(\mathbf{P}=[P_{i}]^{T}\) and \(\mathbf{d}=[d_{i}]^{T}\) (\(\mathbf{P},\mathbf{d}\in\mathbb{R}^{N}\)), one can write the joint likelihood function as  
\[\begin{split}&\Lambda(\mathbf{P},\mathbf{d}|\mathbf{x},b_{i},\beta_{i})=p( \mathbf{P}|\mathbf{x},b_{i})\,  
\[\hat{r}_{i}^{RSS}=d_{0}10^{\frac{P_{0}-P_{i}-b_{i}}{10_{T}}}, \tag{4a}\] \[\hat{r}_{i}^{TOA}=d_{i}-\beta_{i}. \tag{4b}\]  
After squaring (4a) and (4b) and applying simple algebraic manipulations, the following relation was established, written in a matrix notation.  
\[\mathbf{G}\mathbf{\theta}=\mathbf{h}, \tag{5}\]  
where  
\[\mathbf{G}=\begin{bmatrix}\mathbf{a}_{1}^{T}&-\frac{1}{2}\\ \vdots&\vdots\\ \mathbf{a}_{N}^{T}&-\frac{1}{2}\\ \mathbf{a}_{1}^{T}&-\frac{1}{2}\\ \vdots&\vdots\\ \mathbf{a}_{N}^{T}&-\frac{1}{2}\\ \end{bmatrix},\,\mathbf{\theta}=\begin{bmatrix}\mathbf{x}\\ \mathbf{Q}\\ \end{bmatrix},\,\mathbf{h}=\begin{bmatrix}\frac{1}{2}\left(\|\mathbf{a}_{1}\|^{2}-( \hat{r}_{1}^{RSS})^{\frac{2}{7}}\right)\\ \vdots\\ \frac{1}{2}\left(\|\mathbf{a}_{N}\|^{2}-(\hat{r}_{N}^{RSS})^{\frac{2}{7}}\right)\\ \frac{1}{2}\left(\|\mathbf{a}_{1}\|^{2}-(\hat{r}_{1}^{TOA})^{2}\right)\\ \vdots\\ \frac{1}{2}\left(\|\mathbf{a}_{N}\|^{2}-(\hat{r}_{N}^{TOA})^{2}\right)\\ \end{bmatrix},\]  
with \(Q=\|\mathbf{x}\|^{2}\).  
The authors in [18] then considered a noisy environment, and from (5), they obtained a solution to the localization problem by solving the following LS problem:  
\[\hat{\mathbf{\theta}}=\operatorname*{arg\,min}_{\mathbf{\theta}}\left(\mathbf{G}\mathbf{ \theta}-\mathbf{h}\right)^{T}\mathbf{W}\left(\mathbf{G}  
curve of (7) at the point \(\left(\hat{r}_{i}^{TOA},\hat{d}_{i}^{opt}\right|_{\left\|x-\mathbf{a}_{i}\right\|=\hat{ r}_{i}^{TOA}}\right)\) one obtains a better estimate as the intersection of the tangent and the abscissa, i.e.,  
\[\hat{d}_{i}^{NR}=\frac{B_{1i}+B_{2i}\hat{r}_{i}^{TOA}+B_{3i}(\hat{r}_{i}^{TOA})^ {2}+B_{4i}\log\hat{r}_{i}^{TOA}}{B_{2i}+B_{3i}\hat{r}_{i}^{TOA}+\frac{B_{4i}}{ \hat{r}_{i}^{TOA}\ln 10}}. \tag{8}\]  
The estimation can be further improved by repeating this procedure [23].  
Note that perfect knowledge of noise powers and NLOS bias realization is assumed known in [23], which might not be a valid assumption in practice.