# Target Localization via Integrated and Segregated Ranging Based on RSS and TOA Measurements
## 2 Problem Formulation
### JAH Algorithm  
Similarly to [23], the authors in [20] study the problem of range estimation between two sensors. To derive a sub-optimal estimator, the authors in [20] start by taking the derivative of the ML estimator and setting it to zero. After manipulating the result, they obtain  
\[\log\left\|\mathbf{x}-\mathbf{a}_{i}\right\|=-\frac{\gamma^{2}}{a^{2}}\left\|\mathbf{x}- \mathbf{a}_{i}\right\|^{2}+\frac{\gamma^{2}\hat{r}_{i}^{TOA}}{a^{2}}\left\|\mathbf{x} -\mathbf{a}_{i}\right\|-\frac{P_{i}-P_{0}+b_{i}}{a},\;\;i=1,\ldots,N, \tag{9}\]  
where \(\gamma=\frac{\sigma_{n_{i}}}{\sigma_{n_{i}}}\) and \(a=\frac{10\gamma}{\ln 10}\). The right-hand side of (9) is then relaxed into an affine function. However, since there are infinite possible choices to do so, the authors in [20] chose \(n\) \"control points\", \(\delta_{j}\), \(j=1,\ldots,n\), to get a general result, and applied an ordinary least squares approach to minimize the parameters of the affine function. Finally, by assuming that \(\hat{r}_{i}^{TOA}<\delta=\frac{\delta_{3}-\delta_{1}\delta_{2}}{\delta_{2}- \delta_{1}^{2}}\), where \(\delta_{k}=\frac{1}{n}\sum_{j=1}^{n}\delta_{j}^{k}\), \(k=1,2,3\), they derived the following estimator:  
\[\hat{d}_{i}^{JAH}=\frac{1}{\varphi_{i}}W_{0}\left(\varphi_{i}\mathrm{e}^{ \varphi_{i}}\right), \tag{10}\]  
where \(\varphi_{i}=\frac{\gamma^{2}}{a^{2}}\left(\hat{r}_{i}+\delta\right)\), \(\psi_{i}=\frac{\gamma^{2}}{a^{2}}\left(\frac{a}{\gamma^{2}}\left(P_{0}-P_{i}- b_{i}\right)+\delta\right)\), with \(\delta=\delta_{2}-\delta_{1}\delta\) and \(W_{0}(\bullet)\) denoting the principal branch of the Lambert W-function. It is worth mentioning that in [20], uniformly spaced points were chosen within a predefined interval. In addition, the authors in [20] generalized their estimator for the case where the condition \(\hat{r}_{i}^{TOA}<\delta=\frac{\delta_{3}-\delta_{1}\delta_{2}}{\delta_{2}- \delta_{1}^{2}}\) is not met. Furthermore, the authors in [20] assume perfect knowledge of noise powers is available and that all links are LOS, which might not be the case in practice. Even though NR and JAH algorithms were originally designed for estimating the range between two sensors, their generalization to the localization problem is straightforward after one has the range estimates. According to [24], by using the estimates \(\hat{d}_{i}^{NR}\) and \(\hat{d}_{i}^{JAH}\) obtained from [23] and [20] respectively, one can obtain a target location estimate by solving the following problem:  
\[\underset{\mathbf{x}}{\text{minimize}}\;\sum_{i=1}^{N}\omega_{i}\left(\left\|\bm {x}-\mathbf{a}_{i}\right\|^{2}-\hat{d}_{i}^{NR/JAH}\right)^{2}, \tag{11}\]  
where \(\omega_{i}=1-\frac{\hat{d}_{i}^{NR/JAH}}{\sum_{i=1}^{N}\hat{d}_{i}^{NR/JAH}}\). The problem in (11) can be rewritten in a vector form as  
\[\underset{\mathbf{\theta}=\left[\mathbf{x}^{T},\left\|\mathbf{x}\right\|^{2}\right]}{ \text{minimize}}\left\{\left\|\mathbf{\Omega}\left(\mathbf{H}\mathbf{\theta}-\mathbf{g}\right) \right\|^{2}:\mathbf{\theta}^{T}\mathbf{D}\mathbf{\theta}+2\mathbf{f}^{T}\mathbf{\theta}\right\}, \tag{12}\]where \(\Omega=\text{diag}(\omega)\), with \(\omega=\sqrt{\omega_{i}}\), and  
\[\mathbf{H}= \begin{bmatrix}-2\mathbf{a}_{i}^{T},1\\ \vdots\\ -2\mathbf{a}_{N}^{T},1\end{bmatrix} ,\,\mathbf{g}= \begin{bmatrix}\left(\hat{d}_{1}^{NR/ JAH}\right)^{2}-\|\mathbf{a}_{1}\|^{2}\\ \vdots\\ \left(\hat{d}_{N}^{NR/JAH}\right)^{2}-\|\mathbf{a}_{N}\|^{2}\end{bmatrix} ,\,\mathbf{D}= \begin{bmatrix}\mathbf{I}_{p}&\mathbf{0}_{p\times 1}\\ \mathbf{0}_{1\times p}&0\end{bmatrix} ,\,\mathbf{f}= \begin{bmatrix}\mathbf{0}_{p\times 1}\\ -1/2\end{bmatrix} .\]  
Observe that both the objective function and the constraint in (12) are quadratic.