## 2 Mathematical Formulation of Object Localization and Tracking Algorithms
### TOF Estimation
#### 2.3.2 NRLM Algorithm for TOF Estimation
The next step is to calculate the gradient \(\mathbf{\
abla}_{\mathbf{z}_{g}}f(\mathbf{z}_{g})\) as follows:  
\[\mathbf{\
abla}_{\mathbf{z}_{g}}f(\mathbf{z}_{g}) =\Big{[}\dfrac{\partial f(\mathbf{z}_{g})}{\partial z_{g1}}&\dots& \dfrac{\partial f(\mathbf{z}_{g})}{\partial z_{gn}}\Big{]}^{\intercal}=\Big{[} \dfrac{\partial f(\mathbf{z}_{g})}{\partial t_{0}}&\dfrac{\partial f(\mathbf{ z}_{g})}{\partial a_{g}}&\dfrac{\partial f(\mathbf{z}_{g})}{\partial a}& \dfrac{\partial f(\mathbf{z}_{g})}{\partial p}\Big{]}^{\intercal}\] \[=\Big{[}2\sum\limits_{i=1}^{m}r_{i}(\mathbf{z}_{g})\dfrac{ \partial r_{i}(\mathbf{z}_{g})}{\partial t_{0}}&2\sum\limits_{i=1}^{m}r_{i}( \mathbf{z}_{g})\dfrac{\partial r_{i}(\mathbf{z}_{g})}{\partial a_{g}}&2\sum \limits_{i=1}^{where \(\mathbf{G}(\mathbf{z}_{g})_{k,j}=\sum_{i=1}^{m}r_{i}(\mathbf{z}_{g})\frac{\partial^{ 2}r_{i}(\mathbf{z}_{g})}{\partial z_{gk}\partial z_{gj}}\) is defined as the \((k,j)\)th element of matrix \(\mathbf{G}(\mathbf{z}_{g})\). Therefore, the Hessian matrix \(\mathbf{H}(\mathbf{z}_{g})\) can be obtained as the following form:  
\[\mathbf{H}(\mathbf{z}_{g})=2\mathbf{I}_{g}(\mathbf{z}_{g})^{\intercal}\mathbf{ J}_{g}(\mathbf{z}_{g})+\mathbf{G}(\mathbf{z}_{g}) \tag{21}\]  
As a consequence, the recursive formula for the NR algorithm applied to the nonlinear LS problem is as follows:  
\[\mathbf{z}_{g}^{(k+1)}=\mathbf{z}_{g}^{(k)}-(\mathbf{J}_{g}(\mathbf{z}_{g}^{( k)})^{\intercal}\mathbf{J}_{g}(\mathbf{z}_{g}^{(k)})+\mathbf{G}(\mathbf{z}_{g}^{( k)}))^{-1}\mathbf{J}_{g}(\mathbf{z}_{g}^{(k)})\mathbf{r}(\mathbf{z}_{g}^{(k)}) \tag{22}\]  
where the notation \((k)\) and \((k+1)\) denote the \(k\)th and \((k+1)\)th time steps. In general, the term \(\mathbf{G}(\mathbf{z}_{g})\) in (21) is significantly small because it contains the second derivative terms and can be ignored in practical applications. However, ignoring the term \(\mathbf{G}(\mathbf{z}_{g})\) in (21) may cause \(\mathbf{J}_{g}(\mathbf{z}_{g})^{\intercal}\mathbf{J}_{g}(\mathbf{z}_{g})\) to become a singular matrix and make the system crash. This problem can be solved by using the LM modification, which adds a sufficiently large number \(\lambda\) into the diagonal of \(\mathbf{J}_{g}(\mathbf{z}_{g})^{\intercal}\mathbf{J}_{g}(\mathbf{z}_{g})\) such that:  
\[\mathbf{z}_{g}^{(k+1)}=\mathbf{z}_{g}^{(k)}-(\mathbf{J}_{g}(\mathbf{z}_{g}^{( k)})^{\intercal}\mathbf{J}_{g}(\mathbf{z}_{g}^{(k)})+\lambda_{k}\mathbf{I})^{-1} \mathbf{J}_{g}(\mathbf{z}_{g}^{(k)})\mathbf{r}(\mathbf{z}_{g}^{(k)}) \tag{23}\]  
This term \(\lambda_{k}\mathbf{I}\) can make sure that the search always points to the descent direction, and it can be viewed as an approximation to \(\mathbf{G}(\mathbf{z}_{g})\) in (22). Therefore, (23) is the ongoing, recursive formula for the TOF estimation \(t_{0}\). The above procedure also can be applied on the DE model, and (23) is modified as:  
\[\mathbf{z}_{d}^{(k+1)}=\mathbf{z}_{d}^{(k)}-(\mathbf{J}_{d}(\mathbf{z}_{d}^{( k)})^{\intercal}\mathbf{J}_{d}(\mathbf{z}_{d}^{(k)})+\lambda_{k}\mathbf{I})^{-1} \mathbf{J}_{d}(\mathbf{z}_{d}^{(k)})\mathbf{r}(\mathbf{z}_{d}^{(k)}) \tag{24}\]  
where  
\[\mathbf{J}_{d}(\mathbf{z}_{d})_{i,1}= a_{d}(\gamma e^{-\gamma(t_{i}-t_{0})}-\beta e^{-\beta(t_{i}-t_{0})}) \tag{25}\] \[\mathbf{J}_{g}(\mathbf{z}_{g})_{i,2}= -\Big{(}\dfrac{t_{i}-t_{0}}{p}\Big{)}^{\alpha}e^{\big{(}\frac{t_{ i}-t_{0}}{p}\big{)}}\]0 (26) \[\mathbf{J}_{g}(\mathbf{z}_{g})_{i,2}= -\Big{(}\dfrac{t_{i}-t_{0}}{p}\Big{)}^{\alpha}e^{\big{(}\frac{t_{ i}-t_{0}}{p}\big{)}}\]1 (27) \[\mathbf{J}_{g}(\mathbf{z}_{g})_{i,2}= -\Big{(}\dfrac{t_{i}-t_{0}}{p}\Big{)}^{\alpha}e^{\big{(}\frac{t_{ i}-t_{0}}{p}\big{)}}\]2  
to estimate the TOF \(t_{0}\). (23) and (24) are tested with the aforementioned algorithms in this study, and the performances of the simulation and experimental results are compared in Sections 3 and 4, respectively.