# Satellite-Based Localization of IoT Devices Using Joint Doppler and Angle-of-Arrival Estimation
## 6 Experiment and Results
### Localization Simulation and Performance
Furthermore, the GA optimizer is based on the individual's fitness of a population, which is evaluated depending on how well it solves the optimization problem. The fitter individuals have a higher chance of being selected, and a pair of individuals will create new offspring to introduce diversity, thus helping to explore the solution space. Furthermore, random changes are applied to some individuals to introduce additional mutations. The process is repeated for several generations. Similar to the PSW optimizer, it is vital to find the right balance of the parameters that affect the convergence [42]. The GA optimizer is susceptible to premature convergence, where the population stops evolving, thus causing stagnation. Accordingly, SA optimizers are preferred for a global search over PSW and GA optimizers. Figure 17: An example of the joint AoA and Doppler log likelihood function, showing the ground truth point. In addition, the SO optimization process starts with the creation of a surrogate model, often based on the initial evaluations of the objective function. A set of initial samples is selected and evaluated using the surrogate model. The model gets refined as more points are sampled and evaluated. The same model is then utilized to propose new candidate solutions. These solutions are selected based on the predictions made by the surrogate model and evaluated using the true objective function. The proposal and evaluation of candidate solutions is repeated iteratively to converge to the best solution [43]. For the SO optimizer, the quality of surrogate model plays a critical role in the convergence. The success of the method depends heavily on the effectiveness of the surrogate model. Since it requires a well-built model, the SO optimizer is more problem-specific, whereas the SA optimizer is versatile and able to handle various types of optimization. Moreover, deterministic optimization methods involve calculating derivatives of the objective function. The derivatives provide information about the direction and rate of change of the function at a given point. Based on the information, a search direction is determined. The current solution is updated along the chosen search direction [44]. However, deterministic methods struggle in dealing with non-smooth, stochastic, or non-convex objective functions; thus, they can get stuck in local minima. The SA optimizer performs better than other deterministic methods for this stochastic optimization problem due to its ability to avoid the local minimum by allowing a move to a higher cost function with some probability [45]. To cater to the IoT device limitation of short and intermittent transmission, the simulation was run only using a short measurement time (\(\sim\)75 s) from the available satellites. Firstly, we illustrate the localization performance using a stochastic optimizer (simulated annealing) with an increasing number of satellites involved in the localization as depicted in Figure 18. The figure shows that as the number of contributing satellites increases, the localization error decreases. This figure also shows the impact of Doppler error on the localization error where, as expected, a higher Doppler error leads to a lower localization accuracy. To understand the impact of the number of measurement samples taken over separate time instances on the localization error, we depict an example in Figure 19, where the measurements are separated by 5 s. As expected, as more samples are used in the algorithm, the localization error is lower. With a combination of four or more satellites and 15 or more  
Figure 18: The median localization error with increasing satellites for the stochastic optimizer. Different curves represent the Doppler standard deviation \(\sigma_{\mathrm{d}}=5,10,25\), and \(50\) Hz and \(\sigma_{\Phi},\sigma_{\Theta}=1^{\circ}\); the results are averaged over 300 runs.