# Satellite-Based Localization of IoT Devices Using Joint Doppler and Angle-of-Arrival Estimation
## 3 System Model
### Doppler Model  
As we aim to make use of the Doppler shift pattern in the localization framework, it is important to adopt a practical measurement model which can also be empirically validated. Without loss of generality, we consider a stationary (or slowly moving) target where its ground velocity is negligible in comparison with the motion of the satellite with respect to the Earth's fixed inertial frame (ECI). The classical Doppler shift can be found from the radial velocity, which is computed by taking the derivative of the slant distance as follows,  
\[v_{\rm r}=\frac{\rho(t+\Delta t)-\rho(t)}{\Delta t}, \tag{3}\]  
where \(\rho\) is the slant distance between the satellite and the ground IoT device, \(\Delta t\) is a small time step, and \(t\) is the time variable. Accordingly, the true mean classical Doppler shift is obtained as \(\mu_{\rm d}=-\frac{v_{\rm r}}{c}f\), where \(c\) is the propagation speed (light speed in free-space), \(f\) is the center operating frequency, and \(v_{\rm r}\) is the true radial velocity as obtained in (3).  
Figure 2: Example of satellite constellation orbits using Walker-star pattern with 12 orbital planes and 24 satellites distributed equally on a single plane.  
In addition, we also investigated the effect of modeling the Doppler measurements using the relativistic Doppler effect for a receding source as follows  
\[f_{r}=\frac{f}{\gamma(1+v_{\rm r}/c)}-f, \tag{4}\]  
where \(\gamma=1/\sqrt{(1-v^{2}/c^{2})}\) is the Lorentz factor, and \(v\) is the velocity, which can be obtained by differentiating the position vector considering the satellite as the inertial frame of reference. The time dilation term is the difference between the classical and the relativistic Doppler.  
From the plot in Figure 3, we can observe that there is a difference between the two calculations with a 0.62 Hz maximum relative error. Accordingly, in this study, we maintained the usage of the classical Doppler formula for the Doppler measurements.  
We demonstrate in Figure 4 the Doppler patterns for different LEO orbit altitudes. It is evident that the Doppler shift has a distinct pattern at different altitudes. As the satellite approaches the ground transmitter, the Doppler shift curve slope increases, and it is 0 when the distance between the satellite and the ground IoT device is the smallest, as the transverse velocity is null. In contrast, as the satellite recedes, the Doppler shift curve slope decreases. The instance point where the slope changes is called the _inflection point_, which corresponds to the point where the satellite is nearest to the ground IoT device.  
For the measurement noise model of the Doppler, we adopted the Gaussian distribution based on its wide use in the literature [32; 33; 34] and also based on our own direct measurements from the NOAA satellites. In this model, the Doppler error can be formulated as a zero-mean Gaussian distribution with a certain standard deviation, \(\sigma_{\rm d}\) as shown in the upcoming Section 6.1. The error due to atmospheric effects can be excluded from the model due to its negligible effect compared to the satellite's velocity [34; 35]. Thus, the Doppler measurements with the presence of error can be modeled as,  
\[f_{\rm d}\sim\mathcal{N}\Big{(}\mu_{\rm d},\sigma_{\rm d}^{2}\Big{)}. \tag{5}\]  
Figure 3: Comparison between the classical and relativistic Doppler shift frequency measured from a polar orbiting LEO satellite at an altitude of 833 km. The highest relative error is at the inflection point (around 0.62 Hz).