# Satellite-Based Localization of IoT Devices Using Joint Doppler and Angle-of-Arrival Estimation
## 3 System Model
### Angle-of-Arrival Model  
A generalized direction-finding array should be capable of estimating the ground emitter direction (IoT device location) in a 2D (dimensional) angular space. We consider the azimuth, \(\Phi\), and the off-nadir angle, \(\Theta\), centered at the satellite in the north-east-down (NED) frame, where the xNorth axis is parallel to the Earth's surface pointing north along the meridian of longitude, the yEast axis is also parallel to the Earth's surface pointing east along the latitude, and the zDown axis is antiparallel to the Earth's surface normal at the satellite subpoint. The NED frame is shown in Figure 5, where \(\Phi\) is the angle obtained from the xNorth positive axis to the projection of the signal source on the xNorth-yEast plane, and off-nadir angle \(\Theta\) is the angle computed from the zDown axis (i.e., pointing towards the nadir) towards the signal's source.  
Figure 4: Doppler shift frequency at different LEO polar orbit altitudes (ranges from 200 to 1000 km).  
Figure 5: Illustration of the ground IoT device’s angle of arrival measured at satellite in north–east–down (NED) frame. Antenna boresight is assumed to be oriented towards the nadir.  
To convert the Earth-centered Earth-fixed (ECEF) coordinates to the NED coordinates so that the origin is defined at the satellite instead of the Earth's center, we need to apply a coordinate transformation matrix, \(\mathbf{M}\)[36] to the position vector, \(\mathbf{x_{a}}=[x_{a}\;\;y_{a}\;\;z_{a}]\), where \(\mathbf{M}\) is defined as follows:  
\[\mathbf{M}=\begin{bmatrix}-\sin\phi_{S}\cos\lambda_{S}&-\sin\phi_{S}\sin \lambda_{S}&\cos\phi_{S}\\ -\sin\lambda_{S}&\cos\lambda_{S}&0\\ -\cos\phi_{S}\cos\lambda_{S}&-\cos\phi_{S}\sin\lambda_{S}&-\sin\phi_{S}\end{bmatrix}, \tag{6}\]  
where \(\phi_{S}\) is the latitude and \(\lambda_{S}\) is longitude of the satellite and position vector is calculated as \(\mathbf{x_{a}}=\mathbf{M}(\mathbf{x_{G}}-\mathbf{x_{S}})\). \(\mathbf{x_{S}}\) and \(\mathbf{x_{G}}\) are the satellite and ground IoT device coordinates vectors in Earth-fixed Earth-centered frames, respectively.  
For the AoA measurement error, we adopt the common Kent distribution model which (for small errors) can be approximated into Gaussian additive measurement noise for both azimuth and elevation angles as follows [37],  
\[\mathcal{K}(\Phi,\Theta;\kappa,\beta,I_{3}) \approx\] \[\mathcal{N}\bigg{(}\Phi;0,\frac{1}{\kappa-2\beta}\bigg{)}\mathcal{ N}\bigg{(}\Theta;0,\frac{1}{\kappa+2\beta}\bigg{)}, \tag{7}\]  
where \(\kappa\) is the concentration parameter, \(\beta\) is the ovalness parameter such that \(0\leq\beta<\kappa/2\), and \(I_{3}\) is the identity matrix of size 3. In the context of the AoA measurement, \(\kappa\) relates to the spread of the measurements, whereas \(\beta\) represents the ellipticity of the measurement distribution. Alternatively, we can directly define \(\sigma_{\Phi},\sigma_{\Theta}\) as the standard deviation of azimuth and off-nadir measurement errors in the normal distribution, where \(\sigma_{\Phi}=\frac{1}{\kappa-2\beta}\) and \(\sigma_{\Theta}=\frac{1}{\kappa+2\beta}\). Note that in this model, the AoA measurement error due to satellite attitude, refraction, and other errors are not considered. This is to isolate the impact of the measurement uncertainty only.