# Evaluation of 5G Positioning Performance Based on UTDoA, AoA and Base-Station Selective Exclusion
## 2 Signal Model
### Observable Calculation
#### 2.2.2 Observable Based on Angle of Arrival  
The computation process of ranging observables presented in Section 2.2.1 (in terms of physical-layer abstraction) can also be extrapolated to other observables, e.g., angle observable. The computational complexity of the characterization process also scales with the degrees of freedom. For instance, the calculation of angle observables requires an additional dimension for the antenna array orientation with respect to the transmitter, leading to a four-dimensional (4D) interpolator in terms of angle error, antenna array orientation, SNR and propagation probability. In this work, the angular estimation is based on analytical models. Since NLoS measurements are expected to not improve the localization accuracy, this method focuses only on LoS measurements. The Cramer-Rao Bound (CRB) for AoA is used to assess the positioning capabilities in Additive White Gaussian Noise (AWGN) channel, presenting a feasible estimator. The CRB is a well-known lower bound that describes the maximum achievable accuracy of any unbiased estimator in the moderate- to high-SNR region. According to the 5G specifications, the uplink positioning procedure defined in AoA uses the angle of arrival from multiple-arraybase stations to compute the user position. A Uniform Planar Array (UPA) antenna with \(M\) antenna elements in the \(x\)-direction and \(N\) antenna elements in the \(y\)-direction, depicted in Figure 1, is used to estimate AoA (composed of the azimuth and elevation angles). The distance between adjacent antenna elements is assigned to be \(d=\lambda/2\), where \(\lambda=c/f_{c}\) is the received signal wavelength, \(c\) is the speed of light and \(f_{c}\) is the carrier frequency. Sectorized antennas were used, and the orientation of the BS array was chosen so that the array normal vector points approximately in the direction of the target. From Figure 1, one can distinguish the azimuth angle \(\phi\), which the angle between the \(x\)-axis and the orthogonal projection of the vector onto the \(xy\)-plane, and the elevation angle \(\theta\), which is the angle between the vector and its orthogonal projection onto the \(xy\)-plane. The boresight direction of the array is aligned with the positive \(z\)-axis. Knowing the array sectors of each BS sectorized antenna, it is possible to calculate the array direction of BS to UE, which is the angle of incidence at which the signal travels from the UE to the BS. The angle observable computation process starts first with the calculation of the rotation matrix around \(x\) axis for each of the orientations of BS arrays using the formula:  
\[\mathbf{R_{x}}= \begin{bmatrix}1&0&0\\ 0&\cos\zeta&-\sin\zeta\\ 0&\sin\zeta&\cos\zeta\end{bmatrix}  \tag{6}\]  
where \(\zeta\) is the BS array orientation of each of the sectors of the antenna. Then, the calculation of the UE-BS vector's position \(P^{\prime}_{n}\) after rotation is done through the following formula:  
\[\mathbf{P^{\prime}}_{n}= \begin{bmatrix}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\end{bmatrix} = \begin{bmatrix}x-x_{BS,n}\\ y-y_{BS,n}\\ z-z_{BS,n}\end{bmatrix} \mathbf{R_{x}}. \tag{7}\]  
Finally,  
\[v_{n,\zeta}=\cos^{-1}\left(\frac{\mathbf{v}\cdot\mathbf{P^{\prime}}_{n}}{\parallel \mathbf{v}\cdot\mathbf{P^{\prime}}_{n}\parallel}\right) \tag{8}\]  
is computed. This is the angle between \(\mathbf{P^{\prime}}_{n}\) calculated in (7) and \(\mathbf{v}=[0\ 0\ 1]^{T}\), which is the versor of direction of array (i.e., \(z\) axis). The chosen array orientation \(\zeta\) is the one that gives  
Figure 1: A uniform planar array antenna structure. the smallest \(v\) from the results.