# 2D Phase-Based RFID Localization for On-Site Landslide Monitoring
## 3 Theoretical Model
### Phase Error Model : Multipath, Phase Standard Deviation and Radiation Pattern
#### 3.3.2 Two Types of Phase Error  
We define the phase measurement error as the difference between the ideal LOS phase and the full received phase. This error can be divided into two contributions: the phase random deviation \(\sigma_{rdm}\) and the systematic phase bias \(\phi_{b}\), which are both consequences of multipath interference. Let us now consider these two error contributions separately. Previous investigations [18] have shown a direct relationship between antenna received power \(P\)(W) and phase random deviation \(\sigma_{rdm}\)(rad), using the same acquisition configuration (tag, interrogator, and communication protocol):  
\[\sigma_{rdm}=\frac{4\pi f}{c}\cdot 9.5\cdot 10^{-9}/\sqrt{P} \tag{12}\]  
where \(c\) is the light velocity and \(f\) the carrier frequency. This empirical relationship reproduces the phase error value of 0.04 rad used in the previous section. The received power greatly depends on propagation distance, but also on multipath interference, which is why \(\sigma_{rdm}\) is multipath-sensitive. The systematic phase bias \(\phi_{b}\) is defined as the difference between the ideal LOS phase \(\phi_{1}\) and the full received phase \(\phi_{full}\):  
\[\phi_{b}=arg(s_{1}^{2})-arg(s_{full})=\phi_{1}-\phi_{full} \tag{13}\]  
The phase bias obviously depends on multipath behaviour. In phase space, the two error contributions \(\sigma_{rdm}\) and \(\phi_{b}\) can be interpreted, respectively, as a scaling and translation operation on an ideal phase measurement distribution. Indeed, \(\sigma_{rdm}\) represents the width of the measurement error distribution, and the bias \(\phi_{b}\) represents the center of this distribution; compared to the LOS ideal measurement; the true measurement will thus be translated by \(\phi_{b}\) and scaled to a width of \(\sigma_{rdm}\). Assuming Gaussian behaviour for the measurement process, each antenna \(j\) will, hence, present a measurement distribution \(\phi_{j}\) following a normal law:  
\[\phi_{j}=\mathcal{N}(\phi_{b},\sigma_{rdm}^{2})\]  
These considerations can be applied in the phase-localization scheme presented in the previous section via a multi-antenna phase distribution. Let us define the scaling matrix \(S\) and the translation vector \(T\) as follows:  
\[S= \begin{bmatrix}\sigma_{1}&0\\ 0&\sigma_{2}\end{bmatrix} \]  
\[T= \begin{bmatrix}\phi_{b1}&\phi_{b2}\end{bmatrix} ^{\top}\]  
The entries of \(S\) originate from Equation (12) and the entries of \(T\) from Equation (13). They correspond to the values of phase random error and phase bias measured by each antenna (\(N_{a}=2\) in this simple scenario). Note that the phase random deviation values \(\sigma_{j}\) are different for each antenna for geometrical reasons; each antenna is in a different  
Figure 3: Schematic definition of the two-ray multipath model. The orange line represents the line-of-sight path with angle \(i_{1}\) and propagation distance \(r_{1}\). The blue line represents the reflected path with angle \(i_{2}\) and propagation distance \(r_{2}\). \(h_{tag}\) and \(h_{antenna}\) are the tag and antenna heights above ground. location, hence, the multipath and radiation patterns do not yield the same error values. The scaling \(S\) in phase space allows for a definition of the phase covariance matrix \(C_{\phi}\) :  
\[C_{\phi}=S\cdot S^{\top}\]  
\(C_{\phi}\) can be used in the singular value decomposition to compute the displacement error eigenvectors via the displacement covariance matrix \(C_{r}\) (see Equation (9)). The localization spot dimensions are, hence, fully described by the following covariance matrix in displacement space \(C_{r}\) :  
\[C_{r}=(K^{-1})^{\top}C_{\phi}K^{-1} \tag{14}\]  
On the other hand, the translation \(T\) induced by the phase bias corresponds to a translation \(dr_{b}\) in displacement space, obtained by:  
\[dr_{b}=K^{-1}\cdot T \tag{15}\]  
Equations (14) and (15) represent our best attempt to model the deviation from an ideal LOS phase measurement, taking into account the various phase measurement errors, and the geometry of the system.