# Amplitude Modeling of Specular Multipath Components for Robust Indoor Localization
## 2 System and Environment Model
### Position Error Bound  
The channel model in (1) has been used in [27; 38; 47; 48] to analyze multipath-assisted indoor positioning systems and to develop algorithms for it. In [27; 48], the PEB has been derived for the signal model. It quantifies how the delay information of each SMC contributes useful information that can be exploited by a positioning algorithm. In this paper, the PEB is modeled as a spatial field for the current application environment, providing a measure for the achievable positioning accuracy in the given scenario. To develop this \"environment model\", first, the PEB is explained in detail.  
The PEB for the multipath-assisted positioning problem is the Cramer-Rao lower bound on the position error, the square-root of the trace of the upper left 2-by-2 submatrix of the inverse Fisher information matrix (FIM) \(\mathbf{J_{\theta}}\)[48]  
\[\mathcal{P}\{\mathbf{p}\}=\sqrt{\text{tr}\Big{\{}\left[\mathbf{J_{\theta}^{- 1}}\right]_{2\times 2}\Big{\}}}=\sqrt{\text{tr}\Big{\{}\mathbf{J_{p}^{-1}} \Big{\}}} \tag{4}\]  
where \(\theta=\left[\mathbf{p}^{T}\Re\mathbf{\kappa}^{T}\Im\mathbf{\kappa}^{T} \right]^{T}\) is a parameter vector containing the unknown location of the mobile agent, and the real and imaginary parts of the SMC complex amplitudes. \(\mathbf{J_{p}}\) is the equivalent FIM (EFIM) [26]. According to [48], we have  
\[\mathbf{J_{p}}=\frac{8\pi^{2}\beta^{2}}{c^{2}}\sum_{k=1}^{K}\text{SINR}_{k}( \mathbf{p})\mathbf{J_{r}}(\phi_{k}). \tag{5}\]  
in which \(\beta^{2}=\int f^{2}|S(f)|^{2}df\) is the effective bandwidth (\(S(f)\) is the Fourier transform of \(s(t)\)) and \(c\) is the speed of light. The summation shows that the information from individual SMCs is added up. Each SMC contributes information proportional to the \(\text{SINR}_{k}\) and the ranging direction matrix \(\mathbf{J_{r}}(\phi_{k})\)[26]. It is obvious that the more SMC information is available, the lower the PEB. The \(\text{SINR}_{k}(\mathbf{p})\) is defined as  
\[\text{SINR}_{k}(\mathbf{p})=\frac{|\alpha_{k}(\mathbf{p})|^{2}}{N_{0}+T_{p}S_ {v}(\tau_{k};\mathbf{p})} \tag{6}\]  
with \(T_{p}\) being the effective pulse duration [43]. It quantifies the interference of the DM, expressed by the PDP \(S_{v}(\tau_{k};\mathbf{p})\), onto the \(k\)-the SMC with energy \(|\alpha_{k}(\mathbf{p})|^{2}\). The ranging direction matrix describes the geometry of the setup. The delay information of the \(k\)-th SMC is directed towards the \(k\)-th VA, expressed by  
\[\mathbf{J_{r}}(\phi_{k})=\begin{bmatrix}\cos^{2}(\phi_{k})&\cos\phi_{k}\sin \phi_{k}\\ \cos\phi_{k}\sin\phi_{k}&\sin^{2}(\phi_{k})\end{bmatrix} \tag{7}\]  
where \(\phi_{k}\) is the angle from the corresponding VA to the mobile agent at position \(\mathbf{p}\).  
From (6) it can be seen that the range information contributed by the \(k\)-th SMC is quantified by \(\text{SINR}_{k}(\mathbf{p})\). Knowledge of \(\text{SINR}_{k}(\mathbf{p})\), therefore, allows to properly select and weigh the SMCs to be used by positioning algorithms. The idea of this paper is to model the \(\text{SINR}_{k}(\mathbf{p})\) of the SMCs as a spatial field in order to obtain a map of the expected PEB for the current application environment. Such an \"environment map\" will provide the side information needed to make a positioning algorithm more robust.