# Amplitude Modeling of Specular Multipath Components for Robust Indoor Localization
## 2 System and Environment Model  
### System Model  
We consider an indoor wireless communication environment with a fixed physical anchor (PA) at position \(\mathbf{a}_{1}\), as illustrated in Figure 1, and a mobile agent at position \(\mathbf{p}\). The agent is moved along a segmented trajectory. Some VAs \(\mathbf{a}_{k}\) up to order two are also shown, i.e., mirror images of the PA at the corresponding flat surfaces. The VAs establish a geometric model of SMCs originating from these surfaces. The VA-model can be used as a deterministic description of the delay, AOA, and AOD of these SMCs, given the position of the agent.  
We describe the radio channel from the PA located at position \(\mathbf{a}_{1}\) and the agent node located at position \(\mathbf{p}\) via a geometry-based stochastic channel model by:  
\[h(\tau;\mathbf{p})=\sum_{k=1}^{K}\alpha_{k}(\mathbf{p})\delta(\tau-\tau_{k}( \mathbf{p}))+\
u(\tau;\mathbf{p}) \tag{1}\]  
Figure 1: Floor plan of the evaluation scenario. Bold black lines denote walls, thick gray lines represent glass windows, other lines illustrate other materials. One blue cross represents the PA; orange circles denote VAs which were considered in the experimental evaluation. An agent moves along a trajectory segmented into seven parts indicated with distinct colors.  
The first term on the RHS of (1) is the sum of \(K\) SMCs with amplitude \(\alpha_{k}\) and delay \(\tau_{k}\) described as functions of the mobile agent position \(\mathbf{p}\). The delay \(\tau_{k}(\mathbf{p})\) corresponds to the distance between the agent and the PA (for \(k=1\)) or VA (for \(k=2,\ldots,K\)). That is \(\tau_{k}(\mathbf{p})=\|\mathbf{p}-\mathbf{a}_{k}\|/c\), where \(\mathbf{a}_{k}\) is the position of the respective PA or VA, and \(c\) is the speed of light. In Figure 2, the path lengths \(d_{k}(\mathbf{p})=\|\mathbf{p}-\mathbf{a}_{k}\|\) w.r.t various VAs at positions \(\mathbf{a}_{k}\) are plotted on top of the measured CIRs. The second term denotes the diffuse multipath (DM) \(v(\tau;\mathbf{p})\), modelled as a zero-mean Gaussian random process. We assume uncorrelated scattering along the delay axis, hence the autocorrelation function of \(v(\tau;\mathbf{p})\) is given by:  
\[K_{v}(\tau,u;\mathbf{p})=\mathbb{E}\{v(\tau;\mathbf{p})v^{*}(u;\mathbf{p})\}= S_{v}(\tau;\mathbf{p})\delta(\tau-u), \tag{2}\]  
where \(S_{v}(\tau;\mathbf{p})\) is the power delay profile (PDP) of the DM at agent position \(\mathbf{p}\). The DM process is assumed to be quasi-stationary in the spatial domain, i.e., \(S_{v}(\tau;\mathbf{p})\) does not change in the vicinity of \(\mathbf{p}\)[44].  
A base band signal \(s(t)\) is transmitted at carrier frequency \(f_{c}\). The complex envelope of the received signal then read:  
\[r(t;\mathbf{p})=\sum_{k=1}^{K}\alpha_{k}(\mathbf{p})s(t-\tau_{k}(\mathbf{p}))+ (s*v)(t;\mathbf{p})+w(t) \tag{3}\]  
The third term of (3) is the measurement noise \(w(t)\) modelled as additive white Gaussian noise (AWGN) with double-sided power spectral density \(N_{0}/2\). To use (3) for estimating parameters of SMCs, it is useful to minimize autocorrelation side lobes of \(s(t)\). A root-raised-cosine (RRC) pulse is thus considered for the transmitted signal with pulse duration \(T_{s}\). The energy of \(s(t)\) is normalized to one, i.e., \(\int|s(t)|^{2}dt=1\).  
Figure 2: Measured CIRs over the trajectory at a bandwidth of 2 GHz. The blue dash lines indicate expected paths of the SMCs corresponding to VAs up to order two, estimated using the SLAM algorithm from [38; 45] and refined as described in [46]. The locations of VAs are shown in Figure 1.