# Amplitude Modeling of Specular Multipath Components for Robust Indoor Localization
## 2 System and Environment Model
### Radio Environment Map Using Gaussian Process Regression  
According to [34], a spatial field is the distribution of a physical quantity such as RF interference, pollution, temperature, humidity, light intensity, etc., which, if known, will foster the development of a new context-aware application and algorithms. In the context of radio systems, these spatial distributions are referred to as _radio environment maps_ (REMs). In this paper, we are interested in investigating the characteristics of individual SMCs, i.e., the direct link and the ones that resulted from reflections on flat surfaces. Spatial fields are obtained from mapping the agent positions with these SMC characteristics. In the UWB communication system, thanks to its very high time resolution, these SMCs can be resolved and predicted as studied in [43]. The SMC characteristics that are modeled in this paper are the SMC amplitudes and variances, thus the SINRs and the PEB can be predicted using (5) and (6). In Section 5, this information will be exploited to enhance the robustness of a multipath-assisted positioning algorithm. The SMC amplitudes and variances can also be used to predict the channel capacity and bit-error rate. The individual SMC metrics can then be useful for efficient beam selection and beam finding in mmWave wireless systems, exploiting prior information on potentially useful SMCs. To construct the REM for the PEB, we need a spatial field model for the SMC amplitudes and the interference power of DM, c.f. (6). A Gaussian Process model (GPM) is used to learn such an amplitude model from observed data, as described in [43]. In a training phase, the SMC amplitudes \(\hat{\alpha}_{k}(\mathbf{p})\) are estimated from the CIR. Given the known positions of the VAs and the agent, the expected delays \(\tau_{k}(\mathbf{p})=\frac{1}{c}\|\mathbf{a}_{k}-\mathbf{p}\|\) are deduced, yielding  
\[\hat{\alpha}_{k}(\mathbf{p})=\langle s(t-\tau_{k}(\mathbf{p})),r(t;\mathbf{p})\rangle. \tag{8}\]  
Next, the distance dependence is removed from the SMC amplitudes, yielding the normalized amplitude data \(\gamma_{k}(\mathbf{p})=\hat{\alpha}_{k}(\mathbf{p})\|\mathbf{a}_{k}-\mathbf{ p}\|\exp\left(j2\pi\frac{f_{c}}{c}\|\mathbf{a}_{k}-\mathbf{p}\|\right)\), which is assumed to depend on the direction angle \(\phi_{k}(\mathbf{p})=\angle(\mathbf{a}_{k}-\mathbf{p})\), only, describing the antenna pattern and the reflectivity of the reflecting surface [43]. This direction dependence is modeled by means of GPR. Several dataset are obtained, i.e., \(\mathcal{D}_{k}=\left\{\phi_{k}(\mathbf{p}),\gamma_{k}(\mathbf{p})\right\}_{ \mathbf{p}\in\mathcal{P}}\), where \(\mathcal{P}\) is the set of measured positions. Please note that whenever there are overlapping SMCs detected within a CIR, the corresponding amplitudes are neglected/discarded. In this paper, the dataset \(\mathcal{D}_{k}^{\text{abs}}=\left\{\phi_{k}(\mathbf{p}),\left|\gamma_{k}( \mathbf{p})\right|\right\}_{\mathbf{p}\in\mathcal{P}}\) is used, which deals with only the SMC amplitude's abscissa. According to [43], the hyper-parameters of the GPM are \(\theta_{k}^{\text{abs}}=\left[a_{k}^{\text{abs}},\sigma_{k}^{\text{abs}}, \sigma_{v,k}^{\text{abs}},\beta_{k}^{\text{abs}}\right]^{T}\), in which \(a_{k}^{\text{abs}}\) is the coherence angle, \(\sigma_{k}^{\text{abs}}\) is the standard deviation of the correlation kernel, \(\sigma_{v,k}^{\text{abs}}\) is the standard deviation accounting for the DM, and \(\beta_{k}^{\text{abs}}\) is the mean of the abscissa of the k-th SMC amplitude. From these data sets and hyper-parameters \(\theta_{k}^{\text{abs}}\) of the GPM, one can estimate the mean \(\mathbb{E}[\psi(\phi_{k}(\mathbf{p}^{*}))|\mathcal{D}_{k},\theta_{k}^{\text{ abs}}]\) and variance \(\mathbb{V}[\psi(\phi_{k}(\mathbf{p}^{*}))|\mathcal{D}_{k},\theta_{k}^{\text{ abs}}]\) of the normalized SMC amplitudes at any agent position of interest \(\mathbf{p}^{*}\)[43], and hence the \(\text{SINR}_{k}(\mathbf{p}^{*})\) and the PEB \(\mathcal{P}(\mathbf{p}^{*})\), which are the REMs investigated in this paper.