# Gaussian Process Modeling of Specular Multipath Components
## 2 System and Signal Models
### Signal Model  
A baseband signal \(s(t)\) is transmitted at carrier frequency \(f_{\rm c}\). The complex envelope of the received signal then reads  
\[r(t;\mathbf{p})=\sum_{k=1}^{K}\alpha_{k}(\mathbf{p})s(t-\tau_{k}(\mathbf{p}))+(s\ast\
u)(t; \mathbf{p})+w(t), \tag{1}\]  
where the first term describes the superposition of \(K\) specular multipath components (SMCs), the second term describes the convolution of \(s(t)\) and the diffuse/dense multipath component (DMC) \(\
u(\tau;\mathbf{p})\) and the third term is the measurement noise \(w(t)\) modeled as AWGN with a two-sided power spectral density of \(N_{0}\). In order to minimize inter-symbol interference, root raised cosine (RRC) pulses are considered for the transmitted signal, with pulse width \(T_{s}\). The energy of \(s(t)\) is normalized to one, that is, \(\int|s(t)|^{2}dt=1\). The SMCsThe \(k\)th SMC is characterized by delay \(\tau_{k}(\mathbf{p})\) and complex amplitude \(\alpha_{k}(\mathbf{p})\). The delay is modeled deterministically as a function of the (varying) agent position \(\mathbf{p}\), that is, \(\tau_{k}(\mathbf{p})=\frac{1}{c}\|\mathbf{a}_{k}-\mathbf{p}\|=\frac{1}{c}d_{k}(\mathbf{p})\), where \(c\) is the propagation speed. The complex amplitude \(\alpha_{k}(\mathbf{p})\in\mathbb{C}\) is given as  
\[\alpha_{k}(\mathbf{p})=\frac{A(\phi_{k}(\mathbf{p}))\Gamma_{k}(\phi_{k}((\mathbf{p}))}{d_ {k}(\mathbf{p})}{\rm exp}\left\{-j2\pi f_{c}\frac{1}{c}d_{k}(\mathbf{p})\right\}, \tag{2}\]  
where \(A(\phi_{k}(\mathbf{p}))\) comprises the antenna gains of the TX and RX antennas and the path loss at a reference distance \(d_{0}=1\) m. The antenna gains are parameterized by the direction angle \(\phi_{k}(\mathbf{p})=\angle(\mathbf{p}-\mathbf{a}_{k})\) which is defined by the line connecting the agent position \(\mathbf{p}\) and the (virtual) anchor position \(\mathbf{a}_{k}\). We assume that the antenna gains are nearly constant over the observed frequency range and that the antennas are non-dispersive. \(\Gamma_{k}(\phi_{k}(\mathbf{p}))\) describes the reflection coefficient of the (flat) surface corresponding to VA \(\mathbf{a}_{k}\). For the line-of-sight (LOS) component, with no involved reflection, we set the coefficient \(\Gamma_{1}(\phi_{1}(\mathbf{p}))=1\). Equation (2) can be decomposed into a distance-dependent factor and an angle-dependent factor. The former consists of the path-loss of a (reflected) SMC in free space and a distance-dependent phase shift. The latter describes the angle-dependence of the agent and physical anchor antenna patterns (shown in \(A(\phi_{k}(\mathbf{p}))\)) and the dependence of the reflection coefficient \(\Gamma_{k}(\phi_{k}(\mathbf{p}))\) on the incident angle. Since we assume that the agent and physical anchor are at the same height, and the orientations of the agent and physical anchor antennas are fixed throughout the measurement, the incident angle (at the reflecting surface) and the angles at the agent and anchor (used to compute antenna gains) can be represented by one angle \(\phi_{k}(\mathbf{p})=\angle(\mathbf{p}-\mathbf{a}_{k})\) as shown in Figure 1. Thus, (2) can be rewritten as. \[\alpha_{k}(\mathbf{p})=\frac{\gamma_{k}(\phi_{k}(\mathbf{p}))}{d_{k}(\mathbf{p})}{\rm exp} \left\{-j2\pi f_{c}\frac{1}{c}d_{k}(\mathbf{p})\right\}, \tag{3}\]  
where \(\gamma_{k}(\phi_{k}(\mathbf{p}))=|\gamma_{k}(\phi_{k}(\mathbf{p}))|{\rm exp}\big{(}j \angle\gamma_{k}(\phi_{k}(\mathbf{p}))\) denotes the combined angle-dependence of the antenna patterns in \(A(\phi_{k}(\mathbf{p}))\) and the reflection coefficient \(\Gamma_{k}(\phi_{k}(\mathbf{p}))\).