## 3 Background  
### Radio Signal Propagation  
A commonly-used model for radio signal propagation as a function of distance and time is the log-normal shadowing model (LNSM), as shown in the Equation (1). The variation of the RSS in space and time is partly a deterministic and partly a stochastic process [29], because the signal propagates through a multipath fading channel (radio signals from the transmitter arriving at the receiver through multiple paths). The attenuation in the power of the radio signal is defined as the path loss \(PL\) and is caused by many factors, such as distance (free space loss) and multipath propagation effects. In particular, all walls, ceilings and other objects that affect the propagation of radio waves will directly influence the signal strength and the directions from which the radio signals are received. The loss in the RSS can be regarded as a sum of deterministic effects (modeled using a logarithmic function) and stochastic effects (modeled using random Gaussian variables [30]), as shown in the equation below:  
\[PL_{d}=\underbrace{PL_{d_{0}}+10\cdot\eta\log_{10}\left(\frac{d}{d_{0}}\right) }_{\text{Deterministic effect}}+\underbrace{\mathcal{X}_{\sigma}}_{\text{ Stochastic effect}}\qquad\mathrm{[dB]} \tag{1}\]where \(\|d\|=\sqrt{(x_{t}-x_{r})^{2}+(y_{t}-y_{r})^{2}}\) is the Euclidean (\(l^{2}\)-norm) distance between the transmitter at \(p_{t}=(x_{t},y_{t})\) and the receiver at \(p_{r}=(x_{r},y_{r})\), \(PL_{d}\) is the loss (expressed in decibels) in the RSS at distance \(d\), \(PL_{d_{0}}\) is the path loss at a reference distance \(d_{0}\) and \(\eta\geq 2\) is an environment specific propagation constant. The decibel (dB) is a relative unit of power measurement, often used to quantify gain or loss in radio signal power (strength). \(\mathcal{X}_{\sigma}=\chi_{\sigma}+\sum\limits_{i=1}^{N}k_{i}L_{i}\) is the stochastic part of the RSS and is generally termed as a wide-sense stationary (WSS) process (the channel is time invariant). Here, the \(\chi_{\sigma}\) models the stochastic nature of the signal attenuation and can be represented using a zero-mean Gaussian random variable with a variance \(\sigma^{2}\), and \(L_{i}\) is the loss due to walls and ceilings where \(k_{i}\) is a parameter depending on the type of such obstruction. However (as we see later), this generalization is invalid when we include multipath fading effects because the multipath channel is time-variant [10]. Hence, we generalize the stochastic nature of the RSS variations with both shadowing (time-invariant) and multipath fading effects (time-variant). The RSS at the receiver is the difference between the transmitted power \(P_{T}\) and the path loss \(PL_{d}\) over a distance \(d\). Therefore, \(RSS_{d}=P_{T}-PL_{d}\). The path loss \(PL_{d}\) is composed of three main components, as shown in Equation (2);  
\[RSS(d,t)=P_{T}-PL_{d}^{fs}-PL_{d}^{sh}-PL_{(d,t)}^{mp}=RSS_{d_{0}}-\underbrace{ 10\eta\log_{10}(\frac{d}{d_{0}})}_{\text{Free-space path loss}}-\underbrace{ \Psi_{dB}(d)}_{\text{Shadowing}}-\underbrace{\Omega_{dB}(d,t)}_{\text{Multipath}} \tag{2}\]  
Here, \(PL_{d}^{fs}=PL_{d0}^{fs}+10\eta\log_{10}(\frac{d}{d_{0}})\) is the distance-dependent free-space path loss component due to environmental factors, such as the size and the shape of the environment. \(PL_{d}^{sh}\) is the shadow fading component of the path loss accounting for attenuation caused by objects (including walls and ceilings) in the environment. \(PL_{(d,t)}^{mp}\) is the multipath fading component accounting for attenuation caused by dynamics in the multipath channel. Free-space path loss is a slow process (large scale), whereas the fading due to shadowing and multipath effects are relatively medium-scale and fast processes. The path loss in the RSS due to shadowing effects \(PL^{sh}=\psi_{dB}\) is represented using a zero-mean Gaussian distribution (\(\psi\sim\mathcal{N}(0,\sigma_{\psi}^{2})\)) with a variance \(\sigma_{\psi}^{2}\).