# Measurement and Analysis of RSS Using Bluetooth Mesh Network for Localization Applications
## 3 Radio Channel Characteristics and Models
### Path Loss and Shadowing  
According to the electromagnetic wave propagation theory, radio signals attenuate at a rate of 20 dB per decade of distance when propagating in free space, where the line-of-sight path exists between the transmitter and receiver devices and there are no clutters in the environment to induce multipath propagation effects [8]. However, in multipath environments, RSS measurements exhibit rapid fluctuations due to the small-scale fading effects. In contrast, the path loss of radio signals is defined as the difference between the transmitted signal power and the local average power of the received signal while the local average power is determined by averaging multiple RSS measurements over a short time duration and small movements in space to remove the small-scale fading effects [8; 19]. As the receiver moves away from the transmitter, in general, path loss of the received signal increases over distance. Path loss models are used to characterize the correlation between the local average RSS and the distance between the transmitter and receiver devices. Indoor path loss models are typically derived based on measurements [8; 18; 19]. One of the classical radio propagation channel models that has been widely used in practice is the log-normal shadowing path loss model given in the following [8; 19],  
\[L(d) =L_{0}+10\alpha\log_{10}(\frac{d}{d_{0}})+\chi, \tag{1}\] \[\chi \sim\mathcal{N}(0,\,\sigma_{\chi}^{2}),\]  
where the path loss \(L(d)=P_{t}[\mathrm{d}\mathrm{Bm}]-\bar{P}_{r}(d)[\mathrm{d}\mathrm{Bm}]\) is the difference between the transmitted signal power \(P_{t}\) and the average received signal power \(\bar{P}_{r}(d)\), both in dBm unit, for a pair of radio transceivers that are \(d\) meters apart. \(L_{0}\) is the path loss at the reference distance \(d_{0}\), and it is typically determined in the initial system characterization stage. For indoor environments, \(d_{0}\) is normally set to be 1 m. The distance-power gradient (a.k.a. path loss exponent) \(\alpha\) is used to model the average rate at which the path loss increases with distance. The parameter \(\chi\) is a zero-mean Gaussian random variable with standard deviation \(\sigma_{\chi}\), and it is used to model random shadowing effects over many measurement locations which have the same separation distance between the transmitter and receiver but different levels of clutters on the propagation path. The model parameters \(\alpha\) and \(\sigma_{\chi}\) can be determined by applying linear regression to measurement data [8; 19]. The values of \(\alpha\) and \(\sigma_{\chi}\) depend on signal frequency and building characteristics in the surrounding environment. Typical values of \(\alpha\) and \(\sigma_{\chi}\) can be found in references such as [8; 19]. For example, in indoor environments, the value of \(\alpha\) is typically between 1.6 and 1.8 in line-of-sight scenarios, while it is typically between 4 and 6 in no-line-of-sight scenarios [8]. For complex indoor environments, it is also possible to use partitioned path loss models to characterize different coverage regions more accurately with different values of \(\alpha\) and \(\sigma_{\chi}\). In particular, a two-segment partitioned path loss model can be defined as [19],  
\[L(d) =L_{0}+10\alpha_{0}\log_{10}(\frac{d}{d_{0}})+\chi_{0}, \tag{2}\] \[\chi_{0} \sim\mathcal{N}(0,\sigma_{\chi_{0}}^{2}),\]  
for \(d_{0}\leq d\leq d_{1}\), and  
\[L(d) =\bar{L}(d_{1})+10\alpha_{1}\log_{10}(\frac{d}{d_{1}})+\chi_{1}, \tag{3}\] \[\chi_{1} \sim\mathcal{N}(0,\sigma_{\chi_{1}}^{2}),\]  
for \(d>d_{1}\), where \(\bar{L}(d)=E\{L(d)\}\) is the average path loss at the distance \(d\) without shadowing effects. Extensive measurement and modeling efforts are needed to develop accurate path loss and shadowing models while one of the main challenges is that different environments may require dramatically different models, especially in complex indoor environments [19]. The RSS-based ranging approach directly depends on path loss and shadowing models to estimate distance.