# Hybrid Approach for Indoor Localization Using Received Signal Strength of Dual-Band Wi-Fi
## 3 Preliminary  
A free-space PLM can be represented based on the Friis transmission formula [41].  
\[\mathrm{PL_{fs}}=20\log_{10}(d)+20\log_{10}(f)+20\log_{10}(\frac{4\pi}{c})-G_{ \mathrm{Tx}}-G_{\mathrm{Rx}}, \tag{1}\]  
where \(\mathrm{PL_{fs}}\) denotes the free-space loss on the decibel scale, \(d\) is the distance (m) between the transmitter and receiver, \(c\) is the speed of light, \(f\) is the frequency, and \(G_{\mathrm{Tx}}\) and \(G_{\mathrm{Rx}}\) are the antenna gains of the transmitter and receiver, respectively. The PLM in a typical LOS environment is similar to that in free space, and the path-loss difference between the 2.4 and 5.2 GHz bands only has an offset according to the antenna gain and the frequency difference. However, in an NLOS environment, the PLM becomes more complicated. The site-general models are presented in previous research for indoor environments, [6; 25; 26]. Among them, we consider the IEEE 802.11 specification [6], which suggest a 2-slope model for the indoor Wi-Fi systems. The general model can be represented as shown in (2).  
\[r(d)=\begin{cases}r(d_{0})-10\eta_{0}\cdot\log_{10}\frac{d}{d_{0}}+X_{0},\text{ if }d\leq d_{\mathrm{BP}}\\ r(d_{BP})-10\eta_{1}\cdot\log_{10}(d-d_{BP})+X_{1},\text{ if }d>d_{\mathrm{BP}}, \end{cases} \tag{2}\]  
where \(r(d)\) is the RSS at distance \(d\), and \(r(d_{0})\) is the RSS at the reference distance \(d_{0}\). \(d_{\mathrm{BP}}\) is the break-point distance, which divides two regions according to the distance from the transmitter. The two regions are modeled using different path-loss exponents, which are denoted by \(\eta_{0}\) and \(\eta_{1}\), and different shadowing factors, which are denoted by \(X_{0}\) and \(X_{1}\), respectively. This model is not appropriate for positioning, as it is a statistical model for analyzing the quality of service in general indoor environments. When radio waves propagate in a space with obstacles, they are affected by various phenomena, such as reflection, scattering, and diffraction. As a result, the path loss in an NLOS environment becomes extremely complex, and the average attenuation depends on the material of the obstacle and the radio wave frequency. For example, a cement wall, which is the most popular type of wall, shows significant differences in transmission and reflection properties for the 2.4 and 5.2 GHz frequencies [27]. Considering the influence of walls, [PERSON] et al. [28] presented the following effective wall loss model, which is consistent with practical indoor environments:  
\[r(d)=r(d_{0})-10\eta_{f}\cdot\log_{10}(\frac{d}{d_{0}})-\Sigma_{v=1}^{V}W_{v}, \tag{3}\]  
where \(\eta_{f}\) is the path-loss exponent, and \(W_{v}\) is the attenuation caused by the \(v\)-th wall in the propagation path. Focusing on the fact that the wall attenuation differs for each frequency band, the NLOS mode can be estimated. Figure 1 conceptually represents the attenuation tendency of the walls for two frequency bands. In free space, the attenuation in two bands is similar, but the difference in attenuation between two bands increases as there are more obstacles since the attenuation in 5.2 GHz is larger than that in 2.4 GHz for obstacles. In Figure 1, the path loss is expressed according to the number of walls as if the discrete steps are clearly separated; however, in a practical environment, this is not clear because the attenuation varies depending on the wall thickness, material, and map composition. The key point is that the attenuation in the 5.2 GHz band for the same geometric environment tends to be large at a certain rate compared to that in the 2.4 GHz band, and this can be an indicator of the severity of the NLOS condition. We assume that there is a nonlinear relation between the severity of the NLOS condition and the difference in attenuation between two bands.