# An Efficient Indoor Positioning Method with the External Distance Variation for Wireless Networks
## 2 System Models  
The propagation path loss model is used to describe the average power of the received signal or the average loss of the propagation path, which will decrease as the propagation distance increases. A free-space propagation model refers to the signal strength obtained without any obstacles between the transmitter and the receiver, such as satellite communication. The received signal is inversely proportional to the square of the distance. The free-space path loss (FSPL) is expressed as:  
\[\mathrm{FSPL}=\frac{P_{t}}{P_{t}}=\frac{\left(4\pi d\right)^{2}}{\lambda^{2}} \tag{1}\]  
where \(P_{t}\) (Watts) is the signal power at the transmitting point, \(P_{r}\) (Watts) is the signal power at the receiving point, \(\lambda\) (m) is the wavelength, \(d\) (m) is the distance between the transmitting point and the receiving point, and \(\mathrm{c}=3\times 10^{8}\mathrm{m/s}\) is the speed of light. The path loss model demonstrates that the average power of the received signal will attenuate exponentially as the distance increases. This phenomenon is common indoors and outdoors and has been widely used in many documents. Since the average power of the received signal decays exponentially, under normal conditions, the average loss \(PL(d)\) (dB) of the received power caused by the path will be proportional to the \(n\)-th power of the distance. For any transmission distance, the average path loss is expressed by [23]:  
\[PL(d)=\left(\frac{d}{d_{0}}\right)^{n} \tag{2}\]  
where \(PL(d)\) is the average loss of the received power, \(n\) is path loss index, \(d_{0}\) = 1 m is the distance that is usually the distance closer to the transmitting point, \(d\) (m) is the distance between the transmitting point and the receiving point, \(n\) will be changed with the environment, as shown in Table 1.  
It is mainly used to describe the change of the signal over a long distance (or time), and this change is described in a statistical way to estimate the coverage area of radio waves. Regarding the shadowing effect generated by the terrain and the mobile station, the general literature regards the shadowing effect as a random variable. The shadowing effect will make the received signal power a log-normal distribution as shown in Figure 1.  
In the general propagation path loss model, the received power loss caused by the path is influenced by distance. The mobile station will also be affected by neighbors during the movement. Buildings and terrain produce different levels of power loss as expressed by [16; 22]:  
\[PL(dB)=PL(d_{0})+10n\cdot\log\frac{d}{d_{0}}+x_{\sigma} \tag{3}\]  
where \(x_{\sigma}\) (dB) is a log-normal random variable with zero mean and the standard deviation \(\sigma\) = 3-8 dB. \begin{table}
\begin{tabular}{c c} \hline \hline
**Environments** & **Path Attenuation Index (\(n\))** \\ \hline Free space & 2 \\ Metropolitan area & 3–5 \\ Building & 4–6 \\ Factory & 2–3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The value of \(n\) in various environments. The phenomenon of small-scale fading is used to describe the rapid changes of radio signals after a short period of time/distance. These three phenomena exist at the same time in real channels and are rarely used at the same time due to the high complexity of using them at the same time. Generally speaking, if the research goal is the analysis of system capacity, radio wave coverage area, or handover algorithm, most of the time only the propagation path loss model and the large-scale propagation model will be used. The manual algorithm is related to the long-term and large-scale average signal condition of the system; if the research goal is the processing of the receiver's fundamental frequency signal, the small-scale propagation model is used for the fundamental frequency signal processing. When the signal is transmitted from the transmitter through the wireless channel, it then arrives at the receiving point. It encounters various barriers causing electromagnetic waves to produce reflection, refraction, scattering, and diffraction.