# Design and Implementation of an Indoor Localization System Based on RSSI in IEEE 802.11 ax
## 2 Materials and Methods
Here, \(d\) represents the separation between the transmitter (unknown node) and the receiver (anchor nodes), which is fundamental for the localization methods being used. The IEEE 802.15 standards [29] include the following: low power consumption for battery-operated devices; short-range communication (from a few meters to tens of meters); scalability with support for topologies such as star, mesh, and tree topologies; interoperability to ensure compatibility between devices from different manufacturers; and security, with mechanisms like encryption and authentication. Additionally, these standards operate in different frequency bands (such as 2.4 GHz, 900 MHz, 868 MHz, and millimeter waves), offer variable data rates (from a few kbps to several Gbps), and are designed for specific applications like Wireless Personal Area Networks (WPANs), home automation, medical monitoring, industrial networks, and visible light communication. The flexibility and adaptability to various environments and needs are standout features of the IEEE 802.15 family. The IEEE 802.11 ax standard, or Wi-Fi 6, offers several advantages, such as a higher network capacity, improved spectral efficiency, and the ability to handle more devices simultaneously through technologies like OFDMA and MU-MIMO. This makes it ideal for high-density environments. It also improves the performance in congested networks and optimizes the battery life with the Target Wake Time (TWT) feature. However, it has some drawbacks, such as the need for compatible equipment, which can lead to additional costs. Furthermore, its benefits are limited to devices supporting this standard, and it remains susceptible to interference, especially in the 2.4 GHz band. Various localization techniques based on anchor nodes or reference points are used. These methods are essential in wireless communication networks, where precise localization is crucial for calculating distances and characterizing signal propagation. One of the most common approaches is trilateration [30]. This method requires knowing the positions of three anchor nodes, \((X_{1},Y_{1})\), \((X_{2},Y_{2})\), and \((X_{3},Y_{3})\), as well as the distances from the anchor nodes to the unknown node, \(d_{1}\), \(d_{2}\), and \(d_{3}\), respectively. The equation for a circle centered at \((X_{i},Y_{i})\) with the radius \(d_{i}\) is  
\[(x-X_{i})^{2}+(y-Y_{i})^{2}=d_{i}^{2} \tag{9}\]  
Expanding the polynomials for each circle,  
\[(x^{2}-2 xX_{1}+X_{1}^{2})+(y^{2}-2 yY_{1}+Y_{1}^{2}) =d_{1}^{2} \tag{10}\] \[PL=10\log\left(\frac{P_{t}}{P_{r}}\right)=-10\log\left(\frac{G_{t}G_{r}\lambda ^{2}}{(4\pi)^{2}d^{2}}\right) \tag{2}\]0 (11) \[PL=10\log\left(\frac{P_{t}}{P_{r}}\right)=-10\log\left(\frac{G_{t}G_{r}\lambda ^{2}}{(4\pi)^{2}d^{2}}\right) \tag{2}\]1  
Subtracting the equations to eliminate the quadratic terms \(x^{2}\) and \(y^{2}\), we obtain a system of two linear equations. This system can be written in matrix form as  
\[ \begin{bmatrix}A&B\\ D&E\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}C\\ F\end{bmatrix}  \tag{13}\]  
To solve the system, we compute the inverse of the coefficient matrix \(H\):  
\[H= \begin{bmatrix}A&B\\ D&E\end{bmatrix}  \tag{14}\]  
The inverse of \(H\) is given by  
\[H^{-1}=\frac{1}{|H|} \begin{bmatrix}E&-B\\ -D&A\end{bmatrix}  \tag{15}\]  
where the determinant of \(H\) is  
\[|H|=AE-BD \tag{16}\]  
Substituting into the general formula for the unknowns \(x\) and \(y\)  
\[x =\frac{EC-BF}{AE-BD} \tag{17}\] \[y =\frac{-DC+AF}{AE-BD} \tag{18}\]  
Another commonly used localization method is the Min-Max algorithm, a simple and widely used geometric method due to its ease of implementation. The distance \(d_{i}\) from each anchor node to the unknown node is calculated, and a square is drawn centered at the position of each anchor node, with sides of a length \(2\cdot d_{i}\). Then, the intersection of the generated squares is determined to find a common area where the unknown node is located at the center of the resulting quadrilateral.