## 2 RSS Localization Algorithm Description  
The proposed algorithm is based on the path loss model Equation (1) given in Section 1. The equation has two unknowns as mentioned above, i.e., source transmitted power (\(P_{t}\)) and the path loss exponent (\(\alpha\)). Due to this, Equation (1) remains unsolvable. Firstly, to overcome the source transmitted power issue, the source transmitted power is eliminated by using a ratio of distance approach. In ratio of distance approach, one of the nodes in the receiving system is chosen as a reference node. The distance of all other receivers in the receiving systems is divided by the distance of the reference node. In this way, the uncertainty of the source  
Figure 1: PD localization techniques. transmitting power is eliminated. Each node in the receiving system is used as the reference node in turn and a mean estimated location is estimated from all estimated locations for an initially chosen value of path loss exponent. To do this, Equation (1) is converted into distance by re-arranging it in the form of distance as given in Equation (2)  
\[r_{i}=r_{1}\bigg{(}10^{\frac{P_{1}-P_{2}}{100}}\bigg{)} \tag{2}\]  
The coordinates of the receiver that receive the signal transmitted by the source are named as (\(x_{i},y_{i}\)). The RMS distance between the source, and the \(i\)th receiver is given by Equation (3) below:  
\[r_{i}^{2}=\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2} \tag{3}\]  
Then, Equation (2) can be simplified as shown in Equations (4) and (5) below:  
\[p_{i}=10^{\frac{P_{2}}{10}} \tag{4}\]  
\[p_{1}=10^{\frac{P_{1}}{10}} \tag{5}\]  
Equation (2) using Equations (4) and (5) is compared with Equation (3) as shown in Equation (6) below:  
\[\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}=\left(r_{1}\bigg{(}\frac{p_{ i}}{p_{1}}\bigg{)}^{\frac{1}{2}}\right)^{2} \tag{6}\]  
The distance ratio of the reference node to the \(i\)th node is given in Equation (7):  
\[\frac{r_{1}^{2}}{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}}=\bigg{(} \frac{p_{i}}{p_{1}}\bigg{)}^{\frac{2}{2}} \tag{7}\]  
The distance \(r_{1}\) of the reference node from the PD source is given in Equation (8):  
\[r_{1}=\sqrt{\left(x_{1}-x\right)^{2}+\left(y_{1}-y\right)^{2}} \tag{8}\]  
Equation (7) in the ratio form is given in Equation (9):  
\[\frac{\left(x_{1}-x\right)^{2}+\left(y_{1}-y\right)^{2}}{\left(x_{i}-x\right) ^{2}+\left(y_{i}-y\right)^{2}}=\bigg{(}\frac{p_{i}}{p_{1}}\bigg{)}^{\frac{2}{ 2}} \tag{9}\]  
By cross multiplying Equation (9), expanding the square and by rearranging all terms in the form of \(x,y\) and \(z\), where \(z=\ x^{2}+y^{2}\) is an extra variable, a system of matrices in the form of \(AX=b\) is obtained. The coefficients of \(x,y\) and \(z\) for \(i=2\), i.e., the second node in the receiver system are shown respectively, in Equations (10)-(12):  
\[x=\ 2p_{2}^{\frac{2}{2}}x_{2}-2p_{1}^{\frac{2}{2}}x_{1} \tag{10}\]  
\[r_{i}=r_{1}\bigg{(}10^{\frac{P_{1}-P_{2}}{100}}\bigg{)} \tag{2}\]0  
\[r_{i}=r_{1}\bigg{(}10^{\frac{P_{1}-P_{2}}{100}}\bigg{)} \tag{2}\]1  
If there are \(m\) receivers used to receive the signal, the matrix representation of the whole system is shown respectively, in Equations (13)-(15). The algorithm requires at least four nodes and one of the nodes will be used as the reference node. To enhance accuracy, the number of receiving nodes can increase to as many as fulfil the accuracy requirements. This means that multiple equations will be established i.e., an equation for each receiving node in the system, and a matrix form equation will be formed. The relevant matrices are shown below:  
\[A=\left[