## Materials and Methods
### Mathematical Model for TDOA Localization Technique  
A framework for TDOA localization is given below:  
An anonymous source transmitted an altered signal.  
The transmitted signal is captured by three or more receivers that are positioned around the source.  
To calculate the exact position of the source, the time of each captured signal at each receiver is calculated.  
The change in time of signal at each receiver is important, as it is used to calculate the distance variance between receivers.  
The time difference is calculated by hyperbolic lines.  
The connection points of hyperbolas show the target positioned by TDOA.  
Due to an error in calculated TDOA values, the connection points of hyperbolas do not always point out the target. Optimization algorithms are therefore used along with TDOA to correctly locate the target.  
The calculated model for the TDOA localization technique is described below:  
\((a,b)\) represents the exact position of the unknown sender. The sender can be an authorized user or attacker.  
\((a_{0},b_{0})\) represents the position of CRBS.  
\((a_{i},b_{i})\) represents the position of the ith SU.  
\(r_{i}\) indicates the interspace of the unknown sender with the ith SU.  
\(r_{0}\) indicates the interspace of CRBS with the unknown sender.  
\(r_{i,0}\) represents the real distance difference of the PUEA with the ith SU, where CRBS is the static node.  
Real distance difference measurements of the PUEA and the ith SU with the CRBS are given in (2).  
\[\begin{array}{l}r_{i,o}=r_{i}-r_{0}\\ =\sqrt{\left(a-a_{i}\right)^{2}+\left(b-b_{i}\right)^{2}}\\ -\sqrt{\left(a-a_{0}\right)^{2}+\left(b-b_{0}\right)^{2}}\end{array} \tag{2}\]  
where \(r_{i}=\sqrt{\left(a-a_{i}\right)^{2}+\left(b-b_{i}\right)^{2}}\) and \(r_{o}=\sqrt{\left(a-a_{0}\right)^{2}+\left(b-b_{0}\right)^{2}}\).  
Here i ranges from 1,..., N. N is the total number of SUs.  
Values obtained from the TDOA method contain Gaussian noise due to the surroundings. The variations in distance values with Gaussian noise are given in (3).  
\[\widehat{r_{i,0}}=r_{i,o}+n_{i} \tag{3}\]  
Here, \(\widehat{r_{i,0}}\) is the measured range of different distances. \(n_{i}\) is Gaussian error with 0 mean and \(\sigma^{2}\) variance. The fitness function for the localization problem is given in (4).  
\[f(\widetilde{y},\widetilde{z})=\sum_{i=1}^{N}\left(\widehat{r_{i,0}}-\sqrt{ \left(\widetilde{y}-y_{i}\right)^{2}+\left(\widetilde{z}-z_{i}\right)^{2}}+ \sqrt{\left(\widetilde{y}-y_{0}\right)^{2}+\left(\widetilde{z}-z_{0}\right)^{ 2}}\right)^{2} \tag{4}\]  
Here, \((\widetilde{y},\widetilde{z})\) represents the estimated location of the unknown sender, \(\sigma_{i}{}^{2}\) represents the variance at the ith SU, and \(\sigma_{0}{}^{2}\) represents the variance at the CRBS. Equation (5) is used to find the path loss value. The Hata model is used in this paper.  
\[\Delta L_{P}(dB)=\left[44.9-6.55\cdot Log(h)\right]\log\frac{r_{i}}{r_{0}} \tag{5}\]  
Here, \(\Delta L_{P}\) represents the Hata model and h represents the transmitter antenna height. The value of SNR\({}_{i}\) is given in (6).  
\[\mathrm{SNR_{i}}=\mathrm{SNR_{0}}-\Delta L_{P}(dB) \tag{6}\]Here, \(\text{SNR}_{\text{i}}\) represents SNR at the ith SU, and \(\text{SNR}_{0}\) represents the basic SNR at CRBS.