# Hybrid Approach for Indoor Localization Using Received Signal Strength of Dual-Band Wi-Fi
## 4 System model  
In this study, we assumed a Wi-Fi communication environment that uses a dual-band frequency. As shown in Figure 2, when the RSS at the target point \(\mathbf{x}=[x,y]^{\mathrm{T}}\) is measured by three or more APs, the RSS of the \(i\)-th AP is represented as \(\mathbf{r}_{i}=[r_{i}^{2.4},r_{i}^{5.2}]\), where \(r_{i}^{2.4}\) and \(r_{i}^{5.2}\) denote the RSS of 2.4 GHz and that of 5.2 GHz, respectively.  
The proposed hybrid localization scheme is illustrated in Figure 3. The basic structure is the same as that of the typical range-based localization method. The distances between APs and target devices are estimated based on the RSS measured at the ranging stage, and the position of the target node is estimated using the \(n\)-distance estimates of \(n\) AP locations. In the ranging stage, the conventional rule-based method estimates the distance by using the inversion of the PLM, which represents the RSS attenuation according to the distance. For example, using the PLM in (2), the distance estimate \(\tilde{d}\) for the RSS value \(r\) can be calculated as follows.  
\[\hat{d}(r)=\left\{\begin{aligned} d_{0}\cdot 10^{\frac{r(d_{0}) -r}{10\eta_{1}}},\text{ if }r\leq r(d_{\text{BP}})\\ d_{\text{BP}}+10^{\frac{r(d_{\text{BP}})-r}{10\eta_{1}}},\text{ otherwise.}\end{aligned}\right. \tag{4}\]  
Figure 1: Indoor path loss for the different frequency bands.  
Figure 2: Range-based localization scenario.  
In this study, we replaced this ranging part with a regression model that used deep neural networks. The details of the proposed method of using a neural network are explained in the next section. After the distances from each AP are estimated, the positions of the target nodes are specified in the positioning stage. By using the known positions of the fixed APs and the estimated distances, trilateration-based algorithms can be applied. Among the various trilateration-based algorithms, the iterative least-square (ILS) method estimates the target position to be close to the optimal solution [42,43]. The position estimate using ILS can be expressed as follows:  
\[\hat{\mathbf{x}}^{(k)}=\begin{bmatrix}\widehat{\mathbf{x}}^{(k)}\\ \widehat{y}^{(k)}\end{bmatrix}=\hat{\mathbf{x}}^{(k-1)}+\begin{bmatrix}\delta x \\ \delta y\end{bmatrix}, \tag{5}\]  
where  
\[\begin{bmatrix}\delta x\\ \delta y\end{bmatrix}=\begin{bmatrix}\frac{(x_{1}-\widehat{x}^{(k-1)})}{d_{1} ^{(k-1)}}&\frac{(y_{1}-\widehat{y}^{(k-1)})}{d_{1}^{(k-1)}}\\ \vdots&\vdots\\ \frac{(y_{N}-\widehat{x}^{(k-1)})}{d_{N}^{(k-1)}}&\frac{(y_{N}-\widehat{y}^{(k -1)})}{d_{N}^{(k-1)}}\end{bmatrix}^{+}\begin{bmatrix}\widehat{d}_{1}-d_{1}^{(k -1)}\\ \vdots\\ \widehat{d}_{N}-d_{N}^{(k-1)}\end{bmatrix}, \tag{6}\]  
and  
\[d_{i}^{(k-1)}=\sqrt{(x_{i}-x^{(k-1)})^{2}+(y_{i}-y^{(k-1)})^{2}}, \tag{7}\]  
where, \(\hat{\mathbf{x}}^{(k)}\) is the position estimate for the \(k\)-th iteration, and \((x_{i},y_{i})\) is the position of the \(i\)-th AP. The initial position estimate \(\hat{\mathbf{x}}^{(0)}\) can be set as the center of the map. The proposed scheme is a hybrid method that is similar to the existing rule-based method, but only the ranging part was replaced with the data-driven deep learning method. The risk of overfitting was minimized by the data-driven learning part. Ultimately, the position accuracy is increased by improving the ranging accuracy.