# BLE Signal Reception and Localization Performance with Varying Receiver and Beacon Setups
## 4 Overview of Indoor Localization Accuracy Based on the RBF Method  
This section focuses on indoor localization accuracy based on collected data. The Radial Basis Function (RBF)-based methodology used for location estimation relies on  
Figure 10: Received signal strength characteristics received from the ceiling GWs.  
signal strength measurements [13]. This approach employs a matrix framework, where a signal matrix (S) represents the data, a two-dimensional spatial matrix (P) corresponds to the physical layout, and a coefficient matrix captures the positional relationships with the signals. The RBF is utilized as a distance-dependent function, enabling interpolation between known signal points for the localization of new signal measurements.  
The overall process is structured into two distinct phases: modeling and prediction. In the modeling phase, data acquisition is initiated through the collection of RSSI measurements from the sample points within the indoor environment. Subsequently, the obtained data are utilized to compute matrices (G), portraying the RBF effect of each sample point on all others. The modeling phase is finished with the computation of the coefficient matrix (C), which enables the position estimation in the prediction phase.  
\[C=P\cdot G(S)^{-1} \tag{1}\]  
where \(G(S)\), in this study, considers the normalized signal values and represents the Euclidian distances of each sample point from all the other points. \(M\) is the number of sample points. measured \(s\) is the signal strength measurement at a particular point.  
\[G=\begin{bmatrix}g\big{(}\big{\|}s^{1}-s^{1}\big{\|}\big{)}&\cdots&g\big{(} \big{\|}s^{j}-s^{1}\big{\|}\big{)}&\cdots&g\big{(}\big{\|}s^{M}-s^{1}\big{\|} \big{)}\\ \vdots&&\vdots&&\vdots\\ g\big{(}\big{\|}s^{1}-s^{2}\big{\|}\big{)}&\cdots&g\big{(}\big{\|}s^{j}-s^{i} \big{\|}\big{)}&\cdots&g\big{(}\big{\|}s^{M}-s^{i}\big{\|}\big{)}\\ \vdots&&\vdots&&\vdots\\ g\big{(}\big{\|}s^{1}-s^{M}\big{\|}\big{)}&\cdots&g\big{(}\big{\|}s^{j}-s^{M} \big{\|}\big{)}&\cdots&g\big{(}\big{\|}s^{M}-s^{M}\big{\|}\big{)}\end{bmatrix} \tag{2}\]  
The position prediction \(P(s)\) involves the acquisition of RSSI measurements at the test points and the subsequent calculation of a vector \(g(s)\) representing the RBF distances of the measured signal relative to all signals in the database. The actual position estimation is then executed by multiplying this vector with the coefficient matrix (C).  
\[P(s)=C\cdot g(s) \tag{3}\]  
The evaluation of indoor localization errors in this study focuses on three configurations: wall-mounted gateways, ceiling-mounted gateways, and a combined wall and ceiling gateway setup. The performance is systematically examined across five body positions: front pocket, back pocket, top pocket, shoulder, and wrist.  
In the first two configurations, multiple models were constructed using all possible combinations of wall and ceiling gateway data, varying the number of GWs from 2 to 5 in each configuration. In the third configuration, models were built using data from all GWs, considering all possible combinations of wall and ceiling GWs, with the number of GWs varying from 2 to 10.