## 5 Feasibility Study Through System Simulations
### Execution of the Simulations  
The inter-antenna spacing in phased-array antennas is typically half-wavelength. In simple phased-array antennas, the typical element spacing is optimized to the half-wavelength to deal with grating lobes [225; 226; 227]. This may be the case if this kind of antenna is repurposed as a 2D sensor array. However, irrespective of the requirements for communication purposes, it remains to be studied whether the antenna spacing can be exploited to obtain distinctive signal patterns that benefit fingerprinting-based positioning. In this study, we do not claim that the 2D sensor array has to be a phased-array antenna. Therefore, we relax the constraint on the half-wavelength inter-antenna spacing. However, we keep the antenna spacing close to the half-wavelength in a set of one of the cases considered below. In addition, we do not consider the actual antenna design aspects related to the construction of the 2D sensor array. In this context, we are working only with the numeric modules of what would be the equivalent of the RSSs, ignoring the phases and AoA. For a frequency of 2.6 GHz, the half-wavelength is approximately 57.7 mm. In the simulations, we consider an inter-sub-tile spacing that is in the same order of the half-wavelength of 2.6 GHz, but it is not exactly half-wavelength. We choose to initially work with an inter-sub-tile spacing given by \(S_{S}=50\) mm and \(S_{S}=70\) mm, which bounds the \begin{table}
\begin{tabular}{l c c l} \hline \hline
**Layer** & **Size** & **Activation Function** & **Notes** \\ \hline Convolution & 2048 & Rectified linear unit & Input: tensor of fingerprints from \(N_{SS}\times N_{SS}\) \\ (2D) & & & & sub-tiles (2D arrangement) by the size of \\  & & & & the set of fingerprints, \(N_{F}\). \\ Convolution & 2048 & Rectified linear unit & Convolution kernel \(3\times 3\); no padding; stride = 1. \\ (2D) & & & Convolution kernel \(3\times 3\); no padding; stride = 1. \\ Flatten & - & - & - \\ Dense & 2048 & Rectified linear unit & - \\ Dense & \(N_{T}\) & Softmax & Output: tile-class likelihood probability vector of size \(N_{T}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 13: CNN for pattern-matching method 4-MultFingIn-ClassOut. 57.7 mm half-wavelength. In addition, we work with an inter-sub-tile spacing given by \(S_{S}=30\) mm in order to explore the results for a smaller 2D sensor array. We note that in the radio channel model, Equation (7), the frequency contributes to a linear attenuation of the radio signal (in the dB domain). This attenuation is constant for a fixed frequency; therefore, it disappears when normalizing the datasets to the zero mean scaled by the standard deviation. This means that varying the frequency does not change the numeric value of the results in this study. Therefore, the channel model can be thought of as suitable for other frequencies but limited to scenarios with the same power loss coefficient and shadow fading characteristics. In this context, the results could be generalized to the widely used frequency of 2.4 GHz for scenarios with the same characteristics. The half-wavelength of 2.4 GHz is 62.5 mm, which is close to the inter-sub-tile distances considered. Simulations are carried out for different combinations of the parameters \(S_{S}\) (the side length of the sub-tile) and \(N_{SS}\) (the number of sub-tiles per side of the tile), executing the simulation flow presented in Section 5.6. The selected combinations of \(S_{S}\) and \(N_{SS}\) are listed in Table 14. The combinations of \(S_{S}\) and \(N_{SS}\) define different _tile sizes_, and are referred to as _tile arrangements_. The table also lists the total number of tiles required to cover the positioning area, \(N_{T}\), obtained from the function \(FTileCenters()\) (Pseudocode 1), and the number of NN parameters for each pattern-matching method, which are returned by the machine learning library for the selected NN structure.