# Hyperbolic Asynchronous Method of a Radio Navigation Technique
## 4 Theoretical Investigations
### Simulation Software for Theoretical Investigations  
The algorithm of this simulator, based on the proposed method of object localization, was implemented in the Embarcadero XE Builder environment using proprietary software written in C++ language. Taking into account the subject of the research, a form of the main user interface window of this simulator was designed.  
The absolute error \(\delta_{d}[m]\) between the actual position of the user terminal (UT) is described by the coordinates \((x_{UT},y_{UT})\) and its position is determined by simulation calculations described by the coordinates \((x_{UT,all},y_{UT,all})\). This is expressed by the following well-known relationship:  
\[\delta_{d}=\sqrt{\left(x_{UT}-x_{UT,all}\right)^{2}+\left(y_{UT}-y_{UT,all} \right)^{2}}. \tag{17a}\]  
In addition, for selected sets of results of simulation calculations, commonly used accuracy measures have also been used in the form:  
the mean absolute error value \(ME_{\delta}\):  
\[ME_{\delta}=\frac{\sum\limits_{i=1}^{N}\delta_{di}}{N-1}, \tag{17b}\]  
medium square error \(MSE_{\delta}\):  
\[MSE_{\delta}=\frac{\sum\limits_{i=1}^{N}\delta_{di}^{2}}{N-1}, \tag{17c}\]  
root of a mean square error \(RMS_{\delta}\):  
\[RMS_{\delta}=\sqrt{MSE_{\delta}}, \tag{17d}\]  
where \(N\) is the number of crops tested.  
The course of simulation calculations can be presented in an algorithmic form consisting of individual functional blocks, the general form of which is shown in Figure 8, while the part showing the calculation of the position of the localized terminal (UT) is shown in Figure 9.  
The details of the functions of the individual blocks of this algorithm are as follows:  
1. geographical coordinates of the position of the reference stations shall be entered in WGS-84 [24],
2. selection of one of the two modes of operation of the simulator is made by pressing one of the two buttons described in the graphical user interface as the set area and the set route,
3. when the setpoint mode is selected, the position of this area is determined by the entered coordinates of the extreme reference stations, i.e., the minimum and maximum values of their coordinates, which, together with the resolution, are used to determine the rectangular grid nodes which are the locations of the calculated positions of the localized terminal (UT). The resolution is calculated according to the following relation: \[\Delta d_{min}[m]=\frac{v_{EM}\Big{[}\frac{m}{s}\Big{]}}{4B[Hz]}.\] (18) where \(B\) is the bandwidth of the operating band and \(v_{EM}=c\) is the speed of radio wave propagation in a propagation environment.
4. If the mode is selected, the specified route will appear as a sub-window for loading a text file with the coordinates of successive user terminal (UT) locations \((x_{UT},y_{UT})\), which are the reference values calculated during simulation tests.  
Figure 8: Algorithm of a computational simulator—general form.  
The calculation of location coordinates \((x_{UT,all},y_{UT,all})\) is carried out according to the description expressed in relation (12)-(15), which illustrates in more detail the form of the algorithm presented in Figure 9.
6. Then, in area mode, the calculated results are saved to files with the KML extension, in accordance with the Google Earth specification, for further analysis using map images.
7. In route mode, the calculated results are saved to a file with the txt extension for further analysis using a spreadsheet.  
The algorithm of estimating position from part (5) is presented in Figure 9.