# Hyperbolic Asynchronous Method of a Radio Navigation Technique
## 2 Basic of the Hyperbolic Technique  
### TDOA Method  
As it is known, the operation of a hyperbolic radio navigation system, operating in synchronous mode based on the TDOA method, consists of determining the differences in the time \(\Delta t\) of impulse signals coming to the user terminal from individual reference stations. The equivalent of these time differences may be the phase differences \(\Delta\varphi\) of these signals at the antenna of this terminal. They are the starting point for determining the hyperbolic lines, at the intersection of which the terminal is located. This means that in practice the position of the user terminal is determined by measuring the phase differences \(\Delta\varphi\) of the signals received in the user terminal from the main station and three sub-stations forming the structure of the same chain. By converting the differences \(\Delta\varphi\) into differences in the distance \(\Delta d\) of the localized terminal from individual reference stations, individual hyperbolcan be determined as sets of fixed points of difference of these distances, at the intersection of which the user terminal is located. An example of creating such a hyperbolic in two-dimensional form for ground navigation applications is illustrated below in Figure 1 for a single set of two reference stations, i.e., main station \(\mathrm{G}\left(x_{\mathrm{G}},y_{\mathrm{G}}\right)\) and subordinate station \(\mathrm{P}\left(x_{\mathrm{P}},y_{\mathrm{P}}\right)\), where \(\left(x_{\mathrm{G}},y_{\mathrm{G}}\right)\) and \(\left(x_{\mathrm{P}},y_{\mathrm{P}}\right)\) are the known coordinates of their positions and are described in a two-dimensional rectangular coordinate system at this stage of consideration. Using the markings from Figure 1, it is possible to describe the constant difference in distance \(\Delta d(\Delta\varphi)\), which is the basis for determining individual positional hyperbolts, as follows:  
\[\Delta d(\Delta\varphi)=d_{\mathrm{G}}-d_{\mathrm{P}}=c\cdot\Delta t=\sqrt{ \left(x_{\mathrm{G}}-x_{\mathrm{U}T}\right)^{2}+\left(y_{\mathrm{G}}-y_{ \mathrm{U}T}\right)^{2}}-\sqrt{\left(x_{\mathrm{P}}-x_{\mathrm{U}T}\right)^{2 }+\left(y_{\mathrm{P}}-y_{\mathrm{U}T}\right)^{2}}. \tag{1}\]  
where  
* \(d_{\mathrm{G}}\) distance of the localized terminal from the main reference station,
* \(d_{\mathrm{P}}\) distance of this terminal from a subordinate reference station,
* \(\left(x_{\mathrm{U}T},y_{\mathrm{U}T}\right)\) desired coordinates of the location of the user terminal (UT) to be located. In practice, the TDOA method is based on the determination of three differences in the distance of the user terminal from the main reference station and from three individual subordinate reference stations. The differences \(\Delta d_{1}(\Delta\varphi_{1})\), \(\Delta d_{2}(\Delta\varphi_{2})\) and \(\Delta d_{3}(\Delta\varphi_{3})\) can be written down in general terms as \(\Delta d_{i}(\Delta\varphi_{i})\) for \(i\in\left\langle 1,2,3\right\rangle\). Then, based on (1), the above distance differences can be described by the following dependencies:  
\[ \begin{array}{ll}\Delta d_{i}(\Delta\varphi_{i})=d_{\mathrm{G}}-d_{\mathrm{ P}i}&=c\cdot\Delta t_{i}=\\ &=\sqrt{\left(x_{\mathrm{G}}-x_{\mathrm{U}T}\right)^{2}+\left(y_{\mathrm{G}}-y _{\mathrm{U}T}\right)^{2}}-\sqrt{\left(x_{\mathrm{P}i}-x_{\mathrm{U}T}\right)^ {2}+\left(y_{\mathrm{P}i}-y_{\mathrm{U}T}\right)^{2}}\text{ for }i\in\left\langle 1,2,3\right\rangle \end{array}  \tag{2}\]  
The scenario of locating a user terminal using a set of three positional hyperbolts from three pairs of reference stations, at the intersection of which the hyperbol is located, is illustrated in Figure 2.  
Figure 1: A method of determining hyperbolic positional lines for one pair of reference stations. In Figure 3, the time relationships of synchronized impulse signals transmitted from individual reference stations and their exemplary time shift \(\Delta t_{1}\), \(\Delta t_{2}\) and \(\Delta t_{3}\) are shown at the location of the localized terminal.