# Hyperbolic Asynchronous Method of a Radio Navigation Technique
## 3 An Idea of the Asynchronous Radio Localization
### Asynchronous Algorithm  
The operation of this algorithm is illustrated by the example of three equivalent reference stations with known positions \(S_{1}(x_{S1},y_{S1})\), \(S_{2}(x_{S2},y_{S2})\) and \(S_{3}(x_{S3},y_{S3})\) where \((x_{S1},y_{S1})\), \((x_{S2},y_{S2})\) and \((x_{S3},y_{S3})\) express the position of these stations in a two-dimensional rectangular coordinate system. The exemplary arrangement of these stations together with the user localized terminal (UT) is illustrated in Figure 4.  
Each reference station is equipped with a transmitter-receiver radio system with an antenna with omnidirectional characteristics. This makes it possible to send signals to the other reference stations and to the local terminal, also equipped with an omnidirectional antenna. This also allows the terminal to receive signals transmitted by these stations. In addition, each reference station is equipped with a system for measuring the time difference between its reference time and that of  
Figure 4: Example of the arrangement of reference stations with a user terminal (UT) in an asynchronous location system. receiving signals from other reference stations. These signals contain information about the measured time differences, which allows you to determine the degree of synchronization between the reference stations. In addition, the information received at the reference stations is updated and sent out to all stations, including the localized terminal. This issue is described in the next section, in the protocol description of the proposed solution. In Figure 5, there is a sample time description of signals sent from all three reference stations, signals received at the station and signals received at the local user terminal (UT), together with measured time differences. These signals are presented in the form of impulses located on the time axis in places where the maximum autocorrelation function of pilot sequences located in the signals sent and received is present. This allows for an accepted way of accessing the electromagnetic spectrum in the form of DS-CDMA (Direct Sequence Code Division Multiple Access [22]). As shown in Figure 5a, reference stations operate in asynchronous mode. In this case, it is impossible to directly measure the time difference between the stations \(S_{1}\) and \(S_{2}\) (\(\Delta t_{21}\)) and between the stations \(S_{1}\) and \(S_{3}\) (\(\Delta t_{31}\)). It must be calculated based on the data provided, measured and calculated in the localized terminal (UT). Then the position can be calculated (\(x_{UT}\), \(y_{UT}\)) using the following matrix  
Figure 5: Example of the time status of signals sent by individual reference stations, received by the station \(S_{1}\) and in the user terminal, together with measured time differences. equation based on the Chan algorithm [19], formulated for the sample set of reference stations--see relation (5a):  
\[\left[ \begin{array}{c}x_{UT}\\ y_{UT}\\ d_{1}\end{array} \right]=\left\{[\Delta]^{T}\cdot[Q]^{-1}[\Delta]^{-1}\right\} \cdot[\Delta]^{T}\cdot[Q]^{-1}\cdot\left[\Delta_{prim}\right], \tag{12}\]  
where  
\[[\Delta]=-\left[ \begin{array}{c}\Delta x_{21}\Delta y_{21}(c\cdot\Delta t_{21 })\\ \Delta x_{31}\Delta y_{31}(c\cdot\Delta t_{31})\end{array} \right], \tag{13a}\]  
\[[Q]=\left[ \begin{array}{c}1\frac{1}{2}\\ \frac{1}{2}1\end{array} \right], \tag{13b}\]  
\[\left[\Delta_{prim}\right]=\frac{1}{2}\left[ \begin{array}{c}\left(c\cdot \Delta t_{21}\right)^{2}-\left(x_{S2}^{2}+y_{S2}^{2}\right)+\left(x_{S1}^{2}+ y_{S1}^{2}\right)\\ \left(c\cdot\Delta t_{31}\right)^{2}-\left(x_{S3}^{2}+y_{S3}^{2}\right)+\left(x_ {S1}^{2}+y_{S1}^{2}\right)\end{array} \right], \tag{13c}\]  
and  
\[ \begin{array}{c}\Delta x_{21}=x_{S2}-x_{S1},\\ \Delta y_{21}=y_{S2}-y_{S1},\\ \Delta x_{31}=x_{S3}-x_{S1},\\ \Delta y_{31}=y_{S3}-y_{S1},\end{array}  \tag{13d}\]  
The asynchronism (\(\Delta t_{21}\), \(\Delta t_{31}\)) is calculated in the localized UT terminal in the following way--see Figure 5:  
\[