## 5 VTL Implementation in an Asynchronous PL Navigation System
\begin{split}\begin{bmatrix}\rho_{u}^{\mathrm{i}}-^{\mathrm{i}} \
abla^{i}\hat{b}\\ \rho_{u}^{\mathrm{i}}-^{\mathrm{i}}\
abla^{i}\hat{b}\end{bmatrix}=& \begin{bmatrix}a_{x}^{1}&a_{y}^{1}&a_{z}^{1}&1\\ a_{x}^{2}&a_{x}^{2}&a_{z}^{2}&1\\ \vdots&\vdots&\vdots&\vdots\\ a_{x}^{N}&a_{y}^{N}&a_{z}^{N}&1\end{bmatrix}+\varepsilon\\ \rho_{u}^{N}-b^{\mathrm{i}}\end{bmatrix}+\varepsilon\end{split}  \tag{22}\]  
In this form, the receiver clock offset minus the satellite clock offset of PL 1 can be obtained as a navigation solution. Therefore, pseudorange estimation for generation of the VTL control input is possible, as shown in Equation (23). The velocity navigation equation, range-rate measurements and clock drift terms are used in the same way as in Equation (19) to derive the modified equation for the velocity navigation filter:  
\[\rho_{u}^{i}= \left(R^{i}-R_{u}\right)\cdot\hat{e}_{u}^{i}+B_{\rho_{u}}-b^{i}+ \varepsilon_{\rho_{u}}^{i}\] \[\equiv \left(R^{i}-R_{u}\right)\cdot\hat{e}_{u}^{i}+(B_{\rho_{u}}-b^{1}) +^{1}\
abla^{i}\hat{b}+\varepsilon_{\rho_{u}}^{i} \tag{23}\]  
The final navigation equation for VDFLL in PL navigation is obtained as Equation (24), containing eight states including six position and velocity terms, the receiver clock offset minus the satellite clock offset of PL 1 and the receiver clock drift minus the satellite clock drift of PL 1:  
\[\left[ \begin{array}{c}\rho_{u}^{i}-^{1}\
abla^{i}\hat{b}\\ \vdots\\ \rho_{u}^{N}-^{1}\
abla^{N}\hat{b}\\ \hat{\rho}_{u}^{i}-^{1}\
abla^{i}\hat{b}\\ \hat{\rho}_{u}^{N}-^{1}\
abla^{N}\hat{b}\\ \hat{\rho}_{u}^{N}-^{1}\
abla^{N}\hat{b}\\ \end{array} \right]=\left[ \begin{array}{ccccccccc}a_{x}^{1}&a_{y}^{1}&a_{z}^{ 1}&0&0&0&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{x}^{N}&a_{y}^{N}&a_{z}^{N}&0&0&1&0\\ 0&0&0&a_{z}^{1}&a_{y}^{1}&a_{z}^{1}&0&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&a_{z}^{N}&a_{y}^{N}&a_{z}^{N}&0&1\\ \hat{B_{u}}-b^{1}\\ \end{array} \right] \tag{24}\]  
With the new navigation equation and additional measurements, a block diagram of the VTL implementation for our asynchronous PL navigation system is shown in Figure 9. VTL was implemented in a software receiver and the reference receiver supplies satellite clock offset and drift estimates through the Hatch filter. As an implementation issue, it is important that the user and the reference receiver should acquire measurements at the same time in order to make the terms of the left side of Equation (24). To synchronize measurement acquisition time, both receivers are set to trigger measurement acquisition at the each end of C/A code period of the reference PL 1. For GPS, one C/A code period is 1 ms long equivalently converted to about 300 km. So, 1 ms resolution is valid in the use of PLs. Timing offset caused by the difference between a user to PL 1 distance and a reference to PL 1 distance is negligible  
Figure 9: VTL implementation with a new navigation equation model. in a several meters' coverage indoor PL system. If the system would be implemented in a wider area, the timing offset and transport delay from the reference to the user should be compensated using extrapolation of clock model. In this paper, both error terms could be negligible, because this is a meter level system implemented in a small indoor area.