# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 4 The Proposed VTL Architecture
### Navigation Filter Model  
For PLPS, the posterior estimations of the pre-filter can only reflect the error before the end of the current active timeslot. Thus, there is likely to be a time difference among the measurements sampling time of the navigation filter, as shown in the gray shade in Figure 4. Moreover, since the active timeslot of each pseudolite is pseudo-random and does not overlap, the time difference is time-varying. If the posterior estimations of the pre-filter are directly utilized as the measurements of the navigation filter, the measurement errors will increase, reducing the tracking performance of VTL. A predicted measurements module is added into the navigation filter in this paper to solve this problem, as shown in Figure 5. The mathematical model of the predicted measurements module is described as follows. \[\Delta\rho^{i} =\lambda_{c}\cdot\left(\Delta\mathbf{\hat{t}}_{k}^{i}+\beta\Delta\bm {\hat{f}}_{k}^{i}\Delta\mathbf{t}_{k}^{i}+\frac{1}{2}\beta\Delta\mathbf{\hat{a}}_{k}^{ i}\Delta\mathbf{t}_{k}^{i}\right), \tag{32}\] \[\Delta\rho^{i} =\lambda_{r}\cdot\left(\Delta\mathbf{\hat{f}}_{k}^{i}+\Delta\mathbf{ \hat{a}}_{k}^{i}\Delta\mathbf{t}_{k}^{i}\right)\]  
where the superscript \(i\) indicates the \(i\)-th channel. \(\lambda_{c}\) and \(\lambda_{r}\) are the wavelength of pseudorandom-code and radio frequency carrier, respectively. \(\Delta\rho^{i}\) and \(\Delta\rho^{i}\) represent pseudo-range and pseudo-range rate errors that are the measurements of the navigation filter. \(\Delta\mathbf{t}_{k}^{i}\) denotes the intervals between the update time of the pre-filter of each channel and the input sampling time of the navigation filter. The measurement vector of the navigation filter is  
\[\mathbf{Z}_{k,2}=\left[\Delta\rho^{1},\Delta\rho^{2},\cdots,\Delta\rho^{N},\Delta \rho^{1},\Delta\rho^{2},\cdots,\Delta\rho^{N}\right]_{k}^{T}, \tag{33}\]  
where subscript 2 and \(k\) indicate the navigation filter and the \(k\)-th epoch. The system state vector consists of eight errors and is described as  
\[\Delta\mathbf{X}_{k,2}=\big{[}\Delta x,\Delta y,\Delta z,\Delta v_{x},\Delta v_{y},\Delta v_{z},\Delta b,\Delta d\big{]}_{k,2}^{T}, \tag{34}\]  
where \(\Delta x\), \(\Delta y\), \(\Delta z\), and \(\Delta v_{x}\), \(\Delta v_{y}\), \(\Delta v_{z}\) are the receiver position and velocity errors, \(\Delta b\) and \(\Delta d\) are clock bias and drift errors. The system state can be described as a first-order Gauss-Markov process, which is given by  
\[\Delta\mathbf{X}_{k,2}=\mathbf{F}_{k,2}\cdot\Delta\mathbf{X}_{k-1,2}+\mathbf{w}_{k,2}, \tag{35}\]  
where \(\mathbf{w}_{k,2}\) is process noise, and its covariance matrix is \(Q_{k-1,2}\). \(\mathbf{F}_{k,2}\) is the transition matrix, which can be expressed as follows. \[\mathbf{F}_{k-1,2}=\left[ \begin{array}{ccc}\mathbf{I}_{3\times 3}&\mathbf{T}_{f} \cdot\mathbf{I}_{3\times 3}&0_{3\times 2}\\ 0_{3\times 3}&\mathbf{I}_{3\times 3}&0_{3\times 2}\\ 0_{2\times 3}&0_{2\times 3}&\mathbf{K}\\ \end{array} \right]_{8\times 8}, \tag{36}\]where \(T_{f}\) is the frame period of PRPS defined in Equation (3), \(I_{3\times 3}\) represents the third order identity matrix, and \(K\) is  
\[\mathbf{K}=\left[ \begin{array}{cc}1&T_{f}\\ 0&1\end{array} \right]. \tag{37}\]  
The pseudorange equation can be modeled as follows for the \(i\)-th pseudolite. \[\rho_{k}^{i}=r_{k}^{i}+b_{k}+n_{k}^{i}, \tag{38}\]  
where \(\rho_{k}^{i}\) is the corrected pseudorange. \(b\) represents receiver clock bias. \(n_{k}^{i}\) represents pseudo-range noise. \(r_{k}^{i}\) indicates the distance between the receiver and pseudolite, which is given by  
\[r_{k}^{i}=\sqrt{\left(x^{i}-x_{k}\right)^{2}+\left(y^{i}-y_{k}\right)^{2}+ \left(z^{i}-z_{k}\right)^{2}}, \tag{39}\]  
where \(x^{i}\), \(y^{i}\), and \(z^{i}\) represent the position of \(i\)-th pseudolite. \(x_{k}\), \(y_{k}\), \(z_{k}\) indicates the receiver position at \(k\)-th epoch. Because pseudolites are built on the ground, this paper assumes that their positions are fixed.