# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 4 The Proposed VTL Architecture
### Pre-Filter Model  
Because the update period of the states in KF can be time-varying, while that in PLL and DLL can only be fixed, the KF structure is adopted for the IUP pre-filter. The state vector of linear KF describes the system time evolution [40] and is typically defined as an error vector of four parameters: code phase error \(\Delta\tau\) (unit: chips), carrier phase \(\Delta\varphi\) (unit: radians), carrier frequency error \(\Delta f\) (unit: Hz), and carrier frequency rate error \(\Delta a\) (unit: Hz/s). The state vector at the \(k\)-th active timeslot can be described as:  
\[\Delta\mathbf{X}_{k,1}=\left[\Delta\tau,\Delta\varphi,\Delta f,\Delta a\right]_{k}^ {T}, \tag{13}\]  
where subscript 1 indicates the pre-filter. The dynamic system model is related to the intervals, described in Equation (11). \[\Delta\mathbf{X}_{k,1}=\mathbf{F}_{k,1}\cdot\Delta\mathbf{X}_{k-1,1}+\mathbf{w}_{k-1,1}, \tag{14}\]  
where \(\mathbf{F}_{k,1}\) denotes the state transition matrix [39; 41] as:  
\[\mathbf{F}_{k,1}=\left[ \begin{array}{cccc}1&0&\beta T_{k-1}&\frac{1}{2}\beta T_{ k-1}^{2}\\ 0&1&2\pi T_{k-1}&\pi T_{k-1}^{2}\\ 0&0&1&T_{k-1}\\ 0&0&0&1\end{array} \right]_{k}. \tag{15}\]  
It should be emphasized that \(T_{k-1}\) is not constant. Instead, it is calculated according to the pseudo-random active timeslot intervals, avoiding the inaccurate output model of the discriminators, as described in Section 3. In Equation (14), \(\mathbf{w}_{k-1,1}\) indicates the process noise vector, and its covariance matrix is \(\mathbf{Q}_{k-1,1}\) which can be described as [39; 40]:  
\[\mathbf{Q}_{k-1,1}=\mathbf{Q}_{code}+\mathbf{Q}_{LOS}+\mathbf{Q}_{p}+\mathbf{Q}_{f}, \tag{16}\]  
where  
\[\mathbf{Q}_{code}=q_{code}\left[ \begin{array}{cccc}T_{k-1}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\quad\mathbf{Q}_{LOS}=q_{LOS}\left[ \begin{array}{cccc} \frac{T_{k-1}^{5}}{20}\beta^{2}&\frac{T_{k-1}^{5}}{20}\beta^{2}&\frac{T_{k-1} ^{4}}{8}\beta&\frac{T_{k-1}^{3}}{8}\beta&\frac{T_{k-1}^{3}}{6}\beta\\ \frac{T_{k-1}^{2}}{T_{k-1}^{2}}\beta^{2}&\frac{T_{k-1}^{3}}{T_{k-1}^{3}}& \frac{T_{k-1}^{3}}{8}&\frac{T_{k-1}^{3}}{6}\beta\\ \frac{T_{k-1}^{4}}{8}\beta&\frac{T_{k-1}^{4}}{8}&\frac{T_{k-1}^{4}}{8}&\frac{ T_{k-1}^{4}}{8}&\frac{T_{k-1}^{4}}{8}\\ \frac{T_{k-1}^{4}}{8}\beta&\frac{T_{k-1}^{4}}{6}&\frac{T_{k-1}^{4}}{2}&T_{k-1} \end{array} \right], \tag{17}\]  
\[\mathbf{Q}_{p}=S_{p}\omega_{RF}^{2}\cdot\left[ \begin{array}{cccc}T_{k-1}\beta^{ 2}&T_{k-1}\beta&0&0\\ T_{k-1}\beta&T_{k-1}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\quad\mathbf{Q}_{f}=S_{f}\omega_{RF}^{2}\cdot\left[ \begin{array}{cccc}\frac{T_{k-1}^{3}}{3}\beta^{2}&\frac{T_{k-1}^{3}}{3}\beta &\frac{T_{k-1}^{2}}{2}\beta&0\\ \frac{T_{k-1}^{3}}{3}\beta&\frac{T_{k-1}^{3}}{2}&0\\ \frac{T_{k-1}^{2}}{2}\beta&\frac{T_{k-1}^{2}}{2}&T_{k-1}&0\\ 0&0&0&0\end{array} \right], \tag{18}\]  
where \(q_{LOS}\) represents the random walk process driving the line-of-sight acceleration, \(q_{code}\) represents the code/carrier divergence, and \(S_{p}\) and \(S_{f}\) denote the phase and frequency random walk power spectral density due to the receiver oscillator, respectively. \(w_{RF}\) is the carrier frequency. Given the oscillator h-parameters, the clock noise spectral densities are obtained as [39; 42]:  
\[S_{p}=\frac{1}{2}h_{0} \tag{19}\] \[S_{f}=2\pi^{2}h_{-2}\]  
The discriminator outputs of the code phase \(\overline{\Delta\tau}\) and the carrier phase \(\overline{\Delta\phi}\) are used as the measurements in the pre-filter. So, the measurement vector \(\mathbf{Z}_{k,1}\) is:  
\[\mathbf{Z}_{k,1}=\left[ \begin{array}{cc}\overline{\Delta\tau},&\overline{\Delta \phi}\end{array} \right]_{k}^{T}. \tag{20}\]The measurement model can be expressed as follows:  
\[\mathbf{Z}_{k,1}=\mathbf{H}_{1}\cdot\Delta\mathbf{X}_{k,1}+\mathbf{v}_{k,1}, \tag{21}\]  
where \(\mathbf{v}_{k}\) is the measurement noise, and its covariance matrix \(R_{k,1}\) is  
\[\mathbf{R}_{k,1}=\left[ \begin{array}{cc}\sigma_{\Delta\tau}^{2}&0\\ 0&\sigma_{\Delta\phi}^{2}\end{array}