# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 4 The Proposed VTL Architecture
### Pre-Filter Model
\right], \tag{22}\]  
where \(\sigma_{\Delta\tau}^{2}\) and \(\sigma_{\Delta\phi}^{2}\) are the noise standard deviations of the code phase and carrier phase discriminator outputs, respectively. The early minus late envelope and the four-quadrant arctangent are selected as code and carrier discriminators. The measurement variances for the above discriminators are given by Reference [39; 41]:  
\[\sigma_{\Delta\tau}^{2}=\frac{d}{4C/N_{0}\cdot T_{s}}\cdot\left[1+\frac{2}{(2- d)C/N_{0}\cdot T_{s}}\right], \tag{23}\]  
\[\sigma_{\Delta\phi}^{2}=\frac{1}{2C/N_{0}\cdot T_{s}}\cdot\left(1+\frac{1}{2C /N_{0}\cdot T_{s}}\right), \tag{24}\]  
where \(d\) is the spacing between early and late replica codes, and \(C/N_{0}\) is the carrier-to-noise ratio. The relationship between the states and the measurements can be modeled as follows:  
\[\overline{\Delta\tau_{k}} =\frac{1}{T_{s}}\int_{T_{k-1}}^{T_{k-1}+T_{s}}\left(\Delta\tau_{ k-1}+\beta\Delta f_{k-1}t+\frac{1}{2}\beta\Delta a_{k-1}t^{2}\right)dt \tag{25}\] \[=\Delta\tau_{k}-\frac{1}{2}\beta\Delta f_{k}T_{s}+\frac{1}{6} \beta\Delta a_{k}T_{s}^{2}\] \[\overline{\Delta\phi_{k}} =\frac{1}{T_{s}}\int_{T_{k-1}}^{T_{k-1}+T_{s}}\left(\Delta\varphi _{k-1}+2\pi\Delta f_{k-1}t+\pi\Delta a_{k-1}t^{2}\right)dt\] \[=\Delta\varphi_{k}-\pi\Delta f_{k}T_{s}+\frac{1}{3}\pi\Delta a_{ k}T_{s}^{2} \tag{26}\]  
According to Equations (13), (25), and (26), the measurement model can be reconstructed as:  
\[\left[ \begin{array}{c}\overline{\Delta\tau_{k}}\\ \overline{\Delta\phi_{k}}\end{array} \right]_{k}=\left[ \begin{array}{ccc}1&0& -\frac{1}{2}\beta T_{s}&\frac{1}{6}\beta T_{s}^{2}\\ 0&1&-\pi T_{s}&\frac{1}{3}\pi T_{s}^{2}\end{array} \right]\cdot\left[ \begin{array} []{c}\Delta\tau\\ \Delta\varphi\\ \Delta f\\ \Delta a\end{array} \right]_{k}+\mathbf{v}_{k,1}. \tag{27}\]  
Thus, the observation matrix \(\mathbf{H}_{1}\) in Equation (21) is  
\[\mathbf{H}_{1}=\left[ \begin{array}{ccc}1&0&-\frac{1}{2}\beta T_{s}&\frac{1}{6} \beta T_{s}^{2}\\ 0&1&-\pi T_{s}&\frac{1}{3}\pi T_{s}^{2}\end{array} \right]. \tag{28}\]  
After establishing the dynamic and measurement models, the KF including one-step prediction and measurement correction can be implemented [40]. * Prediction \[\Delta\mathbf{\hat{X}}_{k|k-1,1} =\mathbf{F}_{k,1}\cdot\Delta\mathbf{\hat{X}}_{k-1,1}\] (29) \[\mathbf{P}_{k|k-1,1} =\mathbf{F}_{k,1}\cdot\mathbf{P}_{k-1,1}\cdot\mathbf{F}_{k,1}+\mathbf{Q}_{k-1,1},\]  
where \(\Delta\mathbf{\hat{X}}_{k|k-1,1}\) is the priori state estimation, and \(\mathbf{P}_{k|k-1,1}\) is the priori estimation covariance matrix calculated by projecting the error covariance ahead. * Correction\[\mathbf{K}_{k,1} =\mathbf{P}_{k|k-1,1}\mathbf{H}_{k,1}^{T}\Big{(}H_{k,1}\mathbf{P}_{k|k-1,1}\mathbf{H} _{k,1}^{T}+\mathbf{R}_{k,1}\Big{)}^{-1}, \tag{30}\] \[\Delta\mathbf{\hat{X}}_{k,1} =\Delta\mathbf{\hat{X}}_{k|k-1,1}+K_{k,1}\Big{(}\mathbf{Z}_{k,1}-\mathbf{H}_{k,1}\Delta\mathbf{\hat{X}}_{k|k-1,1}\Big{)}.\] (31) \[P_{k,1} =\big{(}\mathbf{I}_{4\times 4}-\mathbf{K}_{k,1}\mathbf{H}_{k,1}\big{)}\mathbf{P}_{k|k -1,1}\]  
In Equations (30) and (31), \(\mathbf{K}_{k,1}\) denotes KF gains, which is the weight of the difference between the received and predicted measurements. The KF gains are then exploited to estimate the posterior state estimation \(\Delta\mathbf{\hat{X}}_{k,1}\) by including the measurement \(\mathbf{Z}_{k,1}\). The posteriori estimation of the covariance matrix can be derived from \(\mathbf{P}_{k|k-1,1}\). Finally, the posterior state estimations \(\Delta\mathbf{\hat{t}}_{k,1}\) and \(\Delta\mathbf{\hat{f}}_{k,1}\) of each channel are input into the navigation filter as measurements.