# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 4 The Proposed VTL Architecture
### Navigation Filter Model
Therefore, the navigation filter can omit the process of estimating the pseudolite position during every epoch. At epoch \(k\), a first-order Taylor's expansion is used to linearize the relationship between the state and measurement vectors. According to Equations (38) and (39), \(\Delta\rho_{k}^{i}\) is  
\[\Delta\rho_{k}^{i}=-\mathbf{I}_{x,k}^{i}\cdot\Delta x_{k}-\mathbf{I}_{y,k}^{i}\cdot \Delta y_{k}-\mathbf{I}_{z,k}^{i}\cdot\Delta z_{k}+\Delta b_{k}, \tag{40}\]  
where \(\mathbf{I}_{k}^{i}=\left[ \begin{matrix}\mathbf{I}_{x}^{i},&\mathbf{I}_{y}^{i},&\mathbf{I}_{z} ^{i}\end{matrix} \right]_{k}^{T}\) is the LOS projection from the receiver to \(i\)-th pseudolite and can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{I}_{x,k}^{i}=\left(x^{i}-x_{k}\right)/r_{k}^{i} \\ \mathbf{I}_{y,k}^{i}=\left(y^{i}-y_{k}\right)/r_{k}^{i}\\ \mathbf{I}_{z,k}^{i}=\left(z^{i}-z_{k}\right)/r_{k}^{i}\end{array} \right.. \tag{41}\]  
Similarly, the pseudorange-rate error \(\Delta\rho_{k}^{i}\) is given by  
\[\Delta\rho_{k}^{i}=-\mathbf{I}_{k}^{i}\cdot\Delta\mathbf{v}_{k}+\Delta d_{k}, \tag{42}\]  
where \(\mathbf{v}_{k}=\left[\Delta v_{x},\Delta v_{y},\Delta v_{z}\right]_{k}^{T}\). According to Equations (40) and (42), the measurement equation can be obtained as:  
\[\mathbf{Z}_{k,2}=\mathbf{H}_{k,2}\cdot\Delta\mathbf{X}_{k,2}+\mathbf{v}_{k,2}, \tag{43}\]  
where \(\mathbf{v}_{k,2}\) is the measurement noise, and its covariance matrix is \(R_{k,2}\). The measurement matrix \(\mathbf{H}_{k,2}\) is given by  
\[\mathbf{H}_{k,2}=\left[ \begin{array}{cccccccc}-I_{x}^{1}&-I_{y}^{1}&-I_{z}^{1}& 0&0&0&1&0\\ -I_{x}^{2}&-I_{y}^{2}&-I_{z}^{2}&0&0&0&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ -I_{x}^{N}&-I_{y}^{N}&-I_{z}^{N}&0&0&0&1&0\\ 0&0&0&-I_{x}^{1}&-I_{y}^{1}&-I_{z}^{1}&0&1\\ 0&0&0&-I_{x}^{2}&-I_{y}^{2}&-I_{z}^{2}&0&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&-I_{x}^{N}&-I_{y}^{N}&-I_{z}^{N}&0&1\end{array} \right]_{2N\times 8}. \tag{44}\]  
Then, the posterior pseudorange and pseudorange-rate error estimations \(\Delta\hat{Y}_{k}\) can be achieved by:  
\[\Delta Y_{k}=H_{k,2}\cdot\Delta X_{k,2}. \tag{45}\]The feedback corrections to the NCO of each channel is calculated as follows:  
\[ \begin{split}\Delta r_{\text{NCO}k}^{i}&=\left(\hat{ \varrho}_{k}^{i}-\tilde{\rho}_{k}^{i}\right)/\lambda_{c}\\ \Delta f_{\text{NCO}k}^{i}&=\left(\hat{\varrho}_{k}^ {i}-\tilde{\rho}_{k}^{i}\right)/\lambda_{r}^{{}^{\prime}}\end{split}  \tag{46}\]  
where \(\tilde{\rho}_{k}^{i}\) and \(\tilde{\rho}_{k}^{i}\) are the estimated and measured pseudo-range at \(k\)-th epoch. \(\hat{\rho}_{k}^{i}\) and \(\tilde{\rho}_{k}^{i}\) are the estimated and measured pseudo-range rate at \(k\)-th epoch. The process of the PM navigation filtering can be summarized as follows, based on the system dynamic and measurement models established above. * Step1: Predict the measurements according to Equation (32) so that they can be sampled at the same time. * Step2: Estimate the priori state: \(\Delta\hat{\mathbf{X}}_{k|k-1,2}=\mathbf{F}_{k,2}\cdot\Delta\hat{\mathbf{X}}_{k-1,2}\). * Step3: Calculate the priori estimation covariance: \(\mathbf{P}_{k|k-1,2}=\mathbf{F}_{k,2}\cdot\mathbf{P}_{k-1,2}\cdot\mathbf{F}_{k,2}+\mathbf{Q}_{k-1,2}\). * Step4: Calculate the Kalman gain: \(\mathbf{K}_{k,2}=\mathbf{P}_{k|k-1,2}\mathbf{H}_{k,2}^{T}\left(\mathbf{H}_{k,2}\mathbf{P}_{k|k-1,2 }\mathbf{H}_{k,2}^{T}+\mathbf{R}_{k,2}\right)^{-1}\). * Step5: Estimate the posterior state: \(\Delta\hat{\mathbf{X}}_{k,2}=\Delta\hat{\mathbf{X}}_{k|k-1,2}+\mathbf{K}_{k,2}\left(\mathbf{Z} _{k,2}-\mathbf{H}_{k,2}\Delta\hat{\mathbf{X}}_{k|k-1,2}\right)\). * Step6: Compute the posterior estimation covariance: \(\mathbf{P}_{k,2}=(\mathbf{I}_{8\times 8}-\mathbf{K}_{k,2}\mathbf{H}_{k,2})\mathbf{P}_{k|k-1,2}\). * Step7: LOS Projection: \(\Delta\hat{\mathbf{Y}}_{k}=\mathbf{H}_{k,2}\cdot\Delta\hat{\mathbf{X}}_{k,2}\). * Step8: Feedback to NCO: as described in Equation (46).