# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 3 Problem Description  
Based on the PRPS structure, this section analyzes the problems existing in the vector tracking process of pseudolite signals. The local replica \(S_{r}^{i}(t)\) of \(i\)-th pseudolite is defined as:  
\[S_{r}^{i}(t)=PN_{r}^{i}(t)\cos(2\pi f_{r}t)\cdot H_{r}^{i}(t), \tag{5}\]  
where \(PN_{r}^{i}\) represents the spreading code replica, \(H_{r}^{i}\) denotes the pseudo-random pulsing signal replica, and \(f_{r}\) is the frequency of local replica. The in-phase (I) integration of prompt correlator can be described in the following formula when the receiver tracks the signal of \(i\)-th pseudolite. \[ \begin{split} I_{p}(k)&=\frac{1}{T_{s}}\int_{t_{k} }^{t_{k}+T_{s}}S^{i}(t)\cdot S_{r}^{i}(t)\ dt\\ &=AD(k)R(\Delta\tau_{k})\operatorname{sinc}(\Delta f_{k}T_{s}) \cos\left[2\pi\Delta f_{k}(t_{k}+\frac{1}{2}T_{s})+\Delta\phi_{0}\right]+n_{I }^{i}\end{split} , \tag{6}\]  
where \(\Delta\tau_{k}\) and \(\Delta f_{k}\) denote the code phase and the frequency errors between the received signal and the local replica in the current epoch, respectively. \(\Delta\phi_{0}\) is the initial carrier phase error. \(R(\cdot)\) is the autocorrelation value of the spreading code, \(T_{s}\) is timeslot period, and \(n_{I}\) indicates the in-phase noise. The starting integration time \(t_{k}\) is  
\[t_{k}=kT_{f}+c_{k}^{i}T_{s}=(kN_{s}+c_{k}^{i})\cdot T_{s}. \tag{7}\]  
The prompt quada-phase (Q) integration of prompt correlator can be expressed as:  
\[Q_{p}(k)=AD(k)R(\Delta\tau_{k})\operatorname{sinc}(\pi\Delta f_{k}T_{s}) \sin\left[2\pi\Delta f_{k}(t_{k}+\frac{1}{2}T_{s})+\Delta\phi_{0}\right]+n_{Q}. \tag{8}\]  
The average carrier phase error measured by the discriminator is as follows. \[\overline{\Delta\phi_{k}}=\arctan\left(\frac{Q_{p}}{I_{p}}\right)\simeq 2\pi \Delta f_{k}(t_{k}+\frac{1}{2}T_{s})+\Delta\phi_{0}. \tag{9}\]  
The discriminator output approximates the average phase error between the received signal and the local replica in the active timeslot, as defined in Equation (9). Thus, \(\overline{\Delta\phi_{k}}\) is affected by \(\Delta f_{k}\), \(T_{s}\), \(\Delta\phi_{0}\), and \(t_{k}\). Since the tracking process replicates the signal parameters by filtering the discriminator outputs, it is vital to establish an appropriate output model. For simple analysis, this paper supposes that the tracking loop does not feedback the estimations to NCO. Thus, \(\Delta f_{k}\) would be constant in this process. \(\Delta\phi_{0}\) is the initial carrier phase error, and \(T_{s}\) is the timeslot period. Both of them are constant, while \(t_{k}\) is varying. Hence, it is necessary to focus on the analysis of the influence of \(t_{k}\) on \(\overline{\Delta\phi_{k}}\). Within each tracking period, the variation of carrier phase discriminator output can be expressed as:  
\[\
abla\overline{\Delta\phi_{k}}=\overline{\Delta\phi_{k}}-\overline{\Delta \phi_{k-1}}=2\pi\Delta f_{k}T_{k-1}, \tag{10}\]  
where  
\[T_{k-1}=t_{k}-t_{k-1}=(N_{s}+c_{k}^{i}-c_{k-1}^{i})\cdot T_{s}. \tag{11}\]  
\(T_{k-1}\) represents the intervals between the last active timeslot and the currently active timeslot, which is time-varying and predetermined by the pseudo-random permutation. However, for continuous CDMA signal, \(T_{k-1}\) is equal to the constant correlation integration time \(T_{f}\). The difference of \(\
abla\overline{\Delta\phi_{k}}\) in tracking continuous CDMA signal and PRPS is shown in Figure 3.  
The horizontal axis in Figure 3 represents the time, and the vertical axis represents the frequency error, so the shaded area represents \(\
abla\overline{\Delta\phi_{k}}\) (ignoring the scaling factor \(2\pi\)). The \(\
abla\overline{\Delta\phi_{k}}\) is time-invariant in tracking continuous CDMA signal shown in Figure 3a. Thus, with a fixed update period, the traditional tracking loop could accurately estimateand correct the frequency error, whereas Figure 3b shows that, when tracking PRPS, the \(\
abla\overline{\Delta\phi_{k}}\) is time-variant because of \(T_{k-1}\).