# An Optimized Vector Tracking Architecture for Pseudo-Random Pulsing CDMA Signals
## 2 Pseudo-Random Pusling Signal Structure  
The signal structure of PRPS is first introduced. The pseudolite positioning system is a ground-based satellite navigation system that uses a low-cost launcher to transmit GNSS-like signals. Therefore, the pseudolite system relies on the spreading code to achieve terminal equipment positioning functions. The difference is that the pseudolite signal adopts the pulsing signal structure to overcome the near-far effect. Figure 1 shows the modulation process of PRPS, which is equivalent to the pulse modulation of a continuous CDMA signal.  
The mathematical model of the received pseudo-random pulsing signal is expressed by the following formula.  
\[S^{i}(t)=\left[A^{i}D^{i}(t)PN^{i}(t-\tau^{i})\cos\left(2\pi(f_{IF}+f_{d}^{i})t +\varphi_{0}^{i}\right)\right]\cdot H^{i}(t-\tau^{i}), \tag{1}\]  
Figure 1: Schematic diagram of PRPS modulation.  
where \(H^{i}(t)\) is the pulse modulation sequences. The pseudo-random code for the \(i\)-th pseudolite \(A^{i}\) represents amplitude of the received signal, \(D^{i}(t)\) denotes the modulated data bits, \(\tau^{i}\) is the \(i\)-th transmitter-receiver time delay, \(f^{i}_{d}\) is the doppler shift, and \(f_{IF}\) and \(\varphi^{i}_{0}\) are the intermediate frequency (IF) and initial carrier phase, respectively.  
The pulse signal \(H^{i}(t)\) at the \(i\)-th transmitter is repeated by the basic pulse scheme \(h^{i}_{0}(t)\), which can be modeled as follows:  
\[H^{i}(t)=\sum_{m=0}^{+\infty}h^{i}_{0}\big{(}t-mT_{p}\big{)}, \tag{2}\]  
where \(T_{p}\) represents the repetition period of \(h^{i}_{0}(t)\), which can be expressed as:  
\[h^{i}_{0}(t)=\sum_{k=0}^{N_{f}-1}\Omega\Big{(}t-kT_{f}-c^{i}_{k}T_{s}\Big{)}. \tag{3}\]  
\(\Omega(t)\) is the single square wave function, which is  
\[\Omega(t)=\left\{\begin{array}{ll}1,&t\in[0,T_{s})\\ 0,&\text{otherwise}\end{array}\right.. \tag{4}\]  
In Equation (3), \(T_{s}\) denotes the duration of timeslot, \(T_{f}\) is one frame time which is equal to \(N_{s}T_{s}\), and \(N_{s}\) is the number of timeslots in one frame. The pulse duty cycle \(d\) is \(1/N_{s}\), and \(N_{f}\) indicates the frame number of the basic pulse pattern, so \(T_{p}=N_{f}T_{f}\). \(c^{i}_{k}\) represents the pulse position index in \(k\)-th frame. Meanwhile, \(c^{i}_{k}\) satisfies the following two requirements:  
1. The value of \(c^{i}_{k}\) ranges from 0 to \(N_{s}-1\).
2. The active timeslots of different pseudolites do not overlap.  
Generally, \(c^{i}_{k}\) is a pseudo-random permutation. It ensures the pulse pseudolite signal has better spectral properties, described in detail in Reference [18]. The pseudolite transmits signals in active timeslots where \(H^{i}(t)\) is 1 and keeps silent in silent timeslots where \(H^{i}(t)\) is 0. Thus, the pseudolite signals would not interfere with each other. The transmitting time of the pulse signal is time-varying and predetermined by a pseudo-random permutation.  
An illustration of the pulse pattern is shown in Figure 2. The horizontal axis represents timeslot index \(c^{i}_{k}\) ranging from 0 to \(N_{s}-1\) in one frame, and the vertical axis represents the frame index \(k\) ranging from 0 to \(N_{f}-1\). The green squares represent the active timeslots indexes.  
Figure 2: Illustration of the pulse pattern.