## 2 Signal and Correlator Model  
Figure 1 shows a generic correlator model, whose input is the undesired (interfering) signal, \(x_{1}(t-\Delta_{k})e^{i2\pi\theta}\), and the desired (victim) signal is \(x_{2}(t)\). There is a differential Doppler frequency, i.e., difference between Doppler frequencies, denoted as \(f\), between the undesired and desired signals. These two signals are correlated by mixing and integration over an arbitrary navigation data bit period, \([kT_{b},(k+1)T_{b})\), where \(T_{b}=20\) ms and \(k=0,1,2,\)....  
To simplify the problem without loss of generality, we focus on an individual undesired signal including a specific spreading code identified by PRN (Pseudo Random Noise) number, \(x_{1}(t)\). This undesired signal can be any one of the undesired signals \(x_{1}(t)\) in the input to the correlator in Figure 1. Therefore, the undesired signal without Doppler or time delay is modelled as:  
\[x_{1}(t)=\sum_{k=-\infty}^{\infty}d_{k}\sum_{u=0}^{19}\alpha_{u}\sum_{v=0}^{20 45}c_{v}p_{T_{c}}(t-[40,920k+2046u+v]T_{c}) \tag{1}\]  
where \(p_{T_{c}}\)(t) is a rectangular pulse of width \(T_{c}\):  
\[p_{T_{c}}(t)=\begin{cases}1&0\leq t<T_{c}\\ 0&else\end{cases} \tag{2}\]  
and \(c_{v}\in\{-1,1\}\) is spreading code chips with a chipping rate \(1/T_{c}=2.046\) MHz. The chip width is \(T_{c}\); \(d_{k}\) is navigation data bits with bit rate \(1/T_{b}=50\) bps, and the bit width is \(T_{b}\). \(\alpha_{u}\) is the NH code adopted for B1-I signal and takes a fixed pattern of (0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0), which spans one navigation data bit. The spreading code here is treated as a random periodic sequence of length  
Figure 1: Generic correlator model.  
2046 chips, with chips independent from each other. The navigation data bit stream is also treated as a random periodic sequence, but with an infinite length.  
In order to facilitate the development in the appendices, spreading code period, \(T\), is introduced, and there is an obvious relationship among \(T_{c}\), \(T\), and \(T_{b}\):\(T_{b}\) = 20\(T\) = 40,920\(T_{c}\). This relationship is illustrated in Figure 2. The receiver local code replica or desired signal is modelled similarly as:  
\[x_{2}(t)=\sum_{k=-\infty}^{\infty}\sum_{l=0}^{2045}c_{k}p_{T_{c}}(t-[2046k+l] T_{c}) \tag{3}\]  
using the same notation as Equation (1). The only difference between the desired and undesired code is that the undesired one has a time delay of \(\Delta\) seconds, which is normally within the range of \(\pm 20\) ms for BDS users on or near the horizon of the Earth. Both signals are cyclostationary, not WSS, and therefore, their characteristics are accurately represented using a two-parameter ACF [16]. Since the period of cyclostationarity is 20 ms, the model derived in the following sections only takes into account time delays ranging from 0 to 20 ms, i.e., \(0\leq\Delta\leq 20\) ms, which is equal to the time delay modulo 20 ms.  
In a real receiver, both the incoming signal and local replica have a Doppler shift. However, for simplicity and without loss of generality, we assume here that only the incoming signal has its Doppler shift, \(f\), while the local replica has zero Doppler. This is simply because the receiver carrier tracking loop is after all, tracking the _differential_ Doppler between the incoming and the local replica, not the absolute Doppler of each signal.