## 3 SSC and Equivalent White Noise Level  
The extent of the undesired signal's impact on the desired signal is characterized by an equivalent white noise level, which is derived from SSC, whose analytical forms are derived in this section. The \(k\)-th correlator output, \(y_{k}\), is a random variable. Its mean value is:  
\[E\{y_{k}\}=\int_{kT_{b}}^{(k+1)T_{b}}E\{x_{1}(t-\Delta)\}E\{x_{2}(t)\}e^{j2\pi f /t}dt=0 \tag{4}\]  
and its variance is:  
\[\begin{array}{l}E\Big{\{}|y_{k}|^{2}\Big{\}}=E\{y_{k}\cdot y_{k}^{*}\}\\ =E\Big{\{}\int_{kT_{b}}^{(k+1)T_{b}}\int_{kT_{b}}^{(k+1)T_{b}}x_{1}(s-\Delta)x_ {1}(t-\Delta)x_{2}(s)x_{2}(t)e^{j2\pi fs}e^{-j2\pi f}tdsdt\Big{\}}\\ =\int_{kT_{b}}^{(k+1)T_{b}}\int_{kT_{b}}^{(k+1)T_{b}}E\{x_{1}(s-\Delta)x_{1}(t -\Delta)\}E\{x_{2}(s)x_{2}(t)\}e^{j2\pi fs}e^{-j2\pi f}dsdt\\ =\int_{kT_{b}}^{(k+1)T_{b}}\int_{kT_{b}}^{(k+1)T_{b}}R_{11}(s-\Delta,t-\Delta) R_{22}(s,t)e^{j2\pi fs}e^{-j2\pi f}dsdt\end{array} \tag{5}\]  
Figure 2: Spreading code, navigation data bit, and NH code relationship for B1-I.  
where the dot within the curly brace means \"dot product\". The development of Equation (5) uses the definition of auto-correlation function of two complex signals. The two-parameter ACF's of both the undesired signal and receiver local replica are derived in Appendix A. The result is as follows:  
\[R_{22}(s,t)=\sum_{p=-\infty}^{\infty}p_{T_{c}}(t-2046 pT_{c}-vT_{c}) \tag{6}\]  
where \(R_{22}(s,t)\) is the two-parameter ACF of the local replica, the integer \(v\) is selected as:  
\[v=\text{mod}(\text{floor}(s/T_{c}),2046) \tag{7}\]  
and:  
\[R_{11}(s-\Delta,t-\Delta)=\sum_{p=0}^{19}p_{T_{c}}(t-\Delta-[40,920k+2046p+v] T_{c}) \tag{8}\]  
where \(R_{11}(s,t)\) is the two-parameter ACF of the undesired signal, \(\Delta\) is the time delay of the undesired signal and the integer \(k\) and \(v\) are selected as:  
\[k=\text{floor}((s-\Delta)/T_{b}) \tag{9}\]  
\[v=\text{mod}(\text{floor}((s-\Delta)/T_{c}),2046) \tag{10}\]  
where function \(\text{mod}(a,m)\) returns the remainder after division of \(a\) by \(m\) where \(a\) is the dividend and \(m\) is the divisor. Function \(\text{floor}(x)\) rounds \(x\) to the nearest integer less than or equal to \(x\).  
The development of Equations (8)-(10) has taken into account the special structure of the B1-I signal which has an NH code modulated on it: in subsection \"Desired signal ACF\" of Appendix A it can be seen that since the NH code is a fixed pattern finite-length code, it is treated as a deterministic variable here.  
Figure 3 illustrates an example of ACF for the desired signal \(R_{22}(s,t)\) with \(s\in[0,T_{c})\). For every fixed value of the first parameter, \(s\), this two parameter ACF is composed of 20 pulses of width \(T_{c}\) with unity amplitude, over arbitrary 20-ms correlation interval, which is equal to one data bit width. For example, if \(s\) falls within the time duration of the repetition of the first chip of the spreading code sequence, the ACF takes on a value of unity during each repetition of this chip and zero elsewhere. The same ACF results if \(s\) fell within repetition of the first chip, e.g., \(s\in[7T,7T+T_{c})\). The ACF for the undesired signal can be obtained by shifting the desired signal's ACF by \(\Delta\).  
The rest of this section is divided into three subsections. Analytical model for variance at the correlator output under three circumstances are derived: (i) data bits of the undesired and desired  
Figure 3: ACF for the undesired signal, which is a pulse train of 20 pulses, each of which has a width of \(T_{c}\). Note that this figure is not to scale.  
signal are aligned, (ii) data bits of these two signals are not aligned by an integer multiples of a code chip, and (iii) data bits of these two signals are misaligned by an arbitrary delay. This variance is then converted to the equivalent white noise level using SSC, which is the metric to measure how the self-interference affects receiver performance [17].