## 4 Characterization and Performance Analysis
### Multipath Rang Error Envelopes  
Figure 1 shows the multipath range error envelopes for Narrow Correlator (NC) and High Resolution Correlator (HRC) discriminators. The NC technique exploits narrow chip spacing between early and late tracking correlators (usually equal to or less than 0.2 chips) instead of using standard correlators with 1 chip spacing. This results in mitigating the effect of multipath on receiver tracking range error as shown in [19]. The HRC discriminator uses more than three correlators in the tracking loop [20]. The idea behind the HRC tracking strategy is to form a linear combination of correlators that results in a correlation function narrower than the standard one. Assuming a coherent tracking procedure, a DLL discriminator for NC and HRC techniques can be defined as follows [2; 30]:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**BPSK(1)**} & \multicolumn{2}{c}{**BOC(1,1)**} \\ \cline{3-6}
**SQM Metric** & **Definition** & \begin{tabular}{c} **Nominal** \\ **Mean** \\ \end{tabular} & \begin{tabular}{c} **Nominal** \\ **Variance** \\ \end{tabular} & \begin{tabular}{c} **Nominal** \\ **Mean** \\ \end{tabular} & \
\begin{tabular}{c} **Nominal** \\ **Variance** \\ \end{tabular} \\ \hline
\begin{tabular}{c} (Single-sided or \\ asymmetric) \\ Ratio Metric \\ \end{tabular} & \(m_{1}=\frac{I_{\text{\tiny{AB}}}}{I_{0}}\) & 0.5 & \(\frac{0.75}{2(C/N_{0})T_{1}}\) & \(-0.5\) & \(\frac{0.75}{2(C/N_{0})T_{1}}\) \\ \hline Delta Metric & \(m_{2}=\frac{(I_{\text{\tiny{AB}}}-I_{\text{\tiny{AB}}})}{I_{0}}\) & 0 & \(\frac{2}{2(C/N_{0})T_{1}}\) & 0 & \(\frac{2}{2(C/N_{0})T_{1}}\) \\ \hline Double Delta Metric & \(m_{3}=\frac{(I_{\text{\tiny{AB}}}-I_{\text{\tiny{AB}}})-(I_{\text{\tiny{AB}}} -I_{\text{\tiny{AB}}})}{I_{0}}\) & 0 & \(\frac{1.6}{2(C/N_{0})T_{1}}\) & 0 & \(\frac{2.4}{2(C/N_{0})T_{1}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definition of SQM metrics and theoretical statistics.  
\[D_{NC}=\left(I_{-d_{trk}/2}-I_{+d_{trk}/2}\right) \tag{5}\]  
\[D_{HRC}=\left(I_{-d_{trk}/2}-I_{+d_{trk}/2}\right)-0.5\left(I_{-d_{trk}}-I_{+d_{ trk}}\right) \tag{6}\]  
where \(d_{trk}=0.2\)\((\)chips\()\) is considered to build up the tracking discriminator functions and extract the tracking range error envelopes for BPSK(1) and BOC(1,1) signaling schemes shown in Figure 1. To this end, a single reflection is considered and then the relative delay of the reflected signal (with respect to LOS signal) is swept through a range of values to evaluate DLL code misalignment and the consequent tracking range errors for in-phase and out-of-phase multipath components. These envelopes are shown in Figure 1 for 3 and 6 dB Signal-to-Multipath Ratio (SMR) values and for different tracking and signaling scenarios. Figure 1a also defines the approximate range of short, medium and long delay multipath considered here. Short-delay multipath is considered for reflected signal delays less than 0.1 chips (or about 30 m for the GPS L1 C/A case); medium-delay multipath is considered for the range of 0.1 to 0.75 chips and long-delay multipath covers the reflected signal delays longer than 0.75 chips.  
According to Figure 1, the error envelopes are almost the same for all tracking strategies and signaling schemes for short-delay multipath. For path delays longer than approximately 0.5 chips, the corresponding tracking range error for the BOC(1,1) modulation is about one third of that of the BPSK(1) signal. This is because of the difference in the shape of correlation peaks for BPSK(1) and BOC(1,1). For the HRC technique, the maximum tracking range error is the same for both signals under the effect of short-delay multipath. Regarding medium-delay multipath, the BPSK(1) outperforms the BOC(1,1) signal as the resulting ranging error is reduced to zero. However, under long-delay multipath, the tracking range error of the BOC(1,1) is significantly lower than that of the BPSK(1) signal.  
Figure 1: _Cont._