# A Low-Ambiguity Signal Waveform for Pseudolite Positioning Systems Based on Chirp
## 4 Numerical Results
### Multipath Resistance  
Tang developed an explicit expression for multipath error envelope (MPEE), which reflects the multipath mitigation performance of GNSS signals, expressed as follows [29]:  
\[\varepsilon(\tau)=\frac{\pm a\int_{-\beta_{f}/2}^{\beta_{f}/2}G_{\mathrm{s}}(f )\sin(2\pi f\tau)\sin(\pi fD)df}{2\pi\int_{-\beta_{f}/2}^{\beta_{f}/2}fG_{ \mathrm{s}}(f)\sin(2\pi fD)[1\pm a\cos(2\pi f\tau)]df}(\mathrm{second}) \tag{20}\]  
where \(\tau\) is multipath delay, \(a\) is the amplitude ratio of multipath to direct path; when the phase difference between direct path and reflected path is 0 or 180 degrees, '\(\pm\)' is '+' or '\(-\)', respectively. Based on (20), Figure 7 computes MPEEs of different modulations for a non-coherent delay DLL with an ELS of 5.86 m and a multipath to direct ratio (MDR) of \(-6\) dB. As the figure shows, the users can achieve smaller MPEEs with a larger PDB. The MPEEs for multipath delay below 7 m and 5 m are practically equal for all the signals with PDBs of 24 MHz and 12 MHz, respectively. For the first situation, with a PDB of 12 MHz, ChPN(5,1) has the smallest MPEEs, followed by ChPN(4,2) and ChPN(3,3); when the PDB increase to 24 MHz, with a multipath delay less than 30 m, ChPN(3,3) has the smallest MPEEs followed by ChPN(5,1) and ChPN(4,2); ChPN(5,1) has the best overall multipath resistance performance. For the second situation, BOC(5,1) has smaller MPEEs than ChPN(5,1) when  
Figure 6: Tracking errors for ChPN(5,1), ChPN(4,2), ChPN(3,3), ChPN(10,1) and BOC(5,1).  
multipath delay below 40 m, MPEE curves of ChPN(5,1) converge much faster than that of BOC(5,1), ChPN(5,1) has a better overall multipath resistance. For the third situation, ChPN(10,1) is better than ChPN(5,1) in MPEEs.  
Figure 8 shows the average worst-case multipath errors (AWME) for various signals, which is drawn with the following model and the same conditions with Figure 7:  
\[\varepsilon_{a}=\frac{1}{\tau}\int_{0}^{\tau}\Bigg{[}\frac{abs\Big{(} \varepsilon(\tau)|_{\varphi=0}\Big{)}+abs\Big{(}\varepsilon(\tau)|_{\varphi=1 80\varphi}\Big{)}}{2}\Bigg{]}d\tau(\text{second}) \tag{21}\]  
where \(\varphi\) carrier phase difference of multipath and direct path. With the PDB increasing, the AWMEs become smaller. For the first situation, ChPN(5,1) has the smallest AWMEs followed by ChPN(4,2) and ChPN(3,3) with a PDB of 12 MHz; when the PDB is 24 MHz, ChPN(3,3) has the smallest AWMEs with a multipath delay below 50 m, ChPN(5,1) owns the smallest AWMEs if multipath delay exceeds 50 m, ChPN(4,2) performs worst over all multipath delay. For the second situation, BOC(5,1) has the smaller AWMEs than ChPN(5,1) with multipath delay less than 80 m and 50 m when the PDBs are 12 MHz and 24 MHz, respectively. For the third situation, ChPN(10,1) has smaller AWMEs than ChPN(5,1). Moreover, due to a PDB of 12 MHz is not sufficient to include the frequency components of ChPN(10,1), the AWME curves of ChPN(10,1) \"jump\" from a high level to low level when the PDB changes from 12 MHz to 24 MHz.  
Figure 8: AWMEs vs multipath delay.  
Figure 7: MPEEs vs multipath delay.