# A Low-Ambiguity Signal Waveform for Pseudolite Positioning Systems Based on Chirp
## 4 Numerical Results
### Tracking Performance  
For the evaluation of the 1-sigma thermal noise jitter of DLL tracking, [28] developed a low bound of tracking error (LBTE), which is independent of specific code tracking circuit design and reflects the ultimate tracking performance. Considering the similarity of the modulation principle of ChPN with BOC signals, the canonical model of code-tracking loop can be employed to evaluate the ChPN tracking performance [26]. The LBTE of the time-of-arrival (TOA) estimate of this model can be written as:  
\[\sigma_{LB}\cong\frac{1}{T_{c}}\sqrt{\frac{B_{L}(1-0.5B_{L}T)}{(2\pi)^{2}C\int _{-\beta_{r}/2}^{\beta_{r}/2}f^{2}[G_{\mathrm{s}}(f)/G_{w}(f)]df}}(\mathrm{chip}) \tag{18}\]  
where \(B_{L}\) is the loop bandwidth, \(T\) is the coherent integration time, \(C\) is the received signal power, \(\beta_{r}\) is the PDB, \(G_{\mathrm{s}}(f)\) is the normalized signal PSD, \(G_{w}(f)\) is the PSD of the sum of noise plus interference.  
Based on (18), Figure 5 can be obtained with the following preconditions: (1) \(B_{L}=0.5\) Hz, \(T=1\) ms, \(G_{w}(f)=N_{0}\), where \(N_{0}\) is the PSD of thermal noise; (2) the same PDB of 30 MHz. Based on Figure 5, the following conclusions can be drawn. For the first situation, ChPN(5,1) owns the smallest LBTEs, ChPN(3,3) has smaller LBTEs than ChPN(4,2) due to there are more high-frequency components in the PDB of 30 MHz. For the second situation, BOC(5,1) has smaller LBTEs than ChPN(5,1). For the third situation, ChPN(10,1) has smaller LBTEs than ChPN(5,1). Furthermore, when the carrier-to-noise ratio (C/N0) increases, either the LBTE itself or the LBTE differences between ChPN signals with different sweep bandwidth or between ChPN and BOC become smaller.  
Figure 4: PSD envelopes of various signals.  
Figure 5: LBTEs for different modulations.  
Assuming non-coherent DLL is used, the tracking precision can be evaluated with the mathematical model presented by [26] as follows:  
\[\sigma=\frac{1}{T_{c}}\sqrt{\frac{B_{1}J_{-\beta_{f}/2}^{\beta_{f}/2}G_{\mathrm{s }}(f)\sin^{2}(\pi fD)df}{(2\pi)^{2}\frac{C}{\mathrm{Nb}}\left(\int_{-\beta_{f}/2 }^{\beta_{f}/2}fG_{\mathrm{s}}(f)\sin(\pi fD)df\right)^{2}}}\times\sqrt{1+ \frac{B_{1}J_{-\beta_{f}/2}^{\beta_{f}/2}G_{\mathrm{s}}(f)\cos^{2}(\pi fD)df}{T \frac{C}{\mathrm{Nb}}\left(\int_{-\beta_{f}/2}^{\beta_{f}/2}fG_{\mathrm{s}}(f) \cos(\pi fD)df\right)^{2}}}(\mathrm{chip}) \tag{19}\]  
where \(D\) is the early-late spacing (ELS), the tracking precision curves of ChPN(5,1), ChPN(4,2), ChPN(3,3), ChPN(10,1) and BOC(5,1) are shown in Figure 6, which is drawn under the conditions that (1) code loop bandwidth is 0.5 Hz; (2) ELS is 5.86 m; and (3) coherent integration time is 1 ms. As we can see from the figure, when the C/N0 increases, either the tracking error itself or the tracking error differences between different signals become smaller. With the PDB increasing, the tracking error become smaller. For the first situation, ChPN(5,1) has the smallest tracking errors, ChPN(3,3) performs better than ChPN(4,2) with a PDB of 24 MHz, and worse than that with a PDB of 12 MHz. For the second situation, BOC(5,1) has a smaller tracking error than ChPN(5,1). For the third situation, ChPN(10,1) has a smaller tracking error than ChPN(5,1) with a PDB of 24 MHz; when the PDB is 12 MHz, however, the larger tracking error happens due to lots of frequency components are wasted which can be easily seen in Figure 4.