# A Low-Ambiguity Signal Waveform for Pseudolite Positioning Systems Based on Chirp
## 4 Numerical Results
### ACF and PSD  
Figure 3 is drawn with infinite front-end bandwidth. For the first situation, as shown in that figure, the ChPN(5,1) has the sharpest ACF main peak among the alternative options, while ChPN(4,2) has a sharper ACF peak than ChPN(3,3) in a relative large time delay and a more obtuse ACF peak in a small time delay. Therefore, ChPN(5,1) may provide the best tracking precision, ChPN(4,2) may provide a better tracking precision than ChPN(3,3) with a small previous detection bandwidth (PDB), and worse tracking precision with a large PDB based on the tracking theory provided by [PERSON] [26; 27]. What's more, ChPN(4,2) has the largest side-peak amplitude, while ChPN(3,3) has the smallest one. For the second situation, BOC(5,1) has sharper ACF main peak than ChPN(5,1), which may lead to better tracking precision. However, the ACF side-peaks of ChPN(5,1) are much lower than that of BOC(5,1), which yields a larger noise margin and thus a lower false locking probability if the traditional E-L discriminator is used for the delay lock loop (DLL). The larger main-peak to maximum-side-peak ratio (MSPR) can reflect the lower ambiguity. The MSPRs of ChPN(5,1) and BOC(5,1) are about 15 dB and 0.91 dB, respectively. Therefore, the ACF peak ambiguity problem can be mitigated by ChPN, which makes it easier to track the correct ACF peak using the traditional E-L discriminator. For the third situation, ChPN(10,1) has a sharper ACF main peak, and more ACF side-peaks, which may lead to better tracking precision and more serious ACF peaks ambiguity, respectively. Owing to the symmetry of the ACF, the \"Bump and Jump\" technique can be employed to achieve the unambiguous tracking of ChPN signal, which will increase the receiver complexity.  
According to (14), the PSD envelopes for ChPN(5,1), ChPN(4,2), ChPN(3,3) and ChPN(10,1), and BOC(5,1) are drawn in Figure 4. As we can see from the figure, some conclusions can be drawn. For the first situation, the PSD of ChPN(5,1) is flatter than the others as well as more high-frequency components, ChPN(4,2) owns more frequency components in the band of 1 MHz to 5 MHz while ChPN(3,3) owns more frequency components in band with higher absolute value of frequency. Hence, ChPN(3,3) may has a better tracking precision than ChPN(4,2) with a large PDB. For the second situation, the PSD of BOC(5,1) contains more high-frequency components than ChPN(5,1), which may yield a better tracking precision. For the third situation, ChPN(10,1) has a flatter PSD envelopes and contains more high-frequency components than ChPN(5,1), which may contribute to better tracking precision.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Situations** & **Signals** & **Characteristics** \\ \hline
1 & ChPN(5,1), ChPN(4,2), ChPN(3,3) & ChPN signals with the same MLB, different sweep bandwidths. \\
2 & ChPN(5,1), BOC(5,1) & Different waveforms with the same code rate and MLB. \\
3 & ChPN(10,1), ChPN(5,1) & ChPN signals with the same code rate and different MLBs. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Analysis situations.  
Figure 3: The auto-correlation of several modulations.