# A Design Method of Code Correlation Reference Waveform in GNSS Based on Least-Squares Fitting
## 3 Results
### Bpsk(1)  
The code correlation reference waveform diagrams for the infinite bandwidth case are shown in Figures 9 and 10. There is little difference between the coherent and non-coherent cases. The coherent waveform is like the W2 waveform, and it presents at every boundary of the original code. The obtained waveform is also formed by four strokes with a symmetrical structure. The linear range of the corresponding S-curve is defined by the width of the center strokes. The strokes from both sides are much narrower, which makes the fitted S-curve quickly vanish to zero. The total area of the coherent waveform is zero. The non-coherent waveform, however, is different in the strobe values, which are not constant. The total area of the non-coherent waveform is also zero. The fitted S-curves for the infinite bandwidth case are presented in Figure 11, compared to the non-coherent W2 and double-delta correlator. Here, we show the S-curve of the coherent discriminator and the S-curve of the non-coherent discriminator. We also show the S-curve of non-coherent W2 and double-delta correlator, which are basically the same. It can be observed that the ideal S-curve is well reproduced by the resulting S-curve. The resulting S-curve returns more quickly to zero than the W2 and double-delta correlator, but slower to the ideal S-curve, which is determined by \(\mu\).  
Figure 8: Comparison of the S-curves of the BPSK(1) signal in different precisions with an 8.184-MHz bandwidth.  
Figure 10: Code correlation reference waveform diagram for the BPSK(1) signal non-coherent discriminator (infinite bandwidth).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scenario** & **Infinite Bandwidth** & **16.368 MHz** & **8.184 MHz** & **4.092 MHz** \\ \hline Fitting range (\(\pm\)chips) & 1 & 1 & 1 & 1 \\ Linear region (\(\pm\)chips) & 0.1 & 0.1 & 0.1 & 0.2 \\ Offset (chips) & 0 & 0.01 & 0.01 & 0.01 \\ Waveform range (chips) & \(-\)0.11\(\sim\)0.11 & \(-\)0.15\(\sim\)0.15 & \(-\)0.2\(\sim\)0.2 & \(-\)0.4\(\sim\)0.4 \\ Gate width (chips) & 0.01 & 0.05 & 0.1 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters for the BPSK(1) signal.  
Figure 9: Code correlation reference waveform diagram for the BPSK(1) signal coherent discriminator (infinite bandwidth).  
The results for the 4.092-MHz case are different from the infinite bandwidth case, because we had to increase \(\upmu\) and the size of the linear region. The resulting correlator diagrams for the 4.092-MHz case are shown in Figures 12 and 13. The corresponding S-curves are presented in Figure 14. Again, the resulting S-curve is more similar to the ideal S-curve with higher peak values and a sharper slope in the linear region, compared to W2 and the double-delta correlator. The ideal S-curve is well reproduced by the fitting result, except the nonzero values outside the linear region.  
Figure 11: Fitted S-curves for the BPSK(1) signal (infinite bandwidth).  
Figure 12: Code correlation reference waveform diagram for the BPSK(1) signal coherent discriminator (4.092-MHz bandwidth).