# A Novel Signal Design and Performance Analysis in NavCom Based on LEO Constellation
## 3 Model Design
### PSD and ACF  
This subsection will analyze the normalized PSD and autocorrelation function of the proposed scheme and then conduct a preliminary comparison and analysis of the signal scheme proposed in this research with the current leading low-orbit satellite signal system. The normalized PSD for the CE-OFDM-PM signal can be described as:  
\[G(f)=\frac{T_{c}}{2N}\sum_{k=1}^{N}\text{sin}c^{2}\bigg{[}\bigg{(}f-\frac{k}{2 T_{c}}\bigg{)}T_{c}\bigg{]}+\text{sin}c^{2}\bigg{[}\bigg{(}f+\frac{k}{2T_{c}} \bigg{)}T_{c}\bigg{]} \tag{8}\]  
where \(\text{sin}(\cdot)\) denotes the sampling function operator. Figure 5 shows the normalized PSD of the proposed signal in different numbers of sub-carriers \(k\). As can be seen from Figure 5, for different sub-carriers, the characteristics of normalized PSD are as follows:  
* The main lobe splits with the increase in \(k\). Figure 5: Comparison of the normalized power spectral density with CE–OFDM–PM under the conditions of \(k=1,2,3,4\). With the increase in parameter k, the main peak of the normalized PSD of the CE–OFDM–PM spread spectrum modulation will be split from the center. This flexible feature is similar to the BOC spread spectrum modulation, and the former has a better main-to-side lobe peak ratio than the latter. * The normalized power spectral density difference between the logarithmic main lobe and the main side lobe is about 26.5 dBW/Hz. * For \(k=1\), the main lobe bandwidth can reach 3.069 MHz, and for \(k\
eq 1\) the bandwidth in the main lobe can reach 2.046 MHz, and the side lobe bandwidth can reach 1.023 MHz. According to the Wiener-Khintchine theorem, the autocorrelation function with the CE-OFDM-PM signal can be expressed as:  
\[R(\tau)=\int\limits_{-B_{r}/2}^{B_{r}/2}\frac{T_{c}}{2N}\sum_{k=1}^{N}\text{ sin}c^{2}\bigg{[}\bigg{(}f-\frac{k}{2T_{c}}\bigg{)}T_{c}\bigg{]}+\text{ sin}c^{2}\bigg{[}\bigg{(}f+\frac{k}{2T_{c}}\bigg{)}T_{c}\bigg{]}\cdot e^{ 2\pi f\tau}df \tag{9}\]  
where \(B_{r}\) denotes the pre-filtering bandwidth for receivers. The ACF of the proposed signal is shown in Figure 6.  
In Figure 6, it can clearly be seen from the waveform above that for the proposed signal, the autocorrelation function has the characteristic of multiple peaks, and for the same value of k, the chip interval of the autocorrelation peak (valley) value is fixed. This will help to lock the characteristics of the autocorrelation peak when waveform transmission distortion occurs in the receiver spreading code tracking stage. With the increase in \(k\), the auto-correlation peak value keeps narrowing, and the period between the peak (valley) values will decrease, causing the receiver to have a higher precision code tracking performance and better multipath resolution. The navigation signal complex carrier (NSCC) is a signal scheme with advanced features in the field of low-orbit satellites. It was the LEO satellite NavCom signal scheme proposed by [PERSON], [PERSON] et al., Academy of Optoelectronics, Chinese Academy of Sciences. Thus, the proposed signal will be compared with NSCC in normalized PSD, as shown in Figure 7.  
Figure 6: Comparison of autocorrelation function with CE–OFDM–PM under the conditions of \(k=1\), \(2\), \(3\), \(4\). For the increase in k, the peaks (valley) are increased in each period, but for a fixed k, the autocorrelation peaks (valleys) are fixed, meaning that this is suitable in the receiver spreading code tracking stage. It can be seen in Figure 7 that, compared to the sidelobes of the normalized power spectral density, both OFDM (64) and NSCC (19) have an excellent main lobe bandwidth performance, and both have remarkable main-sidelobe ratios. (PSR). Although the sidelobe bandwidth of NSCC (19) is slightly better than that of OFDM (64), the latter is a signal scheme that is widely used in wireless communication, while the OFDM signal scheme has a natural compatibility and interoperability. The authors are relatively more optimistic about the theoretical feasibility and engineering application prospects of the CE-OFDM-PM signal scheme.