## 3 The Proposed Delay-Doppler Block Division Multiplexing
### Property Analysis  
In the remainder of this section, we briefly analyze how DDBDM addresses the issues present in existing INAC waveforms for LEO PNT. First, the orthogonality between the navigation and communication components keeps the MI between them low. Similarly to the orthogonality between subcarriers in OFDM, the two-dimensional basis functions in the DD domain are also orthogonal, and this orthogonality remains unchanged in the time domain and TF domain [19; 46]. However, due to high mobility and time-varying Doppler shifts, interference may still exist between navigation PRN codes and communication data [47; 48]. The MI between different signal components can be measured using the spectral separation coefficient (SSC) [49]. The MI becomes lower with a decreasing SSC value; its expression is as follows:  
\[\chi=\int_{-B_{r}/2}^{B_{r}/2}G_{l}(f)G_{s}(f)\mathrm{d}f \tag{17}\]  
where \(B_{r}\) is the front-end bandwidth, \(G_{s}(f)\) is the normalized PSD of the signal, and \(G_{l}(f)\) is the normalized PSD of the interference. Table 1 gives the results of the theoretical analysis of MI between navigation and communication signals under different channel conditions. The number of navigation and communication subcarriers is 500 for both, the number of zero-padded subcarriers is zero, and the other signal parameters are consistent with Section 5.1. The Doppler shift and its rate are zero for the static channel and 0.91 ppm and 0.09 ppm/s for the LEO channel, respectively, in agreement with Section 5. Meanwhile, we give a case where the navigation and communication bands overlap for comparison. This both represents the case of a shared-band waveform, and can be considered as a worst-case scenario for the navigation and communication components of the OTFS modulation in Figure 2 that is affected by Doppler spread. From the table, we can observe that the MI between the navigation and communication signals is lowest in the static channel, and the SSC is only 0.04 dB higher than the former in the LEO channel, which indicates that the MI is limited. In contrast, when the navigation and communication signal bands overlap, the SSC value is 28.4 dB higher than that of the static channel, which leads to severe MI. As a result, the strong communication signal of DDBDM has less of an impact on the ACF of the navigation signal, its MI is much lower than that of the shared-band INAC waveform, and it also avoids the potential MI of the original OTFS. In addition, DDBDM allows for the independent reception of navigation subcarriers, thereby reducing the signal processing overhead without incurring a loss of navigation signal power. The navigation PRN codes and communication data are placed on different DD blocks to separate the two components in the TF domain, as shown in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Condition** & **Static Channel** & **LEO Channel** & **Band Overlap** \\ \hline SSC [dB] & \(-\)106.19 & \(-\)106.15 & \(-\)77.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SSC values under different conditions. Consequently, the navigation receiver utilizes a front-end filter with a bandwidth smaller than that of the INAC signal to receive and process only the navigation signal component. In this case, the ACF can be rewritten as follows:  
\[R^{\prime}{}_{\text{DDBDM}}(\tau)=\frac{\sigma_{X}^{2}}{M_{n}}\frac{\sin(\pi \Delta fM_{n}\tau)}{\sin(\pi\Delta f\tau)}\Big{(}1-\frac{|\tau|}{T}\Big{)}e^{j \pi\Delta f/(M_{n}-1)\tau} \tag{18}\]  
Comparing (7) and (18), it can be found that the difference between \(R^{\prime}{}_{\text{OTFS}}(\tau)\) and \(R^{\prime}{}_{\text{DDBDM}}(\tau)\) lies only in the denominator of the first term in the ACF, because the navigation signal of the former occupies all \(M\) subcarriers, while the latter occupies only \(M_{n}\) subcarriers. The second and fourth terms, on the other hand, depend on the received navigation signal bandwidth \(M_{n}\Delta f\), which is the same for both.