## 3 Typical Applications of ACE-FHBOC Multiplexing
### Three Typical Applications  
For some practical GNSS applications, it is desired to assign different powers to the in-phase component and quadrature-phase component in each sideband, while the total powers of the upper sideband and lower sideband are the same, i.e.,  
\[P_{\rm UI}=P_{\rm LI}=\lambda^{2}P_{\rm UQ}=\lambda^{2}P_{\rm LQ} \tag{13}\]  
This signal can be denoted as ACE-FHBOC(\(\alpha_{M-1}\):\(d\):\(\alpha_{0}\), \(\beta\), \([1,1,\lambda^{2},\lambda^{2}]\)); for convenience, denote this ACE-FHBOC signal as Type-I ACE-FHBOC. By default, suppose \(\lambda\geq 1\)  
Figure 1: Waveforms of the real part and the imaginary part for the example signal. Figure 2: Modulation constellation for the example signal. In the time dimension, by calculating all the zero-crossing points under all 16 states of signal state and \(M\) subcarrier frequencies in a subcarrier cycle according to (9), we can obtain all the possible zero-crossing points with a total number of \(12\times M\) for Type-I ACE-FHBOC. The zero-crossing points in \(\left[0,T_{h}^{i}/4\right)\) are expressed as  
\[\left\{ \begin{array}{l}\psi_{1}^{i}=0\\ \psi_{2}^{i}=\frac{T_{h}^{i}}{2\pi}\arctan(\lambda)\\ \psi_{3}^{i}=\frac{T_{h}^{i}}{4}-\frac{T_{h}^{i}}{2\pi}\arctan(\lambda)\end{array} \right. \tag{14}\]  
where \(T_{h}^{i}=1/f_{h}^{i}\). The other zero-crossing points can be derived by \(\psi_{m}^{i}\), \(T_{h}^{i}/4-\psi_{m}^{i}\), \(T_{h}^{i}/4+\psi_{m}^{i}\), \(T_{h}^{i}/2-\psi_{m}^{i}\), \(T_{h}^{i}/2+\psi_{m}^{i}\)\(3T_{h}^{i}/4-\psi_{m}^{i}\), and \(T_{h}^{i}-\psi_{m}^{i}\), \(m=1,2,3\). In the phase dimension, by calculating \(\Theta(t)\) for the 16 possible signal states, we can obtain all possible modulation phase points of Type-I ACE-FHBOC. The LUT for Type-I ACE-FHBOC is presented in Table 1, where the corresponding modulation phases are provided by (15). \[\Theta(x)=\left\{ \begin{array}{ll}\theta,&x=1,2,3\\ \frac{\pi}{2}+\theta,&x=4,5,6\\ \pi+\theta,&x=7,8,9\\ \frac{3\pi}{2}+\theta,&x=10,11,12\end{array} \right.,\ \ \ \theta=\left\{ \begin{array}{l}\frac{\pi}{2}-\arctan( \lambda)\\ \arctan(\lambda)\\ \frac{\pi}{2}\end{array} \right. \tag{15}\]  
If it is desired to allocate different powers to the two sidebands while the powers of the I and Q components in each sideband are the same, i.e.,  
\[P_{\text{UI}}=\lambda^{2}P_{\text{LI}}=P_{\text{UQ}}=\lambda^{2}P_{\text{LQ}} \tag{16}\]  
then, this signal can be denoted as ACE-FHBOC(\(\alpha_{M-1}\):\(d\):\(\alpha_{0}\), \(\beta\), \(\left[1,\lambda^{2},1,\lambda^{2}\right]\)), defined as Type-II ACE-FHBOC. Similar to Type-I ACE-FHBOC, we can obtain the zero-crossing points in \(\left[0,T_{h}^{i}/4\right)\)  
\[\left\{ \begin{array}{l}\psi_{1}^{i}=\frac{T_{h}^{i}}{2\pi}\arctan(\frac{ \lambda-1}{\lambda+1})\\ \psi_{2}^{i}=\frac{T_{h}^{i}}{8}\\ \psi_{3}^{i}=\frac{T_{h}^{i}}{2\pi}\arctan(\frac{\lambda+1}{\lambda-1})\end{array} \right. \tag{17}\]  
and the modulation phases can also be obtained, expressed as  
\[\Theta(x)=\left\{ \begin{array}{ll}\theta,&x=1,2,3\\ \frac{\pi}{2}+\theta,&x=4,5,6\\ \pi+\theta,&x=7,8,9\\ \frac{3\pi}{2}+\theta,&x=10,11,12\end{array} \right.,\ \ \theta=\left\{ \begin{array}{l} \arctan(\frac{\lambda-1}{\lambda+1})\\ \frac{\pi}{4}\\ \frac{\pi}{2}-\arctan(\frac{\lambda-1}{\lambda+1})\end{array} \right. \tag{18}\]  
According to (17) and (18), we can get the LUT for Type-II ACE-FHBOC. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline sign\([s_{\text{UI}}(t)]\) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) \\ sign\([s_{\text{UQ}}(t)]\) & 1 & 1 & 1 & 1 & \(-1\)For some GNSS requirements, it is not necessary to transmit four different signal components within a frequency band, and a three-signal CEM is a preferable option.