## 2 ACE-FHBOC Multiplexing
We can verify that \(\left|s_{\text{MUX}}(t)\right|^{2}\) remains constant when \(\chi_{\text{I}}^{2}(t)=\chi_{\text{Q}}^{2}(t)\) and is equal to a constant. Therefore, the subcarriers can be reconstructed as \(\widetilde{\chi}_{\text{I}}(t)=\frac{\sqrt{2}}{2}\text{sign}[\chi_{\text{I}} (t)]\)and \(\widetilde{\chi}_{\rm Q}(t)=\frac{\sqrt{2}}{2}{\rm sign}[\chi_{\rm Q}(t)]\). Employing the reconstructed subcarriers clearly demonstrates that the subcarrier frequency does not affect the complex envelope of the multiplexing signal. Then, we can construct the ACE-FHBOC signal as follows:  
\[s_{\rm A}(t)=\frac{\sqrt{2}}{2}\alpha_{\rm I}(t){\rm sign}[\chi_{\rm I}(t)]+{ \rm j}\frac{\sqrt{2}}{2}\alpha_{\rm Q}(t){\rm sign}[\chi_{\rm Q}(t)] \tag{9}\]  
It is easy to verify that the square of the envelope is  
\[|s_{\rm A}(t)|^{2}=\alpha_{\rm I}^{2}(t)+\alpha_{\rm Q}^{2}(t)=\sum_{i}P_{i} \tag{10}\]  
in which the result is a constant. Consistent with the notation for ACE-BOC modulation, an ACE-FHBOC signal can be denoted as ACE-FHBOC(\(f_{h}^{M-1}\):\(f_{d}\):\(f_{h}^{0},f_{c}\), \(p\)), where \(\left\{f_{h}^{0}:f_{d}:f_{h}^{M-1}\right\}\) is the frequency-hopping set, which includes \(M\) possible subcarrier frequencies \(\left[f_{h}^{0},f_{h}^{1},\ldots,f_{h}^{M-1}\right]\). The frequency-hopping set is a monotonically increasing arithmetic sequence; \(f_{d}\) is the common difference, i.e., the minimum frequency-hopping interval, defined as \(f_{d}=f_{h}^{i}-f_{h}^{i-1}\), \(i=1,2,\ldots,M-1\). \(f_{c}\)=\(\left[f_{c}^{\rm UI},f_{c}^{\rm CI},f_{c}^{\rm KU},f_{c}^{\rm LQ}\right]\) denotes the spreading code chip rates of ACE-FHBOC signal components. \(p\)= \(\left[P_{\rm UL},P_{\rm UL},P_{\rm UQ},P_{\rm LQ}\right]\) denotes the power ratio of ACE-FHBOC signal components. Furthermore, other parameters are needed to describe the ACE-FHBOC signal. The hopping rate, denoted as \(f_{\varphi}\), is assumed to be equal to \(f_{m}=1/T_{\rm m}\) but is also allowed to be slower than \(f_{m}\). The frequency-hopping pattern \(g(t)\) controls the frequency hopping of the ACE-FHBOC subcarrier; it should be a pseudorandom and approximately uniform distribution over the frequency channels. For the sake of clarity, the abbreviation ACE-FHBOC(\(\alpha_{M-1}\):\(d\):\(\alpha_{0}\), \(\beta\), \(p\)) is defined to denote an ACE-FHBOC signal, where \(\alpha_{M-1}=f_{h}^{M-1}/f_{0}\), \(d=f_{d}/f_{0}\), \(\alpha_{0}=f_{h}^{0}/f_{0}\), \(\beta=f_{c}/f_{0}\). Notably, ACE-BOC is a special case of ACE-FHBOC modulation in which the frequency-hopping set includes only one single subcarrier frequency; thus, ACE-FHBOC is an extension of ACE-BOC, which inherits the properties of ACE-BOC. The four ACE-FHBOC components have an arbitrary power ratio, and any component can be halted by simply setting the corresponding nominal power \(P_{i}\) to zero, without affecting the constant-envelope property. Considering all the constraints, the time dimension expression for the ACE-FHBOC baseband signal can be constructed as  
\[\left\{ \begin{array}{l}s_{\rm A}(t)=\sqrt{2}/2\alpha_{\rm I}(t){\rm sign}[ \sin(2\pi(f_{0}+g(t)f_{d})t+\varphi_{\rm I}(t))]\\ +{\rm j}\sqrt{2}/2\alpha_{\rm Q}(t){\rm sign}[\sin(2\pi(f_{0}+g(t)f_{d})t+ \varphi_{\rm Q}(t))]\\ M=(\alpha_{M-1}-\alpha_{0})/d+1\\ f_{h}^{i}=(\alpha_{0}+i\times d)\times f_{0},\ i=0,1,2,\ldots,M-1\\ g(t)\in\{f_{h}^{i}/f_{0}\}\end{array} \right. \tag{11}\]  
The waveforms of the real and imaginary parts for \(s_{\rm A}(t)\) are square waves with varying frequencies. Their amplitudes and phases are functions concerning the signal state and the subcarrier frequency. As a complex constant envelope modulation, the phase dimension expression of ACE-FHBOC baseband signal can be written as  
\[ \begin{array}{l}s_{\rm A}(t)=Ae^{{\rm j}\Theta(t)}\\ =A\{\cos[\Theta(t)]+{\rm j}\sin[\Theta(t)]\}\end{array}  \tag{12}\]  
where \(A\) is the constant envelope, \(\Theta(t)={\rm atan2}\{{\rm Im}[s_{\rm A}(t)],{\rm Re}[s_{\rm A}(t)]\}\).