## 2 ACE-FHBOC Multiplexing  
On the premise of an ACE-FHBOC signal having ideal spreading codes and infinite bandwidth, we construct the mathematical models of the signal, and derive its time dimension and phase dimension expressions. Considering the multiplexing of four independent direct-sequence spread spectrum (DSSS) signal components into a dual-sideband integral signal by employing complex frequency-hopping subcarriers, we first combine these four components into two complex quadrature phase-shift keying (QPSK) signals, expressed as  
\[ \begin{cases}s_{\text{L}}(t)=s_{\text{L}\text{I}}(t)+\text{j}s_{\text{LQ}}(t)\\ s_{\text{U}}(t)=s_{\text{U}\text{I}}(t)+\text{j}s_{\text{UQ}}(t)\end{cases}  \tag{1}\]  
where \(s_{\text{I}}(t)\) and \(s_{\text{Q}\text{Q}}(t)\), \(i\in\{\text{L},\text{U}\}\), are the in-phase and quadrature-phase components, respectively. Each component can be written as  
\[s_{i}(t)=\sqrt{P_{i}}\sum_{k=-\infty}^{+\infty}a_{k}^{i}\mu_{i}\Big{(}t-kT_{c}^ {i}\Big{)} \tag{2}\]  
in which \(i\in\{\text{U},\text{L},\text{U},\text{Q},\text{LQ}\}\), \(P_{i}\) is the nominal power of component \(i\), and \(a_{k}^{i}\) is the data-modulated bipolar spreading code chip with chip waveform \(\mu_{i}(t)\), which is a rectangular pulse with a duration of \(T_{c}^{i}\). Then, the input signals value states, dictated as  
\[v=[\text{sign}(s_{\text{U}\text{I}}),\text{sign}(s_{\text{L}\text{I}}),\text {sign}(s_{\text{U}\text{Q}}),\text{sign}(s_{\text{L}\text{Q}})] \tag{3}\]  
With complex frequency-hopping subcarriers, we can construct a complex envelope \(s_{\text{MUX}}(t)\) as  
\[s_{\text{MUX}}(t)=s_{\text{L}}(t)e^{-|2\pi(f_{0}+g(t)f_{d})t}+s_{\text{U}}(t) e^{2\pi(f_{0}+g(t)f_{d})t} \tag{4}\]  
where \(f_{0}\) is the reference frequency, which is typically equal to 1.023 MHz, \(g(t)\) is the frequency-hopping pattern, and \(f_{d}\) is the minimum frequency-hopping interval. Denote the least common multiple of those \(T_{c}^{i}\) as \(T_{m}\). Then, for an ACE-FHBOC signal, we assume that the subcarrier frequency, which hops under the control of \(g(t)\), is constant only in the interval \(t\in[kT_{m},(k+1)T_{m})\). Substitution of (1) into (4) yields  
\[s_{\text{MUX}}(t)=\alpha_{\text{I}}(t)\chi_{\text{I}}(t)+\text{j}\alpha_{ \text{Q}}(t)\chi_{\text{Q}}(t) \tag{5}\]  
where \(\chi_{\text{I}}(t)\)=\(\text{sin}[2\pi(f_{0}+g(t)f_{d})t+\varphi_{\text{I}}(t)]\), \(\chi_{\text{Q}}(t)\)=\(\text{sin}[2\pi(f_{0}+g(t)f_{d})t+\varphi_{\text{Q}}(t)]\), and  
\[ \begin{cases}&\alpha_{\text{I}}(t)=-\sqrt{\left[s_{\text{U}\text{I}}(t)+s_{ \text{L}\text{I}}(t)\right]^{2}+\left[s_{\text{U}\text{Q}}(t)-s_{\text{L} \text{Q}}(t)\right]^{2}}\\ &\alpha_{\text{Q}}(t)=\sqrt{\left[s_{\text{U}\text{I}}(t)-s_{\text{L}\text{I} }(t)\right]^{2}+\left[s_{\text{U}\text{Q}}(t)+s_{\text{L}\text{Q}}(t)\right]^ {2}}\end{cases}  \tag{6}\]  
\[ \begin{cases}&\varphi_{\text{I}}(t)=-\text{atan}2[s_{\text{U}\text{I}}(t)+s_{ \text{L}\text{I}}(t),s_{\text{U}\text{Q}}(t)-s_{\text{L}\text{Q}}(t)]\\ &\varphi_{\text{Q}}(t)=\text{atan}2[s_{\text{U}\text{Q}}(t)+s_{\text{L}\text{Q} }(t),s_{\text{U}\text{I}}(t)-s_{\text{L}\text{I}}(t)]\end{cases}  \tag{7}\]  
The function \(\text{atan}2(\cdot,\cdot)\) in (7) is a four-quadrant arctangent. According to (5) we can derive that the square of the envelope of \(s_{\text{MUX}}(t)\) is  
\[ \begin{split}&\left|s_{\text{MUX}}(t)\right|^{2}=\chi_{\text{I}} ^{2}(t)+\chi_{\text{Q}}^{2}(t)\\ &+2[s_{\text{U}\text{I}}(t)s_{\text{L}\text{I}}(t)-s_{\text{U}\text{Q}}(t )]\times\left[\chi_{\text{I}}^{2}(t)-\chi_{\text{Q}}^{2}(t)\right]\end{split}  \tag{8}\]  
which is not constant. To keep the envelope constant, we need to reconstruct the subcarrier in (5). Let the modified complex envelope be \(\widetilde{s}_{\text{MUX}}(t)\)=\(\alpha_{\text{I}}(t)\widetilde{\chi}_{\text{I}}(t)+\alpha_{\text{Q}}(t) \widetilde{\chi}_{\text{Q}}(t)\), where \(\widetilde{\chi}_{\text{I}}(t)\) and \(\widetilde{\chi}_{\text{Q}}(t)\) are the reconstructed in-phase and quadrature-phase frequency-hopping subcarriers, respectively.