## 3 Inter-Satellite Laser Communication Model Based on Simulink
### Mathematical Model Analysis of Signal Transmission  
Based on the principles of inter-satellite laser communication for space gravitational wave detection and the design of the inter-satellite communication signal scheme presented earlier, a mathematical model for the inter-satellite laser communication signal has been established. At the transmitter, the PRN code signal is expressed as:  
\[u_{\text{PRN}}(t)=\sum\limits_{n=-\infty}^{\infty}a_{n}p(t-nT_{c}) \tag{3}\]  
where \(a_{n}\) represents the PRN code sequence, taking values of +1 or \(-1\) with equal probabilities, and the pulse shape is given by \(p(t)\) with a period of \(\frac{1}{T_{c}}\), where \(T_{c}\) is the chip duration of the PRN code. Similarly, the data code signal is expressed as:  
\[u_{\text{data}}(t)=\sum\limits_{i=-\infty}^{\infty}b_{i}p(t-iT_{d}) \tag{4}\]  
where \(b_{i}\) represents the data code sequence, with values of either 1 or 0, and the pulse shape is given by \(p(t)\) with a period of \(\frac{1}{T_{d}}\), where \(T_{d}\) is the symbol duration of the data code. By performing DS/SS on the data code using the PRN code, the resulting spread signal is:  
\[u_{\text{DS/SS}}(t)=\sum\limits_{i=-\infty}^{\infty}B_{i}\sum\limits_{n=0}^{N- 1}A_{n}p(t-nT_{c}-iNT_{c}) \tag{5}\]  
where \(B_{i}\) is the data code sequence after bipolar transformation, \(A_{n}\) is the PRN code sequence after bipolar transformation (both taking values of +1 or \(-1\)) and \(N\) is the spreading factor, defined as \(N=\frac{T_{c}}{T_{c}}\). Letting \(\sum\limits_{n=-\infty}^{\infty}C_{n}=\sum\limits_{i=-\infty}^{\infty}B_{i} \sum\limits_{n=0}^{N-1}A_{n}\), the spread signal can be rewritten as:  
\[u_{\text{DS/SS}}(t)=\sum\limits_{n=-\infty}^{\infty}C_{n}p(t-nT_{c}) \tag{6}\]  
Here, \(C_{n}\) is the communication ranging code sequence, and \(p(t)\) retains its periodicity with \(\frac{1}{T_{c}}\). The laser carrier for phase modulation is defined as:  
Figure 10: Design of the inter–satellite communication signal scheme. \[u_{\text{carrier}}(t)=A_{c}\sin(\omega_{c}t) \tag{7}\]  
where \(A_{c}\) is the carrier amplitude, determined by the laser's optical power. \(\omega_{c}\) is the carrier angular frequency, determined by the laser's optical frequency. Combining Equations (6) and (7), the optical signal after low-depth phase modulation is given by:  
\[u_{\text{BPSK}}(t)=A_{c}\sin\!\left(\omega_{c}t+m_{\text{pm}}\sum_{n=-\infty}^ {\infty}C_{n}p(t-nT_{c})\right) \tag{8}\]  
where \(m_{\text{pm}}\) is the modulation index applied for both communication and ranging. At the receiver, let the time be \(t^{\prime}=t-\tau\), where \(\tau\) represents the signal delay due to transmission. Since the only terms of the Bessel function with a significant contribution in the optical amplitude are \(J_{0}\) and \(J_{1}\)[28]. After the laser used for communication, ranging and clock noise transmission undergoes heterodyne interference at the receiver, the resulting signal is given by:  
\[ \begin{split} u_{\text{HI}}(t^{\prime})&=\eta G_{ \text{TIA}}J_{0}^{2}(m_{\text{sb}})\sqrt{\gamma P_{\text{LO}}P_{\text{S}}}\sin \left(\omega_{\text{het}}t^{\prime}+\varphi+m_{\text{pm}}\sum_{n=-\infty}^{ \infty}C_{n}p(t^{\prime}-nT_{c})\right)\\ &\quad+\eta G_{\text{TIA}}J_{1}^{2}(m_{\text{sb}})\sqrt{\gamma P_ {\text{LO}}P_{\text{S}}}\sin\!\left(\omega_{\text{sb1}}t^{\prime}+\varphi_{ \text{sb1}}\right)\\ &\quad+\eta G_{\text{TIA}}J_{1}^{2}(m_{\text{sb}})\sqrt{\gamma P_ {\text{LO}}P_{\text{S}}}\sin\!\left(\omega_{\text{sb2}}t^{\prime}+\varphi_{ \text{sb2}}\right)+n(t^{\prime})\end{split}  \tag{9}\]  
where \(\eta\) is the photodiode responsivity. \(G_{\text{TIA}}\) is the total gain of the transimpedance amplifier (TIA). \(J_{0}(m_{\text{sb}})\) and \(J_{1}(m_{\text{sb}})\) are the Bessel function of the first kind, which results from sinusoidal clock sideband modulation with an equivalent modulation index \(m_{\text{sb}}\). \(\gamma\) is the heterodyne efficiency. \(P_{\text{LO}}\) is the local oscillator optical power. \(P_{\text{S}}\) is the signal optical power.