## 2 Modulation Phase Noise Analysis
The phase noise \(\delta\widetilde{\varphi}_{PRN}(f)\) caused by modulation can be estimated and represented based on the PRN amplitude spectral density \(\widetilde{A}_{PRN}(f,f_{het})\) and the root mean square (RMS) of the signal amplitude \(A_{het,rms}\)[11]. \[\delta\widetilde{\varphi}_{PRN}(f)=2\sqrt{2}\cdot\frac{\widetilde{A}_{PRN}(f, f_{het})}{A_{het,rms}} \tag{2}\]  
In the equation, \(f\) represents the observation band frequency, and \(f_{het}\) denotes the heterodyne beat frequency. The estimation model \(\delta\widetilde{\varphi}_{BPSK,main}(f)\) for the main lobe noise of the signal in BPSK modulation is [11]:  
\[\delta\widetilde{\varphi}_{BPSK,main}(f)=2\sqrt{2}\cdot\frac{\widetilde{A}_{ BPSK}(f,f_{het})}{A_{het,rms}} \tag{3}\]  
In Equation (3), \(\widetilde{A}_{BPSK}(f,f_{het})\) can be expressed as:  
\[\widetilde{A}_{BPSK}(f,f_{het})=\sqrt{\frac{\left(A_{het,rms}\cdot m_{prn} \right)^{2}}{2f_{chip}}\cdot\text{sinc}^{2}(\frac{|f-f_{het}|}{f_{chip}})} \tag{4}\]  
\(f_{chip}\) represents the chip rate of the PRN code and \(\text{sinc}(x)\) is:  
\[\text{sinc}(x)=\left\{ \begin{array}{ll}1,&x=0\\ \sin(\pi x)/\left(\pi x\right),&x\
eq 0\end{array} \right. \tag{5}\]Thus, the estimation model for the main lobe noise \(\delta\widetilde{\varphi}_{BPSK,main}(f)\) can be expressed as:  
\[\delta\widetilde{\varphi}_{BPSK,main}(f)=2\frac{m_{pm}}{\sqrt{f_{chip}}}\cdot \text{sinc}(\frac{f}{f_{chip}}) \tag{6}\]  
Secondly, the estimation model for the sidelobe phase noise \(\delta\widetilde{\varphi}_{BPSK,side}(f)\) caused by BPSK modulation is given as [11]:  
\[ \begin{split}\delta\widetilde{\varphi}_{BPSK,side}(f)& =2\frac{\frac{12}{6}(m_{ph})}{f_{1}^{2}(m_{sh})}\cdot\frac{m_{pm}}{\sqrt{f_{ chip}}}\cdot\text{sinc}(\frac{f+\Delta f_{sh}}{f_{chip}})\\ &\approx 0\text{if }f\ll\Delta f_{sb}\text{ and }f_{chip}=\Delta f_{sb}\end{split}  \tag{7}\]  
\(\Delta f_{sb}\) represents the frequency offset from the sidelobe to the main lobe. From Equation (6), it can be seen that the main lobe noise is related to its modulation index and the PRN code chip rate. Taking the source coding design of LISA as an example, \(m_{pm}\) is 0.1 rad, and \(f_{chip}\) is 1.6 MHz [11]. Substituting into Equation (6), and considering \(\text{sinc}(\frac{f}{f_{chip}})=1\) (In the context of gravitational wave detection, the frequency band of interest ranges from \(10^{-3}\) Hz to 1 Hz, meaning \(f\ll f_{chip}\)), the phase noise is approximately 158 \(\upmu\text{rad}/\sqrt{\text{Hz}}\). This value is two orders of magnitude higher than the phase-tracking criterion for gravitational wave detection, which is 1 pm/\(\sqrt{\text{Hz}}\) (about 5.9 \(\upmu\text{rad}/\sqrt{\text{Hz}}\) under a laser wavelength of 1064 nm). Secondly, from Equation (7), it can be seen that the sidelobe noise depends on the PRN and clock sideband modulation parameters. In the case where \(f\ll\Delta f_{sb}\) and \(f_{chip}=\Delta f_{sb}\), the sidelobe noise is approximately 0, where \(\Delta f_{sb}\) is in MHz. Since \(f\ll\Delta f_{sb}\) is always satisfied, it is only necessary to ensure \(f_{chip}=\Delta f_{sb}\) to eliminate the sidelobe noise. Therefore, eliminating the main lobe noise caused by BPSK modulation is key to achieving high-precision phase tracking, which is the main focus of this study.