## 3 Influence of Physical Properties of NS on Communication, Navigation, and TT&C Integration
### Atmospheric Loss in NS
#### 3.3.6 NS Noise  
For very high-frequency signals such as those in the K/Ka bands, the influence of galactic noise can be neglected. We assumed that the main sources of propagation noise in space include the interference from other space radio systems and the equipment itself. This noise is assumed to be additive white Gaussian noise (AWGN), and its power spectral density (PSD) and autocorrelation function (ACF) are as follows [85],  
\[G_{\omega}(\omega)=\frac{N_{0}}{2} \tag{19}\]  
\[R_{\omega}(\tau)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}G_{\omega}(\omega)e^{i \omega\tau}d\omega=\frac{N_{0}}{2}\delta(\tau) \tag{20}\]  
where \(N_{0}\) is a constant and \(\delta\) (\(t\)) is the Dirac shock function.  
Strictly speaking, the white noise defined by Equation (19) is an idealized model, and there is no white noise in practice. However, the noise spectrum is much larger than the bandwidth of the system studied, and its power spectrum is close to a constant in the bandwidth of the system, so it can be regarded as white noise. Now assume that the mean value of noise caused by interference of other radio systems and equipment in space is  
\begin{table}
\begin{tabular}{c c c} \hline
**The Number of Paths** & **Time Delay (\(\mu\)s)** & **Relative Power of the Direct Path (dB)** \\ \hline four paths -1 & 0 & 0 \\ four paths -2 & 10 & \(-1\) \\ four paths -3 & 20 & \(-2\) \\ four paths -4 & 30 & \(-3\) \\ \hline \end{tabular}
\end{table}
Table 4: Simulation parameters of the four-path channel model in NS.  
\begin{table}
\begin{tabular}{c c c} \hline
**The Number of Paths** & **Time Delay (\(\mu\)s)** & **Relative Power of the Direct Path (dB)** \\ \hline direct single path & 0 & 0 \\ double paths & 300 & \(-6\) \\ three paths & 450 & \(-12\) \\ four paths & 600 & \(-18\) \\ \hline \end{tabular}
\end{table}
Table 3: Simulation parameters of different multipath channel models in NS.  
Figure 9: Influence of multipath and Doppler in the NS on signal services. (**a**) BER of different path models under different time delays and powers. (**b**) BER of the four-paths model under different time delays and powers.  
and \(\mu_{x2}\), respectively, and the variance is \(\sigma_{x1}\) and \(\sigma_{x2}\), obey the normal distribution of \(x_{1}\)\(\sim\)\(N_{1}\) (\(\mu_{x1}\), \(\sigma_{2}^{x1}\)) and \(x1\)\(\sim\)\(N_{2}\) (\(\mu_{x2}\), \(\sigma_{2}^{x2}\)), respectively. Then, according to the regeneration of the normal distribution, the sum of two independent normally distributed random variables still obey the normal distribution, that is, the final probability distribution function is as follows,  
\[p(x)=\frac{1}{\sqrt{2\pi(\sigma_{x1}^{2}+\sigma_{x2}^{2})}}e^{-\frac{\left[x -(\mu_{x1}+\mu_{x2})\right]^{2}}{2(\sigma_{1}^{2}+\sigma_{2}^{2})}}\ \ (-\infty<x<\infty) \tag{21}\]  
where \(x=x_{1}+x_{2}\).  
It can be seen that \(x=x_{1}+x_{2}\)\(\sim\) (\(\mu_{x1}+\mu_{x2}\), \(\sigma_{2}^{x1}+\sigma_{2}^{x2}\)). Figure 10 presents the ACF, PSD, and corresponding noise sequence distribution and fitting curve of \(x_{1}\), \(x_{2}\), and \(x\) NS noise sequences generated randomly and all with lengths of 10,000.  
From the simulation results in Figure 10a, it can be seen that when the chip offset is 0, the ACF of the AWGN is the power of the noise, and the correlation is the highest at this time. However, the correlation at other chip offset positions is very small and tends to 0 as the offset increases. In addition, by combining the PSD, data set distribution, and fitting results in Figure 10a,b, it is not difficult to observe that the PSD of the AWGN obeys a uniform distribution, while its amplitude distribution follows a Gaussian distribution, which is in line with the definition of AWGN. Moreover, for the noise random variables of a single Gaussian distribution, their linear combination still follows a Gaussian distribution.