## 3 Ranging Error Source Analysis and Modeling
### Analysis and Modeling of Noise Interference  
In navigation and signal processing theory, noise models are usually based on Gaussian noise. In general, electronic equipment and interference environments, it is accurate enough to model noise as Gaussian noise. However, in some harsh interference environments or extremely complex environments, much noise that affects navigation reception is non-Gaussian noise. Therefore, combined with the application needs of our algorithm, wemodel the noise as multimode Gaussian noise to simulate a more realistic and challenging noise environment. We consider two multimode Gaussian noise models below.  
Since an extreme form of narrowband interference is continuous wave (CW) interference in the form of sines and cosines [31], this model can be used to simulate the interference environment of noise plus narrowband interference; the other form is Gaussian-process-superimposed signal interference.  
(1) For the Gaussian process plus sine or cosine oscillation process, the probability density function of this model is [32]:  
\[p(x)=\frac{1}{{{{(2\pi)}^{m}}\sqrt{2\pi}\sigma_{0}}}{\int_{0}^{2\pi}\cdots\int _{0}^{2\pi}\exp[-\frac{{{(x-\sum\limits_{i=1}^{m}{{B_{i}}\cos{\theta_{i}})}^{2} }}}{{2\sigma_{0}^{2}}}]{d\theta_{1}}\cdots{d\theta_{m}}} \tag{8}\]  
where \(\sigma_{0}^{2}\) is the variance of the Gaussian component, \(B_{i}\) is the amplitude of the \(i\)-th sine or cosine oscillating signal and \(\theta_{i}\) is the phase of the \(i\)-th signal. We obtain the average power of noise from Equation (8) as:  
\[P=\sum\limits_{i=1}^{m}{\frac{{B_{i}^{2}}}{2}+\sigma_{0}^{2m}},\;i\in N \tag{9}\]  
At this time, we define the signal-to-noise ratio (SNR) as the ratio of the average power of the signal to the average power of the noise. The average power of the signal is \(P_{S}\); then:  
\[\text{SNR}=\frac{P_{S}}{P}=\frac{P_{S}}{\sum\limits_{i=1}^{m}{\frac{{B_{i}^{2 }}}{2}+\sigma_{0}^{2m}}},\;i\in N \tag{10}\]  
(2) For the interference between Gaussian-process-superimposed signals, the probability density function of this model is [33]:  
\[p(x)=\frac{1}{{\sqrt{2\pi}\sigma_{0}}}{\sum\limits_{i=0}^{m}{q_{i}}\exp[-\frac {{{{(x-b_{i})}^{2}}}}{{2\sigma_{0}^{2}}}]} \tag{11}\]  
where \(\sum\limits_{i=1}^{m}{q_{i}}=1\) is the probability of inter-signal interference and its distribution can be a uniform or binomial distribution [32].  
For Equation (11), we can find the average power of noise as:  
\[P=\sum\limits_{i=1}^{m}{q_{i}}{b_{i}^{2}}+\sigma_{0}^{2},\;i\in N \tag{12}\]  
At this time:  
\[\text{SNR}=\frac{P_{S}}{P}=\frac{P_{S}}{\sum\limits_{i=1}^{m}{q_{i}}{b_{i}^{2 }}+\sigma_{0}^{2}},\;i\in N \tag{13}\]  
Comparing Equations (9) and (12) or (10) and (13), we can find that when \(i=1\) and \(B_{i}=\pm B\), the multimode interference at this time is dual-modal interference and they have general expressions. The corresponding SNR can be expressed by the following equation:  
\[\text{SNR}=\frac{P_{S}}{P}=\frac{P_{S}}{\beta^{2}+\sigma_{0}^{2}},\;\beta=b,B \tag{14}\]  
We further sort out:  
\[\text{SNR}=\frac{P_{S}}{P}=\frac{1}{(\frac{\beta^{2}}{\sigma_{0}^{2}}+1)} \times\frac{P_{S}}{\sigma_{0}^{2}}=\frac{1}{(C_{S}+1)}\times\text{SNR}_{0},\; \;\;\beta=b,B \tag{15}\]where \(C_{\beta}\) is the ratio of interference noise to Gaussian noise intensity, which reflects the relative strength of interference noise and dual-mode Gaussian noise. It is also the existence of \(C_{\beta}\) that leads to a decrease in the entire SNR and SNR\({}_{0}\) is the well-known Gaussian noise, that is, single-mode Gaussian noise. Therefore, the larger \(C_{\beta}\) is, the greater the impact on the SNR of the entire system.