## 2 Methods
### Model for Eliminating the Mixed Noise Based on Linear Signal Filtration  
The random process, which may appear as white noise \(\xi(t)\), has a constant power spectral density \(S_{\eta}(\omega)\) throughout the frequency domain, and its average power (dispersion) is infinite. The model of a normal random process based on linear filtering is based on two principles of the theory of random processes:  
1. the result of an arbitrary linear transformation of a Gaussian random process is a random Gaussian process; and,
2. the power spectral densities of random processes at the input \(S_{in}(\omega)\) and output \(S_{out}(\omega)\) of a linear filter with a frequency \(H(i\omega)\) response are related by: \[S_{out}(\omega)=\left|H(i\omega)\right|^{2}S_{in}(\omega)\] (1) \[H(i\omega)=\left\{ \begin{array}{l}1,|\omega-\omega^{\prime}|\leq\Delta \omega/2\\ 0,|\omega-\omega^{\prime}|>\Delta\omega/2\end{array} \right.\] (2) where \(\omega^{\prime}\) and \(\Delta\omega\) are the average frequency and filter bandwidth in the frequency band \([\omega^{\prime}-\Delta\omega/2,\omega^{\prime}+\Delta\omega/2]\). To solve the linear filter modeling problem, it is necessary that the power spectral density \(S_{\eta}(\omega)\) for a random process can be described by a fractional rational function of the variable \(\omega\). If white noise \(\xi(t)\) enters at the input of the linear filter, then the spectral power density of the random process at the input will be equal to:  
\[S\eta(\omega)=\frac{N_{0}}{2},|\omega|<\infty \tag{3}\]  
Subsequently, the function of the spectral power density at the output of the linear filter can be represented in a fractional rational form:  
\[S_{\xi}(\omega)=\frac{b_{M}\omega^{2M}+b_{M-1}\omega^{2(M-1)}+\cdots+b_{0}}{a _{N}\omega^{2N}+a_{N-1}\omega^{2(N-1)}+\cdots+a_{0}} \tag{4}\]  
where \(a_{N}\) and \(b_{M}\) are polynomial roots in the numerator and denominator of the spectral density function for the power of white noise \(S_{\eta}(\omega)\). The algorithm for modeling a Gaussian random process with specified spectral properties is based on transmitting white noise \(\xi(t)\) with a single spectral density \(S_{\eta}(\omega)=1\) through a linear filter whose frequency response corresponds to the expression (Figure 1):  
\[H(s)=\frac{F(s)}{G(s)} \tag{5}\]  
where  
\[F(s)=\sqrt{\frac{b_{M}}{a_{N}}}\prod_{m=1}^{M}(s+f_{m}) \tag{6}\]  
\[G(s)=\prod_{n=1}^{N}(s+g_{n}) \tag{7}\]  
\[g_{n}=ia_{n},n=\overline{1,N} \tag{8}\]  
\[f_{m}=i\beta_{m},m=\overline{1,M} \tag{9}\]  
where \(a_{N}\) and \(b_{M}\) are positive numbers in the power spectral density function of a random stationary process \(H(s)\), \(s\) is a complex variable that corresponds to the actual signal frequency \(\omega=i\,s\). If white noise \(\xi(t)\) with a single spectral density \(N_{0}/2=1\) is applied to the input of such a linear filter, then the signal will have a normal distribution at the output of the filter. Subsequently, the frequency characteristic of the filter \(H(i\omega)=\big{|}H(i\omega)\big{|}\) and impulse characteristic of the linear filter take the following form:  
\[h(t)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\big{|}H(i\omega)\big{|}e^{ i\omega t}d\omega \tag{10}\]  
When a digital model of a signal with noise \(\xi(t)\) is created, two general properties of white noise must be taken into account: the constant power spectral density \(S_{\eta}(\omega)\) in the frequency domain and the statistical independence of time samples taken at arbitrary points in time \(\Delta t\). It is known that when a continuous signal \(x(t)\) is sampled with a sampling period \(\Delta t\), the spectral function of the sampled signal becomes periodic, with a period of \(1/\Delta t\).. Therefore, if the spectral power density \(S_{\eta}(\omega)\) of the discrete white noise \(\xi(t)\) is constant in the interval \([-0.5/\Delta t\); \(0.5/\Delta t]\) and equal to \(N_{0}/2\), then it will be constant over the entire frequency domain. Figure 1: Functions of the power spectral density of a linear filter.