## 2 Principle of Separating Multiple Signal Sources Based on Selected Bispectra  
Supposing that \(\textbf{a}=[a_{1},\cdots,a_{L}]^{T}\) is a random vector, \(r\)th-order cross-cumulant means accumulant between different processes, and the \(r\)th-order cross-cumulants of \(\mathbf{a}\) are defined as [16; 18; 19]:  
\[\begin{array}{cc}C_{k_{1}+\cdots+k_{L}}&=c_{r}\!\left\{\!\!\!\!\!\!\!\!\!\! \!\!  
where \(j\) is the imaginary unit, i.e., \(j^{2}=-1\), and \(E\left[\cdot\right]\) represents mathematical expectation. \(r\)th-order auto-cumulant means the cumulant of the process itself, and the \(r\)th-order autocumulants of a random process \(\{b[n]\}\) are defined as:  
\[C_{rb}=c_{r}\{b[n],b[n+\tau_{1}],\cdots,b[n+\tau_{r-1}]\}, \tag{3}\]  
where \(\tau_{i}\)(\(i=1\), \(2\), \(\cdots\), \(r-1\)) denotes the time delay, and \([n]\) is the sampling point. When the order of the cumulants is no less than three, the cumulants are defined as high-order cumulants with the multidimension Fourier transform. Denoting the real signal observed by an antenna as \(\{x[n]\}\), the signal sampling rate is \(f_{s}\) and the mean value of \(\{x[n]\}\) is zero. By the definition of high-order cumulants, the third-order cumulants of \(\{x(n)\}\) can be calculated as:  
\[C_{3x}(\tau,\rho)=E[x[n]x[n+\tau_{1}]x[n+\tau_{2}]], \tag{4}\]  
Consequently, the bispectrum of \(\{x[n]\}\) is defined as 2D Fourier transformation of the third-order cumulants, i.e.,  
\[B(\omega_{1},\omega_{2})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}C_{3x}( \tau,\rho)e^{-j(\omega_{1}\tau_{1}+\omega_{2}\tau_{2})}d\tau_{1}d\tau_{2}, \tag{5}\]  
The steps for the estimation of discretized bispectra shown in [13; 17] are as follows. The signal data is split into \(K\) sections with \(M\) observed samples in each section, denoted as \(x^{(i)}(1)\), \(x^{(i)}(2)\), \(\cdots\), \(x^{(i)}(M)\). Let \(i=1\), \(2\), \(\cdots\), \(K\), the data between the two adjacent sections can coincide. Then, calculate the discretized Fourier transformation coefficient for each section of signal data:  
\[X^{(i)}(\lambda)=\frac{1}{M}\underset{n=1}{\overset{M}{\sum}}x^{(i)}[n]e^{-j 2\pi(n-1)\lambda/M}, \tag{6}\]  
where \(\lambda=0\), \(1\), \(\ldots\), \(M/2\); then, bispectra for each section of signals can be estimated as:  
\[\overline{b}^{(i)}(\lambda_{1},\lambda_{2})=\frac{1}{\Delta^{2}}\underset{i_ {1}=-L}{\overset{L}{\sum}}\underset{i_{2}=-L}{\overset{L}{\sum}}X^{(i)}( \lambda_{1}+i_{1})X^{(i)}(\lambda_{2}+i_{2})X^{(i)*}(-\lambda_{1}-\lambda_{2}- i_{1}-i_{2}), \tag{7}\]  
where \(\Delta=f_{s}/N_{0}\), and \(L\) and \(N_{0}\) satisfy \(M=\left(2L+1\right)N_{0}\). The bispectra estimation of the whole signal can be calculated as the mean bispectra of all sections:  
\[\overline{B}(\lambda_{1},\lambda_{2})=\frac{1}{K}\underset{l=1}{\overset{K}{ \sum}}\overline{b}^{(i)}(\lambda_{1},\lambda_{2}), \tag{8}\]  
Denote the estimated bispectra as \(\overline{B}(\omega_{1},\omega_{2})\),  
\[\omega_{i}=\left(\frac{2\pi f_{s}}{N_{0}}\right)\lambda_{i}\ \ \ \ \ i=1,2\ \, \tag{9}\]  
where \(\xi=(\omega_{1},\omega_{2})\) is a bifrequency, \(\overline{B}(\xi)=\overline{B}(\omega_{1},\omega_{2})\). \(\left\{\overline{B}^{(l)}_{k}(\xi)\right\}_{k=1,2,\cdots N_{l}}\) is the set of sample bispectra, where the superscript \(l\) means the \(l\)th type of signals; \(k\) is the \(k\)th group of the \(l\)th type of observed signals, and \(N_{l}\) is the number of observed signals of the \(l\)th type.