## 5 Proposed Blind Approach  
The \(n\)th-order polyspectrum is defined as the \((n-1)\)-dimensional discrete-time Fourier transform of the \(n\)th-order cumulants \(C_{n,y}(\tau_{1},\tau_{2},\ldots,\tau_{n})\), i.e.,  
\[S_{n,y}(\omega_{1},\omega_{2},\ldots,\omega_{n-1}) = \sum\limits_{\tau_{1}=-\infty}^{+\infty}\cdots\sum\limits_{\tau_ {n-1}=-\infty}^{+\infty}C_{n,y}(\tau_{1},\tau_{2}, \tag{19}\] \[\ldots,\tau_{n-1})exp\Big{(}-\sum\limits_{i=1}^{n-1}\omega_{i}\tau_{i }\Big{)}.\]  
The second-order polyspectrum, Fourier transform of the (16), (called power spectrum) is linked to the cubic system transfer function \(H(\omega,\omega,\omega)\) by  
\[S_{2,y}(\omega) = T.F\Big{\{}C_{2,y}(\tau)\Big{\}} \tag{20}\] \[= \phi_{2}H(\omega,\omega,\omega)H(-\omega,-\omega,-\omega),\]  
where, \(\phi_{2}=\gamma_{6,x}-\gamma_{3,x}^{2}\).  
The fifth-order polyspectrum, Fourier transform of Equation (18), of the cubic system output is linked to the system transfer function by  
\[S_{5,y}(\omega_{1},\omega_{2},\omega_{3},\omega_{4}) = T.F\Big{\{}C_{5,y}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\Big{\}} \tag{21}\] \[= \phi_{5}H\Big{(}\omega_{1},\omega_{1},\omega_{1}\Big{)}H\Big{(} \omega_{2},\omega_{2},\omega_{2}\Big{)}\] \[\times H\Big{(}\omega_{3},\omega_{3},\omega_{3}\Big{)}H\Big{(}\omega_{4 },\omega_{4},\omega_{4}\Big{)}\] \[\times H\Big{(}-\sum\limits_{i=1}^{4}\omega_{i},-\sum\limits_{i=1}^{4} \omega_{i},-\sum\limits_{i=1}^{4}\omega_{i}\Big{)},\]  
where \(\phi_{5}=\gamma_{15,x}-5\gamma_{3,x}\gamma_{12,x}-10\gamma_{6,x}\gamma_{9,x} +20\gamma_{3,x}^{2}\gamma_{9,x}+30\gamma_{3,x}\gamma_{6,x}^{2}-60\gamma_{3,x} ^{3}\gamma_{6,x}+24\gamma_{3,x}^{5}\).  
Defining \(\omega=\sum\limits_{i=1}^{4}\omega_{i}\), (20) becomes  
\[S_{2,y}\Big{(}\sum\limits_{i=1}^{4}\omega_{i}\Big{)} = \phi_{2}H\Big{(}\sum\limits_{i=1}^{4}\omega_{i},\sum\limits_{i=1}^ {4}\omega_{i}\Big{)} \tag{22}\] \[\times H\Big{(}-\sum\limits_{i=1}^{4}\omega_{i},-\sum\limits_{i=1}^{4} \omega_{i},-\sum\limits_{i=1}^{4}\omega_{i}\Big{)}.\]  
From (21) and (22), we obtain  
Using (27) and (28), we obtain the following expression which links the kernels of cubic system and fifth-order cumulants only:  
\[\sum\limits_{i=0}^{q}h(i,i,i)C_{5,y}(2q-i,q-i,q-i,\tau_{4}-i) \tag{29}\] \[= C_{5,y}(q,0,0,q)h(\tau_{4}-q,\tau_{4}-q,\tau_{4}-q),\]  
where  
\[q\leq\tau_{4}\leq 2q. \tag{30}\]  
Thus, (29) and (30) are combined in the following matrix form  
\[\left[\begin{array}{cccc}C_{5,y}(2q-1,q-1,q-1,q-1)&\ldots&C_{5,y} (q,0,0,0)\\ C_{5,y}(2q-1,q-1,q-1,q-1,q)-\psi&\ldots&C_{5,y}(q,0,0,1)\\ \vdots&\ddots&\vdots\\ C_{5,y}(2q-1,q-1,q-1,2q-1)&\ldots&C_{5,y}(q,0,0,q)-\psi\end{array}\right]\] \[\times\left[\begin{array}{cccc}h(1,1,1)\\.\\.\\ h(i,i,i)\\.\\.\\.\\ h(q,q,q)\end{array}\right]=\left[\begin{array}{cccc}\psi-C_{5,y}(2q,q,q,q) \\ -C_{5,y}(2q,q,q,q+1)\\.\\.\\.\\.\\.\\.\\.\\.\\.\\ -C_{5,y}(2q,q,q,2q)\end{array}\right], \tag{31}\]  
where, \(\psi=C_{5,y}(q,0,0,q)\).  
Equation (31) can be rewritten in other compact form as  
\[MH=D, \tag{32}\]  
where, \(H=[h(1,1,1)\,\ldots,\ h(q,q,q)]^{T}\) is a column vector of kernels of the cubic system with size \(q\), \(D=[\psi-C_{5,y}(2q,q,q,q),\ -C_{5,y}(2q,q,q,q+1)\,\ldots,\ -C_{5,y}(2q,q,q,2q)]^{T}\) is a column vector of size \((q+1)\) and \(M\) the matrix of size \((q+1)\times q\) elements. The least square solution of (32) is expressed by  
\[\widehat{H}=(M^{T}M)^{-1}M^{T}D \tag{33}\]