## Appendix A  
In the case of cubic systems, the fifth-order cumulants of the stationary random output signal, \(\{y(k)\}\), are given by:  
\[\begin{array}{rcl}\text{C}_{5,y}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})&=& Cum[y_{1},y_{2},y_{3},y_{4},x_{5}]=Cum[y(k),y(k+\tau_{1}),y(k+\tau_{2}),y(k+\tau_{3}),y (k+\tau_{4})]\\ &=& E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\sum_{j=0}^{q}h(j,j,j)x^{3}(k+\tau_ {1}-j)\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau_{2}-l)\\ &\times&\sum_{m=0}^{q}h(m,m,m)x^{3}(k+\tau_{3}-m)\sum_{n=0}^{q}h(n,n,n)x^{3}(k +\tau_{4}-n)\Big{]}-5E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\Big{]}\\ &\times& E\Big{[}\sum_{j=0}^{q}h(j,j,j)x^{3}(k+\tau_{1}-j)\sum_{l=0}^{q}h(l,l,l )x^{3}(k+\tau_{2}-l)\sum_{m=0}^{q}h(m,m,m)x^{3}(k+\tau_{3}-m)\\ &\times&\sum_{n=0}^{q}h(n,n,n)x^{3}(k+\tau_{4}-n)\Big{]}-10E\Big{[}\sum_{i=0}^ {q}h(i,i,i)x^{3}(k-i)\sum_{j=0}^{q}h(j,j,j)x^{3}(k+\tau_{1}-j)\Big{]}\\ &\times& E\Big{[}\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau_{2}-l)\sum_{m=0}^{q}h(m,m,m)x ^{3}(k+\tau_{3}-m)\sum_{n=0}^{q}h(n,n,n)x^{3}(k+\tau_{4}-n)\Big{]}\\ &+& 20E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\Big{]}E\Big{[}\sum_{j=0}^ {q}h(j,j,j)x^{3}(k+\tau_{1}-j)\Big{]}\\ &\times& E\Big{[}\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau_{2}-l)\sum_{m=0}^{q}h(m,m,m)x ^{3}(k+\tau_{3}-m)\sum_{n=0}^{q}h(n,n,n)x^{3}(k+\tau_{4}-n)\Big{]}\\ &+& 30E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\Big{]}E\Big{[}\sum_{j=0}^ {q}h(j,j,j)x^{3}(k+\tau_{1}-j)\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau_{2}-l)\Big{]} \\ &\times& E\Big{[}\sum_{m=0}^{q}h(m,m,m)x^{3}(k+\tau_{3}-m)\sum_{n=0}^{q}h(n,n,n)x ^{3}(k+\tau_{4}-n)\Big{]}\\ &-& 60E\Big{[}\sum_{l=0}^{q}h(i,i,i)x^{3}(k-i)\Big{]}E\Big{[}\sum_{j=0}^ {q}h(j,j,j)x^{3}(k+\tau_{1}-j)\Big{]}\\ &\times& E\Big{[}\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau_{2}-l)\Big{]}E\Big{[}\sum_{m=0} ^{q}h(m,m,m)x^{3}(k+\tau_{3}-m)\sum_{n=0}^{q}h(n,n,n)x^{3}(k+\tau_{4}-n)\Big{]} \\ &+& 24E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\Big{]}E\Big{[}\sum_{j=0}^ {q}h(j,j,j)x^{3}(k+\tau_{1}-j)\Big{]}E\Big{[}\sum_{l=0}^{q}h(l,l,l)x^{3}(k+\tau _{2}-l)\Big{]}\\ &\times& E\Big{[}\sum_{m=0}^{q}h(m,m,m)x^{3}(k+\tau_{3}-m)\Big{]}E\Big{[}\sum_{m=0} ^{q}h(n,n,n)x^{3}(k+\tau_{4}-n)\Big{]}\end{array}\]\[C_{5,y}(\tau_{1},\tau_{2},\tau_{3},\tau_{4}) = \sum_{i=0}^{q}h(i,i,i)h(i+\tau_{1},i+\tau_{1},i+\tau_{1})\] \[\times h(i+\tau_{2},i+\tau_{2},i+\tau_{2})h(i+\tau_{3},i,\] \[\tau_{3},i+\tau_{3})h(i+\tau_{4},i+\tau_{4},i+\tau_{4})\] \[\times \Big{(}E\Big{[}x^{15}(k-i)\Big{]}-5E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{12}(k-i)\Big{]}-10E\Big{[}x^{6}(k-i)\Big{]}\] \[\times E\Big{[}x^{9}(k-i)\Big{]}+20E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{3}(k-i)\Big{]}E\Big{[}x^{9}(k-i)\Big{]}\] \[+ 30E\Big{[}x^{3}(k-i)\Big{]}E\Big{[}x^{6}(k-i)\Big{]}\] \[\times E\Big{[}x^{6}(k-i)\Big{]}-60E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{3}(k-i)\Big{]}E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{6}(k-i)\Big{]}+24E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{3}(k-i)\Big{]}E\Big{[}x^{3}(k-i)\Big{]}\] \[\times E\Big{[}x^{3}(k-i)\Big{]}E\Big{[}x^{3}(k-i)\Big{]}\Big{)}\]  
Under the assumption that the input sequence \(x(k)\) is i.i.d zero mean, stationary, non-Gaussian with \(\gamma_{n,x}=E[x^{n}(k)]\
eq 0\), \(\forall\ n=3,\ 6,\ 9,\ 12,\ 15\) and (A2), the fifth-order cumulants and the diagonal kernels of cubic systems are linked by the following expression:  
\[C_{5,y}(\tau_{1},\tau_{2},\tau_{3},\tau_{4}) = \Big{(}\tau_{15,x}-5\tau_{3,x}\tau_{12,x}-10\tau_{6,x}\tau_{9,x}\] \[+ 20\gamma_{3,x}^{2}\gamma_{9,x}+30\gamma_{3,x}\gamma_{6,x}^{2}-60 \gamma_{3,x}^{3}\gamma_{6,x}\] \[+ 24\gamma_{3,x}^{5}\Big{)}\sum_{i=0}^{q}h(i,i,i)h(i+\tau_{1},i+ \tau_{1},i+\tau_{1})h(i+\tau_{2},i+\tau_{2})\] \[\times h(i+\tau_{3},i+\tau_{3},i+\tau_{3})h(i+\tau_{4},i+\tau_{4},i+\tau _{4})\]