## 4 Theoretical Tools of HOC  
In this section, we focus on the theoretical development of HOC, which present the main relationships linking diagonal kernels of cubic systems and the cumulants of the output that received random signals up to the fifth-order for the purpose to use the HOC to identify the cubic system described by (10). The starting point for all linear-quadratic blind methods is the Leonov-Shiryaev formula [57; 58], which links the different order of cumulants to the moments. The latter formula allows expressing any cumulant as a function of moments of lower or equal orders and is well-known in the case of non-delayed cumulants [59]. It is assumed above that the output signal is stationary and ergodic and these cumulants do not depend on time but on the difference time between the instant of observation. The \(r\)th-order cumulants of the output random signal are linked to the moments, where the order \(p\) is inferior or equal to \(r\), by the following formula of [PERSON] and [PERSON] [57; 58]:  
\[Cum[y_{1},\ldots,y_{r}] = \sum(-1)^{k-1}(k-1)!E\Big{[}\prod_{i\in\
u_{1}}y_{i}\Big{]}E\Big{[} \prod_{j\in\
u_{2}}y_{j}\Big{]}\ldots E\Big{[}\prod_{k\in\
u_{p}}y_{k}\Big{]}, \tag{11}\]  
The summation extends over all ensembles \(\{v_{1},v_{2},\ldots,v_{p}:1\leq p\leq r\}\) forming a partition \(\{1,2,\ldots,r\}\). In this formula, \(k\) is the number of elements that compose the partition. For the second-order terms, both partitions are possible (1, 2) and (1) (2). Thus, we have the following expression:  
\[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\] \[= E[y_{1}y_{2}]-E[y_{1}]E[y_{2}].\]  
For the stationary random output signal, \(\{y(t)\}\), its second-order cumulants, in the cubic system case, become  
\[C_{2,y}(\tau_{1}) = Cum[y_{1},y_{2}]=Cum[y(k),y(k+\tau_{1})] \tag{13}\] \[= E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)\sum_{j=0}^{q}h(j,j,j)\] \[\times x^{3}(k+\tau_{1}-j)\Big{]}-E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k -i)\Big{]}\] \[\times E\Big{[}\sum_{j=0}^{q}h(j,j,j)x^{3}(k+\tau_{1}-j)\Big{]}\]  
Figure 1: Nonlinear cubic system. \[C_{2,y}(\tau_{1}) = E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k-i)h(i+\tau_{1},i+\tau_{1},\] \[i+\tau_{1})x^{3}(k-i)\Big{]}-E\Big{[}\sum_{i=0}^{q}h(i,i,i)x^{3}(k- i)\Big{]}\] \[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\]0  
\[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\]1 \[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\]2 \[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\]3  
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 \((15)\), the second-order cumulants and the diagonal kernels of cubic systems are linked by the following expression  
\[Cum[y_{1},y_{2}] = (-1)^{0}(0)!E[y_{1}y_{2}]+(-1)^{2-1}(2-1)!E[y_{1}]E[y_{2}] \tag{12}\]4  
In the case of fifth-order cumulants, we have 52 possible partitions of 7 different types:  
* 1 partition of type: (1, 2, 3, 4, 5) with \(k=1\);
* 5 partitions of type: (1) (2, 3, 4, 5) with \(k=2\);
* 10 partitions of type: (1) (2) (3, 4, 5) with \(k=2\);
* 10 partitions of type: (1) (2) (3, 4, 5) with \(k=3\);
* 15 partitions of type: (1) (2, 3) (4, 5) with \(k=3\);
* 10 partitions of type: (1) (2) (3) (4, 5) with \(k=4\);
* 1 partition of type: (1) (2) (3) (4) (5) with \(k=5\).