## 6 Performance Results  
To study the effectiveness and analyze the convergence of the proposed blind approach under various signal-to-noise levels (SNR), defined as \(\text{SNR}\ =\ E[y_{0}^{2}(k)]/E[n_{G}^{2}(k)]\). We consider 200 Monte Carlo runs and use two cubic systems. To measure the precision of cubic kernel identification with respect to the true values, we define the NMSE for each run as  
\[\text{NMSE}=\frac{\sum\limits_{i=0}^{q}\left(h(i,i,i)-\widehat{h}(i,i,i) \right)^{2}}{\sum\limits_{i=0}^{q}h^{2}(i,i,i)}, \tag{34}\]  
where \(\widehat{h}(i,i,i)\) are the estimated kernels in each run, and \(h(i,i,i)\) are the true kernels in the cubic system. System I: The third-order cubic system is  
\[\left\{ \begin{array}{l}y_{0}(k)=x^{3}(k)-0.15x^{3}(k-1)-0.35x^{3}(k-2)\\ \qquad+0.90x^{3}(k-3)\\ \\ y(k)=y_{0}(k)+n_{G}(k)\end{array} \right. \tag{35}\]  
With reference to (10), in this case, we have \(h(0,0,0)=1\), \(h(1,1,1)=-0.15\), \(h(2,2,2)=-0.35\) and \(h(3,3,3)=0.90\). Then, we try to increase the system order to test robustness of our approach. We consider the fifth-order cubic system given by  
System II:  
\[\left\{ \begin{array}{l}y_{0}(k)=x^{3}(k)-0.25x^{3}(k-1)+0.36x^{3}(k-2)\\ \qquad-0.688x^{3}(k-3)+0.56x^{3}(k-4)\\ \qquad+0.227x^{3}(k-5)\\ \\ y(k)=y_{0}(k)+n_{G}(k)\end{array} \right. \tag{36}\]  
With reference to (10), in this case, we have \(h(0,0,0)=1\), \(h(1,1,1)=-0.25\), \(h(2,2,2)=0.36\), \(h(3,3,3)=-0.688\), \(h(4,4,4)=0.56\) and \(h(5,5,5)=0.227\). The simulation results of the estimated cubic kernels, mean and the standard deviations (Std. Dev.) of system I, over 200 Monte Carlo runs with non-Gaussian sample length input \(N=2400\) for various SNR are shown in Table 1. To analyze the convergence of this proposed blind approach to identify the nonlinear system I with high levels of noise (i.e., SNR = 0 dB), we represent the fluctuations around the means of the estimated nonlinear cubic kernels in Figures 2-4, respectively. To illustrate this convergence and to test and validate it, these fluctuations are compared with those obtained using the adaptive algorithm such as CIM-LLAD (\(\mu=0.01\), \(\rho=0.0001\), \(\tau=1.2\), \(\sigma=0.02\), \(M=1024\)) developed in [56]. The proposed blind method has lower fluctuations around the true values of the identifying cubic kernels when compared with the CIM-LLAD algorithm of [56]. Figure 2: Estimated cubic kernel, \(h(1,1,1)=-0.15\), for system I. Table 1 compares the different techniques. Clearly, the standard deviations obtained with the proposed approach are smaller than for the CIM-LLAD algorithm proposed in [56]. Moreover, the proposed approach uses only the output and can identify the cubic system blindly without any information about the input model with a good precision. The main reason is that the proposed method uses the fifth-order cumulants which exploit the maximum information by calculating the cumulants of the output making it more robust when we use a small sample size of non-Gaussian signal input. Additionally, Gaussian noise had a minor effect on the estimated cubic kernels when using the proposed approach as indicated in Table 1.  
The advantages of the proposed method are even more clear when we look to the NMSE, especially for high noise levels (i.e., SNR = 0 dB). This is well supported by the