## 6 Performance Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Approach** & \(\widehat{h}(i,i,t)\pm Std\) & **0 dB** & **8 dB** & **16 dB** & **24 dB** \\ \hline \multirow{3}{*}{AlgCum5} & \(\widehat{h}(1,1,1)\pm Std\) & \(-0.1237\pm 0.0554\) & \(-0.1246\pm 0.0365\) & \(-0.1278\pm 0.0347\) & \(-0.1297\pm 0.0318\) \\  & \(\widehat{h}(2,2,2)\pm Std\) & \(-0.3442\pm 0.0573\) & \(-0.3446\pm 0.0293\) & \(-0.3464\pm 0.0205\) & \(-0.3456\pm 0.0198\) \\  & \(\widehat{h}(3,3,3)\pm Std\) & \(0.8905\pm 0.0372\) & \(0.8964\pm 0.0290\) & \(0.8982\pm 0.0275\) & \(0.8980\pm 0.0265\) \\  & \(NMSE\) & \(4.1833\times 10^{-4}\) & \(3.5208\times 10^{-4}\) & \(2.6065\times 10^{-4}\) & \(2.2156\times 10^{-4}\) \\ \hline \multirow{3}{*}{CIMLLAD [56]} & \(\widehat{h}(1,1,1)\pm Std\) & \(-0.1843\pm 0.2376\) & \(-0.1353\pm 0.1882\) & \(-0.1704\pm 0.2087\) & \(-0.1300\pm 0.1907\) \\  & \(\widehat{h}(2,2,2)\pm Std\) & \(-0.3880\pm 0.2272\) & \(-0.3457\pm 0.1874\) & \(-0.3788\pm 0.2206\) & \(-0.3611\pm 0.1974\) \\ \cline{1-1}  & \(\widehat{h}(3,3,3)\pm Std\) & \(0.8836\pm 0.2223\) & \(0.8629\pm 0.1762\) & \(0.8950\pm 0.2161\) & \(0.8773\pm 0.1903\) \\ \cline{1-1}  & \(NMSE\) & \(15\times 10^{-4}\) & \(8.2349\times 10^{-4}\) & \(6.5195\times 10^{-4}\) & \(5.2984\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: True and estimated (means \(\pm\) Std. Dev.) cubic kernels of the system I under 200 Monte Carlo runs. Figure 4: Estimated cubic kernel, \(h(3,3,3)=0.90\), for system I.  
Figure 3: Estimated cubic kernel, \(h(2,2,2)=-0.35\), for system I.  
existing literature as indicated above. Indeed, the additive Gaussian noise will almost vanish in the higher cumulants domain (superior to second-order) and the proposed approach, which uses fifth-order cumulants can achieve excellent identification kernels in very noisy environments compared to the CIM-LLAD algorithm. To illustrate the differences in the performance between the proposed and existing methods, we calculate the corresponding NMSE for various SNR and input signal sample sizes. The corresponding identification results are depicted in Figure 5.  
Clearly, the NMSE and standard deviation values obtained using the developed approach are much lower than those for the CIM-LLAD algorithm, for all considered values of SNR and input data sizes, being able to improve significantly the identification of cubic kernels of the system blindly, even in very noisy environments and/or small sample sizes (e.g., SNR = 0 dB, \(N=400\)). To study the convergence of the proposed fifth-order cumulants-based approach more clearly, an additional set of simulation results is represented in Figures 6 and 7. It describes through system I the behavior distribution of NMSE and indicates the effect of the Gaussian noise on the estimated cubic kernels in noisy environments for various SNR levels over 200 Monte-Carlo runs. Figure 8 concerns the differences between the true and estimated output signal using the proposed blind approach (AlgCum5) and the CIM-LLAD method. We take MSE on the true and estimated outputs as criterion of comparison, i.e.,  
\[MSE=\frac{1}{N}\sum_{n=0}^{N}\left[y(n)-\widehat{y}(n)\right]^{2}, \tag{37}\]  
where \(y(n)\) is the true output signal of the cubic system considered and \(\widehat{y}(n)\) is the estimated output signal of the cubic system to be identified. We conclude from Figure 8 and the results summarized in Table 2 using MSE criterion that no noticeable differences between true output signal of the system I and their corresponding estimates using the proposed blind approach. On the other hand, for the CIM-LLAD method, we have an MSE about 12 times higher. Figure 8: True and estimated output signals for System I.  
Figure 7: Behavior of the distributions of estimated NMSE using the CIM-LLAD method in system I. The obtained results of nonlinear system II with non-Gaussian input sample of length \(N=2400\) and different SNR levels are given in Table 3. Clearly, we can make remarks similar to those made in the case of system I.