## 1 Introduction
A concrete application of higher-order statistics theory with more experimental discussion was developed by [PERSON] et al. [50], to identify machine faults through vibration measurements using higher-order statistics-based methods, such as a nonparametric phase-bearing and a parametric linear or nonlinear modeling approach. Another application was developed in [51] to model the delay of video-packets transmission by identification of quadratic nonlinear systems using fourth-order cumulants-based blind approaches, which were compared to the Levenberg-Marquardt algorithm, least mean square (LMS) and recursive least square (RLS). A HOC was also used by [PERSON] et al. [52] to develop both extending approaches, developed for the linear and nonlinear case, which uses fourth-order only and combined third- and fourth-order cumulants, to identify the diagonal parameters of nonlinear quadratic systems. Numerical simulations were presented to verify the theoretical results with comparison to the exiting method in terms of the normalized mean-squared error (NMSE) and the fluctuations around the true kernels. Motivated by reductions in computational requirements and the mathematical tractability of nonlinear system identification issue, [PERSON] et al. [41] used a Hammerstein series to build a new method to identify a specific time invariant nonlinear system when the input is a non-Gaussian stationary signal. As previously indicated, several authors considered the issue of nonlinear system identification using HOC theory, but most works focused on second-order or quadratic system identification, and the order of cumulants was limited to fourth-order cumulants. The fifth-order received little attention up to now. Indeed, crosscumulants up to the fifth-order were used by [PERSON] et al. [39] to identify the input-output quadratic system. Cubic system identification has also received so far little attention. Indeed, the Volterra kernels of the cubic system identification method using higher-order moments were developed by [PERSON] and [PERSON] in [53]. The authors showed that the proposed method reduces the complexity of Volterra kernel identification compared to the non-Gaussian and non-i.i.d. input case. The authors of [54] used the TOM from the extending SOM to identify the nonlinear cubic system. It was shown that more statistical knowledge can be obtained in the third-order statistics domain for blind system identification. In [55], mixed third- and fourth-order cumulants were used to extend linear HOC-based methods to nonlinear systems. In general, when we increase the nonlinearity degree in the model, the performances of methods relying on cumulants with small orders can degrade substantially in the presence of Gaussian noise. Therefore, we increase the order of the proposed approach by considering the fifth-order. In this framework, we consider cubic nonlinear systems, which can be used to describe a wide range of nonlinear systems, requiring only a small number of kernels in the Volterra series. The main contributions of this work are the development of new fifth-order cumulants which exploit the maximum information of the output cubic system without the need of training signals, i.e., blindly. The methodology of the proposed blind approach is based on the main theoretical tools of HOC using the link between Fourier transform of second-order and fifth-order cumulants in the case of cubic systems which are established to build a new formula which relates the second- and fifth-order cumulants by using the inverse Fourier transform. Based on this formula, which relates the second- and fifth-order cumulants, we can propose an approach based only on fifth-order under some assumptions for the kernels of systems and the properties of the autocorrelation function.