This paper deals with the analysis of a third-order tensor composed of a fourth-order output cumulants used for blind identification of a second-order Volterra-Hammerstein series. It is demonstrated that this nonlinear identification problem can be converted in a multivariable system with multiequations having the form of Ax By c. The system may be solved using several methods. Simulation results with the Iterative Alternating Least Squares IALS algorithm provide good performances for different signal-to-noise ratio SNR levels. Convergence issues using the reversibility analysis of matrices A and B are addressed. Comparison results with other existing algorithms are carried out to show the efficiency of the proposed algorithm. 2 Mathematical Problems in Engineering of the signal. This function is then included in the cost function yielding to an augmented Lagrangian function. It has been demonstrated that this approach gives good identification results for a nonlinear systems. However, this approach is still sensitive to additive Gaussian noise because the 2nd-order moment is used as a constraint. Authors, in 7 , overcame this sensitivity by using 4th-order cumulants as a constraint instead of 2nd-order moments in order to smooth out the additive Gaussian noise. But the proposed approach which is based on a simplex-genetic algorithm becomes so long and computationally complex. The main drawback of identification with Volterra series lies on the parametric complexity and the need to estimate a very big number of parameters. In many cases, Volterra series identification problem may be well simplified using the tensor formulation 10-12, 14. Authors, in 10 , used a parallel factor PARAFAC decomposition of the kernels to derive Volterra-PARAFAC models yielding an important parametric complexity reduction for Volterra kernels of order higher than two. They proved that these models are equivalent to a set of parallel Wiener models. Consequently, they proposed three adaptive algorithms for identifying these proposed Volterra-PARAFAC models for complex-valued input/output signals, namely, the extended complex Kalman filter, the complex least mean square CLMS algorithm, and the normalized CLMS algorithm. In this paper, the algorithm derived in 14 is extended to be applied to blind identification of a general second-order Volterra-Hammerstein system. The main idea is to develop a general expression for each direction slices of a cubic tensor and then express the tensor slices in an unfolded representation. The three-dimensional tensor elements are formed by the fourth-order output cumulants. This yields to an Iterative Alternating Least Square IALS algorithm which has the benefit over the original Volterra filters in terms of implementation and complexity reduction. A convergence analysis based on matrices reversibility study is given showing that the proposed IALS algorithm converges to optimal solutions in the least mean squares sense. Furthermore, some simulation results and comparisons with different existing algorithms...
This paper is devoted to the blind identification problem of a special class of nonlinear systems, namely, Volterra models, using a real-coded genetic algorithm (RCGA). The model input is assumed to be a stationary Gaussian sequence or an independent identically distributed (i.i.d.) process. The order of the Volterra series is assumed to be known. The fitness function is defined as the difference between the calculated cumulant values and analytical equations in which the kernels and the input variances are considered. Simulation results and a comparative study for the proposed method and some existing techniques are given. They clearly show that the RCGA identification method performs better in terms of precision, time of convergence and simplicity of programming.
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