Stochastic gradient based third-order Volterra system identification by using nonlinear Wiener adaptive algorithm

The nonlinear Wiener stochastic gradient adaptive algorithm for third-order Volterra system identification application with Gaussian input signals is presented. The complete self-orthogonalisation procedure is based on the delay-line structure of the nonlinear discrete Wiener model. The approach diagonalises the autocorrelation matrix of an adaptive filter input vector which dramatically reduces the eigenvalue spread and results in more rapid convergence speed. The relationship between the autocorrelation matrix and cross-correlation matrix of filter input vectors of both nonlinear Wiener and Volterra models is derived. The algorithm has a computational complexity of O(M³) multiplications per sample input where M represents the length of memory for the system model, which is comparable to the existing algorithms. It is also worth noting that the proposed algorithm provides a general solution for the Volterra system identification application. Computer simulations are included to verify the theory.