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Finite model order asymptotic analysis in Wiener system identification

In this paper, we derive the linear part's asymptotic covariance matrices under white noise and coloured noise respectively in the Wiener system. The model orders do not exist in these two asymptotic covariance forms. And we use some reproducing kernel function constructed by a group of orthogonal basis functions to replace the model order. So when some priori information about Wiener system are known, these two asymptotic covariance matrices approximate their true sample values accurately. When some assumption conditions are relaxed, the cross spectrum can be decomposed as a constant matrix multiply its spectrum. Based on this conclusion, the asymptotic covariance matrices are generalised to other simplified cases. Additionally, based on these asymptotic covariance matrices, we construct an optimisation problem with respect to the input power spectrum. By solving this optimisation problem with some constraint conditions, the optimal input power spectrum used to excite the Wiener system is obtained.