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Regularized system identification using orthonormal basis functions

Most of existing results on regularized system identification focus on regularized impulse response estimation. Since the impulse response model is a special case of orthonormal basis functions, it is interesting to consider if it is possible to tackle the regularized system identification using more compact orthonormal basis functions. In this paper, we explore two possibilities. First, we construct reproducing kernel Hilbert space of impulse responses by orthonormal basis functions and then use the induced reproducing kernel for the regularized impulse response estimation. Second, we extend the regularization method from impulse response estimation to the more general orthonormal basis functions estimation. For both cases, the poles of the basis functions are treated as hyperparameters and estimated by empirical Bayes method. Then we further show that the former is a special case of the latter, and more specifically, the former is equivalent to ridge regression of the coefficients of the orthonormal basis functions.