Differentiation by the cardinal spline wavelet and its application to the estimation of a transfer function

In this paper, we consider the differentiation by a wavelet with the scaling function given by the cardinal B-spline and its application to the estimation of a transfer function. As the cardinal B-spline consists of a Riesz base, we can define its conjugate function definitely. In this paper, we propose a calculation method of the conjugate function by the inverse finite Fourier transform. Using the conjugate scaling function given by the numerical data table, we calculate a finite expansion series in a nested subspace of the multiresolution analysis generated by the scaling function. In particular; we can show that the Gibbs' phenomenon is not aroused at the discontinuity points of a function. Next, we define a several order differential filter from the wavelet expansion formula by the property of the cardinal B-spline. Using these differential filters, we propose an identification method of a transfer function. In order to demonstrate the property and effectiveness of the proposed method, some numerical simulations are presented.