Abstract
In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for band-limited signals is presented. The ill-posedness is analyzed by the Shannon sampling theorem. The convergence of the regularized derivative interpolation is studied by the combination of a regularized Fourier transform and the Shannon sampling theorem. The error estimation is given, and high-order derivatives are also considered. The algorithm of the regularized derivative interpolation is compared with derivative interpolation using some other algorithms.
Highlights
The computation of the derivative is widely applied in engineering, signal processing, and neural networks [1,2,3]
Example 5 In this example, we show how the square error depends on the regularization parameter α and give the optimal α
6 Conclusion The interpolation formula obtained by differentiating the formula of the Shannon sampling theorem is not stable
Summary
The computation of the derivative is widely applied in engineering, signal processing, and neural networks [1,2,3]. We describe the problem of finding the derivative of band-limited signals by the Shannon sampling theorem [5]. In [7], Marks presented an algorithm to find the derivative of band-limited signals by the sampling theorem:. In [10], the series to approximate a band-limited function and its derivative is given, but the ill-posedness and noise are not considered. We measure the approximation error in the L∞-norm on any finite interval in R and assume additive l∞ noise. It is not necessary for the step size h of the samples to be close to 0
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