Abstract
In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The convergence of the regularized derivative interpolation is studied. The numerical results are given and compared with derivative interpolation using the Tikhonov regularization method. The regularized derivative interpolation in this paper is more accurate in computation.
Highlights
IntroductionWe present the problem of finding the derivative of non-bandlimited signals by the sampling theorem
The computation of the derivative is widely applied in science and engineering [1].we present the problem of finding the derivative of non-bandlimited signals by the sampling theorem.Definition 1: Suppose a function f ∈ L2 (R), its Fourier transform fis: = f (ω ) F= ( f )(ω)∫+∞ f (t ) e−iωtdt, ω ∈ R −∞ (1)Definition 2: A function f ∈ L2 (R) is said to be Ω -band-limited if f (ω ) = 0, for every ω ∉[−Ω, Ω]
In [6], a regularized derivative interpolation formula is presented for Ω -bandlimited functions
Summary
We present the problem of finding the derivative of non-bandlimited signals by the sampling theorem. Definition 2: A function f ∈ L2 (R) is said to be Ω -band-limited if f (ω ) = 0 , for every ω ∉[−Ω, Ω] F (ω ) eiωtdω, a.e. For band-limited signals, we have the Shannon sampling theorem [2] [3]. In [4], Marks presented an algorithm to find the derivative of band-limited signals by the sampling theorem:. In [6], a regularized derivative interpolation formula is presented for Ω -bandlimited functions. A regularized derivative interpolation formula will be presented for nonbandlimited functions. In the case non-bandlimited functions, the error estimate is different and the step size h of the samples is necessary to be close to zero
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