2000 character limit reached
Some remarks on regularized Shannon sampling formulas (2407.16401v2)
Published 23 Jul 2024 in math.NA and cs.NA
Abstract: The fast reconstruction of a bandlimited function from its sample data is an essential problem in signal processing. In this paper, we consider the widely used Gaussian regularized Shannon sampling formula in comparison to regularized Shannon sampling formulas employing alternative window functions, such as the sinh-type window function and the continuous Kaiser-Bessel window function. It is shown that the approximation errors of these regularized Shannon sampling formulas possess an exponential decay with respect to the truncation parameter. The main focus of this work is to address minor gaps in preceding papers and rigorously prove assumptions that were previously based solely on numerical tests. In doing so, we demonstrate that the sinh-type regularized Shannon sampling formula has the same exponential decay as the continuous Kaiser-Bessel regularized Shannon sampling formula, but both have twice the exponential decay of the Gaussian regularized Shannon sampling formula. Additionally, numerical experiments illustrate the theoretical results.
- M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1972.
- Á. Baricz and T.K. Pogány. Functional inequalities for modified Struve functions II. Math. Inequal. Appl., 17:1387–1398, 2014.
- A. H. Barnett. Aliasing error of the exp(β1−z2)𝛽1superscript𝑧2\exp(\beta\sqrt{1-z^{2}})roman_exp ( italic_β square-root start_ARG 1 - italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) kernel in the nonuniform fast Fourier transform. Appl. Comput. Harmon. Anal., 51:1–16, 2021.
- Flatiron Institute nonuniform fast Fourier transform libraries (FINUFFT). http://github.com/flatironinstitute/finufft.
- L. Chen and H. Zhang. Sharp exponential bounds for the Gaussian regularized Whittaker–Kotelnikov–Shannon sampling series. J. Approx. Theory, 245:73–82, 2019.
- I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992.
- I. Daubechies and R. DeVore. Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order. Ann. of Math. (2), 158:679–710, 2003.
- H. G. Feichtinger. New results on regular and irregular sampling based on Wiener amalgams. In K. Jarosz, editor, Function Spaces, Proc Conf, Edwardsville/IL (USA) 1990, volume 136 of Lect. Notes Pure Appl. Math.,pages 107––121. New York, 1992.
- H. G. Feichtinger. Wiener amalgams over Euclidean spaces and some of their applications. In K. Jarosz, editor, Function Spaces, Proc Conf, Edwardsville/IL (USA) 1990, volume 136 of Lect. Notes Pure Appl. Math., pages 123––137. New York, 1992.
- I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, New York, 1980.
- D. Jagerman. Bounds for truncation error of the sampling expansion. SIAM J. Appl. Math., 14(4):714–723, 1966.
- On regularized Shannon sampling formulas with localized sampling. Sampl. Theory Signal Process. Data Anal., 20: Paper No. 20, 34 pp., 2022.
- On numerical realizations of Shannon’s sampling theorem. Sampl. Theory Signal Process. Data Anal., 22: Paper No. 13, 33 pp., 2024.
- V. A. Kotelnikov. On the transmission capacity of the ether and wire in electrocommunications. In J.J. Benedetto and P.J.S.G. Ferreira (eds.) Modern Sampling Theory: Mathematics and Application, pages 27–45. Birkhäuser, Boston, 2001. Translated from Russian.
- R. Lin and H. Zhang. Convergence analysis of the Gaussian regularized Shannon sampling formula. Numer. Funct. Anal. Optim., 38(2):224–247, 2017.
- Optimal learning of bandlimited functions from localized sampling. J. Complexity, 25(2):85–114, 2009.
- F. Natterer. Efficient evaluation of oversampled functions. J. Comput. Appl. Math., 14(3):303–309, 1986.
- F. Oberhettinger. Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer, Berlin, 1990.
- J. R. Partington. Interpolation, Identification, and Sampling. Clarendon Press, New York, 1997.
- Numerical Fourier Analysis. Second edition, Birkhäuser/Springer, Cham, 2023.
- D. Potts and M. Tasche. Continuous window functions for NFFT. Adv. Comput. Math. 47(2): Paper 53, 34 pp., 2021.
- L. Qian. On the regularized Whittaker–Kotelnikov–Shannon sampling formula. Proc. Amer. Math. Soc., 131(4):1169–1176, 2003.
- L. Qian. The regularized Whittaker-Kotelnikov-Shannon sampling theorem and its application to the numerical solutions of partial differential equations. PhD thesis, National Univ. Singapore, 2004.
- L. Qian and D.B. Creamer. Localization of the generalized sampling series and its numerical application. SIAM J. Numer. Anal. 43(6):2500–2516, 2006.
- L. Qian and H. Ogawa. Modified sinc kernels for the localized sampling series. Sampl. Theory Signal Image Process. 4(2):121–139, 2005.
- T. S. Rappaport. Wireless Communications: Principles and Practice. Prentice Hall, New Jersey, 1996.
- G. Schmeisser and F. Stenger. Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process., 6(2):199–221, 2007.
- C. E. Shannon. Communication in the presence of noise. Proc. I.R.E., 37:10–21, 1949.
- T. Strohmer and J. Tanner. Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling. SIAM J. Numer. Anal., 44(3):1071-1094, 2006.
- Complex analytic approach to the sinc-Gauss sampling formula. Japan J. Ind. Appl. Math., 25:209–231, 2008.
- E.T. Whittaker. On the functions which are represented by the expansions of the interpolation theory. Proc. R. Soc. Edinb., 35:181–194, 1915.