×

Localized sampling in the presence of noise. (English) Zbl 1095.41020

The reconstruction of a band-limited function \(f(x)\) with bandwith \(\sigma>0\) i.e., a function in \(L^2(R)\) whose Fourier transform vanishes ouside the interval \([-\sigma,\sigma]\), making use of the Shannon sampling theorem, is a typical problem in data processing. In this note, the so-called “localized sampling” which is based upon the Shannon sampling theorem is used in order to approximate a band-limited function in the presence of noise. Error bounds are deduced with applications to various types of error when the sampled values are obtained via real-world acquisition devices.

MSC:

41A80 Remainders in approximation formulas
41A30 Approximation by other special function classes
65D25 Numerical differentiation
65G99 Error analysis and interval analysis
94A24 Coding theorems (Shannon theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zayed, A. I., Advances in Shannon’s Sampling Theory (1993), CRC Press: CRC Press Boca Raton · Zbl 0868.94011
[2] Qian, L. W., On the regularized Whittaker-Kotel’nikov-Shannon sampling formula, Proc. Amer. Math. Soc., 131, 4, 1169-1176 (2003) · Zbl 1018.94004
[3] L.W. Qian, D.B. Creamer, A modification of the sampling series with a Gaussian multiplier, Sampl. Theory Signal Image Process.—An International Journal (in press); L.W. Qian, D.B. Creamer, A modification of the sampling series with a Gaussian multiplier, Sampl. Theory Signal Image Process.—An International Journal (in press) · Zbl 1137.41355
[4] Butzer, P. L.; Lei, J. J., Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal., 4, 2, 245-255 (2000) · Zbl 1089.94503
[5] Sun, W. C.; Zhou, X. W., Average sampling in spline subspaces, Appl. Math. Lett., 15, 233-237 (2002) · Zbl 0998.94518
[6] Aldroubi, A.; Sun, Q. Y.; Tang, W. S., Nonuniform average sampling and reconstruction in multiply generated shift-invariant spaces, Constr. Approx., 20, 2, 173-189 (2004) · Zbl 1049.42017
[7] Wiley, R. G., Recovery of bandlimited signals from unequally spaced samples, IEEE Trans. Commun., 26, 1, 135-137 (1978) · Zbl 0372.94013
[8] Butzer, P. L.; Splettstösser, W., On quantization, truncation and jitter errors in the sampling theorem and its generalizations, Signal Process., 2, 2, 101-112 (1980)
[9] Atreas, N.; Bagis, N.; Karanikas, C., The information loss error and the jitter error for regular sampling expansions, Sampl. Theory Signal Image Process., 1, 3, 261-276 (2002) · Zbl 1052.94012
[10] Butzers, P., A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition, 3, 186-212 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.