BIT Numerical Mathematics

, Volume 56, Issue 3, pp 807–832

# Bounds for truncation and perturbation errors of nonuniform sampling series

Article

## Abstract

This paper introduces a comprehensive study of truncation and perturbation errors of nonuniform sampling, interpolation, representations. We derive pointwise as well as global estimates associated with the truncated nonuniform sampling formulae. The study of the perturbation error involves the estimation of the amplitude error which results from replacing the actual samples by approximate ones, as well as the jitter error arising when alternative nodes replace the original nodes, is carried out with rigorous error bounds. Numerical and geometrical verifications of the accuracy of our estimates are exhibited.

## Keywords

Nonuniform sampling theorems Truncation  Amplitude and jitter errors

## Mathematics Subject Classification

30D10 41A05 41A30 94A20

## Notes

### Acknowledgments

The authors wish to thank Alexander von Humboldt foundation for the Grant 3.4-JEM/142916 and 3.4-8131KAT/1039259.

## References

1. 1.
Annaby, M.H., Asharabi, R.M.: Truncation, amplitude and jitter errors on $$\mathbb{R}$$ for sampling series derivatives. J. Approx. Theory. 163, 336–362 (2011)
2. 2.
Berrut, J.-P.: Barycentric formulae for cardinal (SINC-) interpolants. Numer. Math. 54, 703–718 (1989) [Erratum in Numer. Math. 55 (1989), 747]Google Scholar
3. 3.
Berrut, J.-P.: First applications of a formula for the error of finite sinc interpolation. Numer. Math. 112, 341–361 (2009)
4. 4.
Birkhoff, G., MacLane, S.: Survey of Modern Algebra. The Macmillan Co., New York (1947)
5. 5.
Boas, R.P.: Entire Functions. Academic Press, New York (1954)
6. 6.
Brown Jr., J.L.: Bounds for truncation error in sampling expansion of bandlimited signals. IEEE Trans. Inf. Theory IT-15, 440–444 (1969)Google Scholar
7. 7.
Butzer, P.L.: A survey of the Whitteker–Shannon sampling theorem and some of its extensions. J. Math. Res. Expo. 3, 185–212 (1983)
8. 8.
Butzer, P.L., Engels, W., Scheben, U.: Magnitude of the truncation error in sampling expansions of bandlimited signals. IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 906–912 (1982)Google Scholar
9. 9.
Butzer, P.L., Higgins, J.R., Stens, R.L.: Sampling Theory of Signal Analysis, Development of Mathematics 1950–2000. Birkhäuser, Basel (2000)
10. 10.
Butzer, P.L., Splettstösser, W.: On quantization, truncation and jitter errors in the sampling theorem and its generalizations. Signal Process. 2, 101–112 (1980)
11. 11.
Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. Nonuniform sampling. Inf. Technol. Transm. Process. Storage, pp. 17–121. Kluwer/Plenum, New York (2001)Google Scholar
12. 12.
Butzer, P.L., Splettstösser, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math.-Verein. 90, 1–70 (1988)
13. 13.
Eldar, Y.C., Oppenheim, A.V.: Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples. IEEE Trans. Signal Process. 48, 2864–2875 (2000)
14. 14.
Gautschi, W.: Barycentric formulae for cardinal (SINC-) interpolants by Jean-Paul Berrut. Numer. Math. 87, 791–792 (2000)
15. 15.
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam, Oxford (2007)
16. 16.
Helms, H.D., Thomas, J.B.: Truncation error of sampling theorem expansion. Proc. IRE 50, 179–184 (1962)
17. 17.
Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis Foundations. Oxford University Press, Oxford (1996)
18. 18.
Hinsen, G.: Explicit irregular sampling formulas. J. Comput. Appl. Math. 40, 177–198 (1992)
19. 19.
Jagerman, D.: Bounds for truncation error of the sampling expansion. SIAM J. Appl. Math. 14, 714–723 (1966)
20. 20.
Jenq, Y.C.: Digital spectra of nonuniformly sampled signals: fundamentals and high-speed waveform digitizers. IEEE Trans. Instrum. Meas. 37, 245–251 (1988)
21. 21.
Li, X.M.: Uniform bounds for sampling expansions. J. Approx. Theory 93, 100–113 (1998)
22. 22.
Margolis, E., Eldar, Y.C.: Nonuniform sampling of periodic bandlimited signals. IEEE Trans. Signal Process. 56, 2728–2745 (2008)
23. 23.
McArthur, K.M., Bowers, K.L., Lund, J.: The Sinc method in multiple space dimensions: model problems. Numer. Math. 56, 789–816 (1989)
24. 24.
Piper Jr., H.S.: Bounds for truncation error in sampling expansios of finite energy bandlimited signals. IEEE Trans. Inf. Theory IT-21, 482–485 (1975)Google Scholar
25. 25.
Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integrals and Series, vol. 15. Gordon and Breach, New York (1986)
26. 26.
Splettstösser, W., Stens, R.L., Wilmes, G.: On approximation by the interpolatin series of G. Valiron. Funct. Approx. Comment. Math. 11, 39–56 (1981)
27. 27.
Tao, R., Li, B.-Z., Wang, Y., Aggrey, G.K.: On sampling of band-limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56, 5454–5464 (2008)
28. 28.
Yao, K., Thomas, J.B.: On truncation error bounds for sampling representations of bandlimited signals. IEEE Trans. Aerosp. Electron. Syst. AES-2, 640–647 (1966)Google Scholar
29. 29.
Yen, J.L.: On nonuniform sampling of bandwidth-limited signals. IEEE Trans. Circuit Theory 3, 251–257 (1956)