Abstract
Unimodular waveforms x are constructed on the integers with the property that the autocorrelation of x is one at the origin and zero elsewhere. There are three different constructions: exponentials of the form \(e^{2 \pi i n^\alpha \theta},\) sequences taken from roots of unity, and sequences constructed from the elements of real Hadamard matrices. The first is expected and elementary and the second is based on the construction of Wiener. The third is the most intricate and is really one of a family of distinct but structurally similar waveforms. A natural error estimate problem is posed for the last construction. The analytic solution is not as useful as the simulations because of the inherent counting problems in the construction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslander, L., Barbano, P.E.: Communication codes and Bernoulli transformations. Appl. Comput. Harmon. Anal. 5(2), 109–128 (1998)
Benedetto, J.J.: A multidimensional Wiener-Wintner theorem and spectrum estimation. Trans. Amer. Math. Soc. 327(2), 833–852 (1991)
Benedetto, J.J.: Harmonic Analysis and Applications. CRC, Boca Raton (1997)
Benedetto, J.J., Donatelli, J.J.: Ambiguity function and frame theoretic properties of periodic zero autocorrelation waveforms. IEEE J. Special Topics Signal Process. 1, 6–20 (2007)
Helleseth, T., Kumar, P.V.: Sequences with Low Correlation. Handbook of Coding Theory, vol. I, II, pp. 1765–1853. North-Holland, Amsterdam (1998)
Kerby, R.: The correlation function and the Wiener-Wintner theorem in higher dimension. Ph.D. thesis, University of Maryland, College Park (1990)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley-Interscience [Wiley], New York (1974)
Levanon, N., Mozeson, E.: Radar Signals. Wiley Interscience, IEEE Press, New York (2004)
Long, M.L.: Radar Reflectivity of Land and Sea. Artech House, London (2001)
Mow, W.H.: A new unified construction of perfect root-of-unity sequences. In: Proc. IEEE 4th International Symposium on Spread Spectrum Techniques and Applications (Germany), pp. 955–959, Mainz, September 1996
Nathanson, F.E.: Radar Design Principles—Signal Processing and the Environment. SciTech, Mendham (1999)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Stimson, G.W.: Introduction to Airborne Radar. SciTech, Mendham (1998)
Ulukus, S., Yates, R.D.: Iterative construction of optimum signature sequence sets in synchronous CDMA systems. IEEE Trans. Inform. Theory 47(5), 1989–1998 (2001)
Verdú, S.: Multiuser Detection. Cambridge University Press, Cambridge (1998)
Wiener, N.: Generalized harmonic analysis. Acta Math. 55, 117–258 (1930)
Wiener, N., Wintner, A.: On singular distributions. J. Math. Phys. 17, 233–246 (1939)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yuesheng Xu.
Rights and permissions
About this article
Cite this article
Benedetto, J.J., Datta, S. Construction of infinite unimodular sequences with zero autocorrelation. Adv Comput Math 32, 191–207 (2010). https://doi.org/10.1007/s10444-008-9100-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-008-9100-9