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Constructions and a Generalization of Perfect Autocorrelation Sequences on ℤ

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Multiscale Signal Analysis and Modeling

Abstract

Low autocorrelation signals have fundamental applications in radar and communications. We construct constant amplitude zero autocorrelation (CAZAC) sequences x on the integers ℤ by means of Hadamard matrices. We then generalize this approach to construct unimodular sequences x on ℤ whose autocorrelations A x are building blocks for all functions on ℤ. As such, algebraic relations between A x and A y become relevant. We provide conditions for the validity of the formulas A x+y =A x +A y .

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Notes

  1. 1.

    For the necessary measure theory and definitions of Borel and Lebesgue measure we refer to [7].

References

  1. Auslander L, Barbano PE (1998) Communication codes and Bernoulli transformations. Appl Comput Harmon Anal 5(2):109–128

    Article  MathSciNet  MATH  Google Scholar 

  2. Bell DA (1966) Walsh functions and Hadamard matrices. Electron Lett 2:340–341

    Article  Google Scholar 

  3. Benedetto JJ (1991) A multidimensional Wiener–Wintner theorem and spectrum estimation. Trans Am Math Soc 327(2):833–852

    Article  MathSciNet  MATH  Google Scholar 

  4. Benedetto JJ (1997) Harmonic analysis and applications. CRC Press, Boca Raton, FL

    Google Scholar 

  5. Benedetto JJ, Benedetto RL, Woodworth JT (2011) Björk CAZACs: theory, geometry, implementation, and waveform ambiguity behavior. Preprint, to be submitted

    Google Scholar 

  6. Benedetto JJ, Benedetto RL, Woodworth JT (2012) Optimal ambiguity functions and Weil’s exponential sum bounds. J. Fourier Analysis and Applications 18:471–487

    Article  MathSciNet  MATH  Google Scholar 

  7. Benedetto JJ, Czaja W (2009) Integration and modern analysis. Birkhäuser Boston Inc., Boston, MA

    Book  MATH  Google Scholar 

  8. Benedetto JJ, Datta S (2010) Construction of infinite unimodular sequences with zero autocorrelation. Adv Comput Math 32(2):191–207

    Article  MathSciNet  MATH  Google Scholar 

  9. Benedetto JJ, Donatelli JJ (2007) Ambiguity function and frame theoretic properties of periodic zero autocorrelation waveforms. IEEE J Special Topics Signal Process 1:6–20

    Article  Google Scholar 

  10. Björck J, Saffari B (1995) New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries. C R Acad Sci 320:319–324

    MATH  Google Scholar 

  11. Christensen O (2003) An introduction to frames and riesz bases. Birkhäuser, Basel

    MATH  Google Scholar 

  12. Datta S (2007) Wiener’s generalized harmonic analysis and waveform design. Ph.D. thesis. University of Maryland, College Park, Maryland

    Google Scholar 

  13. Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics, Philadelphia, PA

    Book  MATH  Google Scholar 

  14. Hadamard J (1893) Résolution d’une question relative aux déterminants. Bulletin des Sciences Mathématiques 17:240–246

    Google Scholar 

  15. Helleseth T, Kumar PV (1998) Sequences with low correlation. In: Handbook of coding theory, Vol. I, II. North-Holland, Amsterdam, pp 1765–1853

    Google Scholar 

  16. Horadam KJ (2007) Hadamard matrices and their applications. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  17. Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, Cambridge (Corrected reprint of the 1985 original)

    Google Scholar 

  18. Kerby R (1990) The correlation function and the Wiener–Wintner theorem in higher dimension. Ph.D. thesis. University of Maryland, College Park

    Google Scholar 

  19. Kerby R (2008) Personal communication

    Google Scholar 

  20. Levanon N, Mozeson E (2004) Radar signals. Wiley Interscience, IEEE Press

    Book  Google Scholar 

  21. Long ML (2001) Radar reflectivity of land and sea. Artech House, Boston

    Google Scholar 

  22. Mow WH (1996) A new unified construction of perfect root-of-unity sequences. In: Proceedings of IEEE 4th International Symposium on Spread Spectrum Techniques and Applications (Germany), pp 955–959

    Google Scholar 

  23. Nathanson FE (1999) Radar design principles—signal processing and the environment. SciTech Publishing Inc., Mendham, NJ

    Google Scholar 

  24. Paley REAC (1933) On orthogonal matrices. J Math Phys 12:311–320

    Google Scholar 

  25. Richards MA, Scheer JA, Holm WA (eds.) (2010) Principles of modern radar: basic principles. SciTech Publishing Inc., Raleigh, NC

    Google Scholar 

  26. Schipp F, Wade WR, Simon P (1990) Walsh series, an introduction to dyadic harmonic analysis. Taylor & Francis

    Google Scholar 

  27. Stein EM, Weiss G (1971) Introduction to fourier analysis on euclidean spaces. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  28. Stimson GW (1998) Introduction to airborne radar. SciTech Publishing Inc., Mendham, NJ

    Google Scholar 

  29. Sylvester JJ (1867) Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to newton’s rule, ornamental tile-work, and the theory of numbers. Phil Mag 34:461–475

    Google Scholar 

  30. Ulukus S, Yates RD (2001) Iterative construction of optimum signature sequence sets in synchronous CDMA systems. IEEE Trans Inform Theor 47(5):1989–1998

    Article  MathSciNet  MATH  Google Scholar 

  31. Verdú S (1998) Multiuser detection. Cambridge University Press, Cambridge, UK

    MATH  Google Scholar 

  32. Viterbi AJ (1995) CDMA: principles of spread spectrum communication. Addison-Wesley, Pearson, Boston

    MATH  Google Scholar 

  33. Walsh JL (1923) A closed set of normal orthogonal functions. Am J Math 45:5–24

    Article  MATH  Google Scholar 

  34. Wiener N (1930) Generalized harmonic analysis. Acta Math (55):117–258

    Article  MathSciNet  MATH  Google Scholar 

  35. Wiener N (1988) The Fourier integral and certain of its applications. Cambridge Mathematical Library (Reprint of the 1933 edition. With a foreword by Jean-Pierre Kahane). Cambridge University Press, Cambridge

    Google Scholar 

  36. Wiener N, Wintner A (1939) On singular distributions. J Math Phys 17:233–246

    Google Scholar 

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Acknowledgements

The first named author gratefully acknowledges the support of ONR Grant N00014-09-1-0144 and MURI-ARO Grant W911NF-09-1-0383. The second named author gratefully acknowledges the support of AFOSR Grant FA9550-10-1-0441.

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Authors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto

  2. Department of Mathematics, University of Idaho, Moscow, ID, 83844, USA

    Somantika Datta

Authors
  1. John J. Benedetto
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  2. Somantika Datta
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Corresponding author

Correspondence to John J. Benedetto .

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Editors and Affiliations

  1. , Department of Mathematics, Ohio University, Ohio University 1, Athens, 45701, Ohio, USA

    Xiaoping Shen

  2. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614, Illinois, USA

    Ahmed I. Zayed

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Benedetto, J.J., Datta, S. (2013). Constructions and a Generalization of Perfect Autocorrelation Sequences on ℤ . In: Shen, X., Zayed, A. (eds) Multiscale Signal Analysis and Modeling. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4145-8_8

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  • DOI: https://doi.org/10.1007/978-1-4614-4145-8_8

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  • Print ISBN: 978-1-4614-4144-1

  • Online ISBN: 978-1-4614-4145-8

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