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Optimal Ambiguity Functions and Weil’s Exponential Sum Bound

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Abstract

Complex-valued periodic sequences, u, constructed by Göran Björck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A p (u). The Björck sequences, which are defined on ℤ/pℤ for p>2 prime, are unimodular and have zero autocorrelation on (ℤ/pℤ)∖{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is \(|A_{p}(u)| \leq2/\sqrt{p} + 4/p\) outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power of Weil’s exponential sum bound, which, in turn, is a consequence of his proof of the Riemann hypothesis for finite fields. Such bounds are not only of mathematical interest, but they have direct applications as sequences in communications and radar, as well as when the sequences are used as coefficients of phase-coded waveforms.

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References

  1. Alltop, W.O.: Complex sequences with low periodic correlations. IEEE Trans. Inf. Theory 26, 350–354 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bell, M.R., Monrocq, S.: Diversity waveform signal processing for delay-Doppler measurement and imaging. Digit. Signal Process. 12(2/3), 329–346 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Benedetto, J.J., Donatelli, J.J.: Ambiguity function and frame theoretic properties of periodic zero autocorrelation waveforms. IEEE J. Spec. Top. Signal Process. 1, 6–20 (2007)

    Article  Google Scholar 

  6. Benedetto, J.J., Konstantinidis, I., Rangaswamy, M.: Phase coded waveforms and their design—the role of the ambiguity function. IEEE Signal Process. Mag. 26, 22–31 (2009)

    Article  Google Scholar 

  7. Benedetto, J.J., Benedetto, R.L., Woodworth, J.T.: Björck CAZACs: theory, geometry, and waveform ambiguity behavior. Preprint (2011)

  8. Björck, G.: Functions of modulus one on \(\textbf{Z}_{p}\) whose Fourier transforms have constant modulus. In: A. Haar Memorial Conference, Vols. I, II, Budapest, 1985. Colloq. Math. Soc. János Bolyai, vol. 49, pp. 193–197. North-Holland, Amsterdam (1987)

    Google Scholar 

  9. Björck, G.: Functions of modulus one on ℤ n whose Fourier transforms have constant modulus, and cyclic n-roots. In: Proc. of 1989 NATO Adv. Study Inst. on Recent Advances in Fourier Analysis and its Applications, pp. 131–140 (1990)

    Google Scholar 

  10. Chu, D.C.: Polyphase codes with good periodic correlation properties. IEEE Trans. Inf. Theory 18, 531–532 (1972)

    Article  MATH  Google Scholar 

  11. Frank, R.L., Zadoff, S.A.: Phase shift pulse codes with good periodic correlation properties. IRE Trans. Inf. Theory 8(6), 381–382 (1962)

    Article  Google Scholar 

  12. Golomb, S.W., Gong, G.: Signal Design for Good Correlation. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  13. Gurevich, S., Haddani, R., Sochen, N.: The finite harmonic oscillator and its applications to sequences, communication, and radar. IEEE Trans. Inf. Theory 54, 4239–4253 (2008)

    Article  Google Scholar 

  14. Haagerup, U.: Orthogonal maximal abelian ∗-subalgebras of the n×n matrices and cyclic n-roots. In: Operator Algebras and Quantum Field Theory, Rome, 1996, pp. 296–322. Int. Press, Cambridge (1997)

    Google Scholar 

  15. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Clarendon Press, Oxford (1965)

    Google Scholar 

  16. Helleseth, T., Kumar, P.V.: Sequences with low correlation. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, Vols. I, II, pp. 1765–1853. North-Holland, Amsterdam (1998)

    Google Scholar 

  17. Herman, M.A., Strohmer, T.: High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57, 2275–2284 (2009)

    Article  MathSciNet  Google Scholar 

  18. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)

    MATH  Google Scholar 

  19. Jacobsthal, E.: Über die darstellung der primzahlen der form 4n+1 als summe zweier quadrate. J. Reine Angew. Math. 132, 238–245 (1907)

    MATH  Google Scholar 

  20. Klauder, J.R.: The design of radar signals having both high range resolution and high velocity resolution. Bell Syst. Tech. J. 39, 809–820 (1960)

    Google Scholar 

  21. Klauder, J.R., Price, A.C., Darlington, S., Albersheim, W.J.: The theory and design of chirp radars. Bell Syst. Tech. J. 39, 745–808 (1960)

    Google Scholar 

  22. Levanon, N., Mozeson, E.: Radar Signals. Wiley Interscience, IEEE Press, New York (2004)

    Book  Google Scholar 

  23. Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82(4), 365–377 (1997)

    MathSciNet  MATH  Google Scholar 

  24. 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), September, pp. 955–959 (1996)

    Chapter  Google Scholar 

  25. Popovic, B.M.: Generalized chirp-like polyphase sequences with optimum correlation properties. IEEE Trans. Inf. Theory 38(4), 1406–1409 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Popovic, B.M.: Fourier duals of Björck sequences. In: SETA, pp. 253–258 (2010)

    Google Scholar 

  27. Richards, M.A., Scheer, J.A., Holm, W.A. (eds.): Principles of Modern Radar. SciTech Publishing, Raleigh (2010)

    Google Scholar 

  28. Saffari, B.: Some polynomial extremal problems which emerged in the twentieth century. In: Twentieth Century Harmonic Analysis—A Celebration, Il Ciocco, 2000. NATO Sci. Ser. II Math. Phys. Chem., vol. 33, pp. 201–233. Kluwer Academic, Dordrecht (2001)

    Google Scholar 

  29. Saffari, B.: Oral and email communications (2004–2010)

  30. Salié, H.: Über die Kloostermanschen Summen S(u,v;q). Math. Z. 34(1), 91–109 (1932)

    Article  MathSciNet  Google Scholar 

  31. Skolnik, M.I.: Introduction to Radar Systems. McGraw-Hill, New York (1980)

    Google Scholar 

  32. Strohmer, T., Heath, R.W. Jr.: Grassmannian frames with applications to coding and communications. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Turyn, R.J.: Sequences with small correlation. In: Error Correcting Codes, Proc. Sympos. Math. Res. Center, Madison, Wis., 1968, pp. 195–228. Wiley, New York (1968)

    Google Scholar 

  34. Vakman, D.E.: Sophisticated Signals and the Uncertainty Principle in Radar. Springer, New York (1969)

    Google Scholar 

  35. Weil, A.: On some exponential sums. Proc. Natl. Acad. Sci. USA 34, 204–207 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  36. Weil, A.: Sur les courbes algébriques et les variétés qui s’en déduisent. In: Actualités Sci. et Ind. no. 1041. Hermann, Paris (1948)

    Google Scholar 

  37. Woodward, P.M.: Theory of radar information. IEEE Trans. Inf. Theory 1(1), 108–113 (1953)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support of various grants. For the first-named author, the grants are ONR Grant N00014-09-1-0144 and MURI-ARO Grant W911NF-09-1-0383. For the second named author, the grant is NSF Grant DMS-0901494. The third-named author was supported by the Norbert Wiener Center as recipient of the Daniel Sweet Undergraduate Research Fellowship. Further, at the time of the first-named author’s presentation of their results at SampTA2011 in Singapore, Professor Bruno Torresani kindly pointed out two references on which we comment in Sect. 1.2. Finally, and although not represented explicitly in this paper, we have benefitted from expert advice on hardware implementation by Drs. Michael Dellomo, Joseph Lawrence, and George Linde.

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Authors and Affiliations

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

    John J. Benedetto & Joseph T. Woodworth

  2. Department of Mathematics, Amherst College, Amherst, MA, 01002, USA

    Robert L. Benedetto

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  1. John J. Benedetto
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  2. Robert L. Benedetto
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  3. Joseph T. Woodworth
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Correspondence to John J. Benedetto.

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Communicated by Thomas Strohmer.

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Benedetto, J.J., Benedetto, R.L. & Woodworth, J.T. Optimal Ambiguity Functions and Weil’s Exponential Sum Bound. J Fourier Anal Appl 18, 471–487 (2012). https://doi.org/10.1007/s00041-011-9204-3

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