Advertisement

The Finite Heisenberg-Weyl Groups in Radar and Communications

  • S. D. Howard
  • A. R. Calderbank
  • W. Moran
Open Access
Research Article
Part of the following topical collections:
  1. Frames and Overcomplete Representations in Signal Processing, Communications, and Information Theory

Abstract

We investigate the theory of the finite Heisenberg-Weyl group in relation to the development of adaptive radar and to the construction of spreading sequences and error-correcting codes in communications. We contend that this group can form the basis for the representation of the radar environment in terms of operators on the space of waveforms. We also demonstrate, following recent developments in the theory of error-correcting codes, that the finite Heisenberg-Weyl groups provide a unified basis for the construction of useful waveforms/sequences for radar, communications, and the theory of error-correcting codes.

Keywords

Radar Information Technology Quantum Information Spreading Sequence Unify Basis 

References

  1. 1.
    Folland GB: Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ, USA; 1989.zbMATHGoogle Scholar
  2. 2.
    Miller W: Topics in harmonic analysis with applications to radar and sonar. In Radar and Sonar, Part I, IMA Volumes in Mathematics and Its Applications. Edited by: Blahut R, Miller W, Wilcox C. Springer, New York, NY, USA; 1991.Google Scholar
  3. 3.
    Auslander L, Tolimieri R: Radar ambiguity functions and group theory. SIAM Journal on Mathematical Analysis 1985, 16(3):577–601. 10.1137/0516043MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mackey GW: Some remarks on symplectic automorphisms. Proceedings of the American Mathematical Society 1965, 16: 393–397. 10.1090/S0002-9939-1965-0177064-5MathSciNetCrossRefGoogle Scholar
  5. 5.
    Segal IE: Transforms for operators and symplectic automorphisms over a locally compact abelian group. Mathematica Scandinavica 1963, 13: 31–43.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Weil A: Sur certains groupes d'opérateurs unitaires. Acta Mathematica 1964, 111: 143–211. 10.1007/BF02391012MathSciNetCrossRefGoogle Scholar
  7. 7.
    Richman MS, Parks TW, Shenoy RG: Discrete-time, discrete-frequency, time-frequency analysis. IEEE Transactions on Signal Processing 1998, 46(6):1517–1527. 10.1109/78.678465CrossRefGoogle Scholar
  8. 8.
    Tolimieri R, An M: Time-Frequency Representations. Birkhäuser Boston, Boston, Mass, USA; 1998.zbMATHGoogle Scholar
  9. 9.
    Bottomley GE: Signature sequence selection in a CDMA system with orthogonal coding. IEEE Transactions on Vehicular Technology 1993, 42(1):62–68. 10.1109/25.192388CrossRefGoogle Scholar
  10. 10.
    Yang K, Kim Y-K, Vijay Kumar P: Quasi-orthogonal sequences for code-division multiple-access systems. IEEE Transactions on Information Theory 2000, 46(3):982–993. 10.1109/18.841175MathSciNetCrossRefGoogle Scholar
  11. 11.
    Welti G: Quaternary codes for pulsed radar. IEEE Transactions on Information Theory 1960, 6(3):400–408. 10.1109/TIT.1960.1057572CrossRefGoogle Scholar
  12. 12.
    Golay M: Complementary series. IEEE Transactions on Information Theory 1961, 7(2):82–87. 10.1109/TIT.1961.1057620MathSciNetCrossRefGoogle Scholar
  13. 13.
    Budisin SZ, Popovic BM, Indjin IM: Designing radar signals using complementary sequences. IEEE Transactions on Aerospace and Electronic Systems 1985, 21(2):170–179.MathSciNetGoogle Scholar
  14. 14.
    Macwilliams FJ, Sloane NJA: The Theory of Error Correcting Codes. North-Holland Elsevier, Amsterdam, The Netherlands; 1983.zbMATHGoogle Scholar
  15. 15.
    Calderbank AR, Cameron PJ, Kantor WM, Seidel JJ:Open image in new window-Kerdock codes, orthogonal spreads, and extremal euclidean line-sets. Proceedings of the London Mathematical Society 1997, 75(2):436–480. 10.1112/S0024611597000403MathSciNetCrossRefGoogle Scholar
  16. 16.
    Reiter H: Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, London, UK; 1968.zbMATHGoogle Scholar
  17. 17.
    Renes JM, Blume-Kohout R, Scott AJ, Caves CM: Symmetric informationally complete quantum measurements. Journal of Mathematical Physics 2004, 45(6):2171–2180. 10.1063/1.1737053MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wilcox CH: The synthesis problem for radar ambiguity functions. In Radar and Sonar, Part I, IMA Series in Mathematics and Its Applications. Volume 32. Springer, New York, NY, USA; 1991:229–260.Google Scholar
  19. 19.
    Feichtinger HG, Kozek W: Quantization of TF lattice-invariant operators on elementary LCA groups. In Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis. Edited by: Feichtinger HG, Strohmer T. Birkhäuser, Boston, Mass, USA; 1998:233–266. chapter 7CrossRefGoogle Scholar
  20. 20.
    Delsarte P, Goethals JM, Seidel JJ: Bounds for systems of lines and Jacobi polynomials. Philips Research Reports 1975, 30(3):91–105.zbMATHGoogle Scholar
  21. 21.
    Lemmens PWH, Seidel JJ: Equiangular lines. Journal of Algebra 1973, 24(3):494–512. 10.1016/0021-8693(73)90123-3MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hoggar SG: 64 lines from a quaternionic polytope. Geometriae Dedicata 1998, 69(3):287–289. 10.1023/A:1005009727232MathSciNetCrossRefGoogle Scholar
  23. 23.
    Strohmer T, Heath RW Jr.: Grassmannian frames with applications to coding and communication. Applied and Computational Harmonic Analysis 2003, 14(3):257–275. 10.1016/S1063-5203(03)00023-XMathSciNetCrossRefGoogle Scholar
  24. 24.
    Perelomov AM: Generalized Coherent States and Their Applications. Springer, Berlin, Germany; 1986.CrossRefGoogle Scholar
  25. 25.
    Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.CrossRefGoogle Scholar
  26. 26.
    Perelomov AM: Coherent states for arbitrary Lie group. Communications in Mathematical Physics 1972, 26(3):222–236. 10.1007/BF01645091MathSciNetCrossRefGoogle Scholar
  27. 27.
    Brown EH: Generalizations of kervaire's invariant. Annals of Mathematics 1972, 95: 368–383. 10.2307/1970804MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pottie GJ, Calderbank AR: Channel coding strategies for cellular radio. IEEE Transactions on Vehicular Technology 1995, 44(4):763–770. 10.1109/25.467960CrossRefGoogle Scholar
  29. 29.
    Costas JP: A study of a class of detection waveforms having nearly ideal range-Doppler ambiguity properties. Proceedings of the IEEE 1984, 72(8):996–1009.CrossRefGoogle Scholar
  30. 30.
    Kerdock AM: A class of low rate nonlinear binary codes. Information and Control 1972, 20(2):182–187. 10.1016/S0019-9958(72)90376-2MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hammons AR Jr., Kumar PV, Calderbank AR, Sloane NJA, Solé P:TheOpen image in new window-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Transactions on Information Theory 1994, 40(2):301–319. 10.1109/18.312154MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wootters WK, Fields BD: Optimal state-determination by mutually unbiased measurements. Annals of Physics 1989, 191(2):363–381. 10.1016/0003-4916(89)90322-9MathSciNetCrossRefGoogle Scholar
  33. 33.
    Davis JA, Jedwab J: Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes. IEEE Transactions on Information Theory 1999, 45(7):2397–2417. 10.1109/18.796380MathSciNetCrossRefGoogle Scholar
  34. 34.
    Shapiro HS: Extremal problems for polynomials and power series, Sc.M. thesis. Massachusetts Institute of Technology, Cambridge, Mass, USA; 1951.Google Scholar
  35. 35.
    Golay M: Multislit spectrometry. Journal of the Optical Society of America June 1949, 39: 437–444. 10.1364/JOSA.39.000437CrossRefGoogle Scholar
  36. 36.
    Budisin SZ: New complementary pairs of sequences. Electronics Letters 1990, 26(13):881–883. 10.1049/el:19900576CrossRefGoogle Scholar
  37. 37.
    Craigen R: Complex golay sequences. Journal of Combinatorial Mathematics and Combinatorial Computing 1994, 15: 161–169.MathSciNetzbMATHGoogle Scholar
  38. 38.
    Calderbank AR, Rains EM, Shor PW, Sloane NJA: Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory 1998, 44(4):1369–1387. 10.1109/18.681315MathSciNetCrossRefGoogle Scholar

Copyright information

© Howard et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • S. D. Howard
    • 1
  • A. R. Calderbank
    • 2
  • W. Moran
    • 3
  1. 1.Defence Science and Technology OrganisationEdinburghAustralia
  2. 2.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  3. 3.Department of Electrical and Electronic EngineeringThe University of MelbourneVictoriaAustralia

Personalised recommendations