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Unit time-phase signal sets: Bounds and constructions

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Abstract

Digital signals are complex-valued functions on Z n . Signal sets with certain properties are required in various communication systems. Traditional signal sets consider only the time distortion during transmission. Recently, signal sets taking care of both the time and phase distortion have been studied, and are called time-phase signal sets. Several constructions of time-phase signal sets are available in the literature. There are a number of bounds on time signal sets (also called codebooks). They are automatically bounds on time-phase signal sets, but are bad bounds. The first objective of this paper is to develop better bounds on time-phase signal sets from known bounds on time signal sets. The second objective of this paper is to construct four series of time-phase signal sets, one of which is optimal.

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References

  1. Allop, W.O.: Complex sequences with low periodic correlations. IEEE Trans. Inf. Theory 26(3), 350–354 (1980)

    Article  Google Scholar 

  2. Bajwa, W.U., Calderbank, R., Mixon, D.G.: Two are better than one: fundamental parameters of frame coherence. Appl. Comput. Harmon. Anal. 33(1), 58–76 (2012) Preprint. arXiv:1103.0435v2 (2011)

  3. Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley-Interscience (1998)

  4. Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: Z 4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. 75(3), 436–480 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conway, J.H., Harding R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)

    Article  MATH  Google Scholar 

  6. Ding, C.: Codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006)

    Article  MATH  Google Scholar 

  7. Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, C., Yin, J.: Signal sets from functions with optimum nonlinearity. IEEE Trans. Commun. 55(5), 936–940 (2007)

    Article  Google Scholar 

  9. Grassl, M.: Computing equiangular lines in complex space. In: Calmet, J., Geiselmann, W., Muller-Quade, J. (eds.) Beth Festschrift, LNCS 5393, pp. 89–104 (2008)

  10. Gurevich, S., Hadani, R., Sochen, N.: The finite harmonic oscillator and its applications to sequences, communication and radar. IEEE Trans. Inf. Theory 54(9), 4239–4253 (2008)

    Article  MathSciNet  Google Scholar 

  11. Heath, R.W. Jr., Strohmer,T., Paulraj, A.J.: On quasi-orthogonal signature for CDMA systems. IEEE Trans. Inf. Theory 52(3), 1217–1226 (2006)

    Article  MathSciNet  Google Scholar 

  12. Helleseth, T., Kumar, V.P.: Sequences with low correlation. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. II, pp. 1765–1853. Elsevier, Amsterdam (1998)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Howard, S.D., Calderbank, A.R., Moran, M.: The finite Heisenberg-Weyl groups in radar and communications. EURASIP J. Appl. Signal Process. 2006 1–12 (2006)

    MathSciNet  Google Scholar 

  15. Karystinos, G.N.: Optimum binary signature set design and short-data-record adaptive receivers for CDMA communications. Ph.D Thesis, University of New York at Buffalo (2003)

  16. Kretschmer, F.F., Lewwis, B.L., Jr.: Doppler properties of polyphase coded pulse compression waveforms. IEEE Trans. Aero. Elec. Syst. 19(4), 521–531 (1983)

    Article  Google Scholar 

  17. Levenstein, V.I.: Bounds on the maximal cardinality of a code with bounded modules of the inner product. Soviet Math. Dokl. 26, 526–531 (1982)

    Google Scholar 

  18. Li, W.C.W.: Number theory with applications. In: Series on University Mathematics, vol. 7 (1995)

  19. Love, D.J., Heath, R.W., Strohmer, T: Grassmannian meamingforming for multiple-input multiple-output wireless systems. IEEE Trans. Inf. Theory 49(10), 2735–2747 (2003)

    Article  Google Scholar 

  20. Nelson, J.L., Temlyakov, V.N.: On the size of incoherent systems. J. Approx. Theory 163, 1238–1245 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2007)

    Article  MathSciNet  Google Scholar 

  22. Sarwate, D.: Meeting the Welch bound with equality. In: Proc. of the Sequences and their Applications, SETA’ 98, pp. 79–102. Springer, London (1999)

    Chapter  Google Scholar 

  23. Scott, A.J., Grassl, M.: SIC-POVMs: a new computer study. J. Math. Phys. 51, 042203 (2010)

    Article  MathSciNet  Google Scholar 

  24. Sidelnikov, V.M.: On mutual correlation of sequences. Probl. Kibern. 24, 15–42 (1971)

    MathSciNet  Google Scholar 

  25. Strohmer, T., Heath, R.W. Jr.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Welch, L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zauner, G.: Quantendesigns: Grundzuege einer nichtkommutativen Designtheorie. Dissertation, Universitaet Wien (1999)

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Acknowledgements

This work was done while some of the authors were attending the Summer School for Mathematical Foundations of Coding Theory and Cryptography hosted by the Beijing International Center for Mathematical Research. The authors wish to thank the Center for its support in many aspects.

C. Ding is supported by the Hong Kong Research Grants Council under Project No. 601311. K. Feng and R. Feng are supported by the NSFC Grant No. 10990011. K. Feng is also supported by the Tsinghua National Lab. for Information Science and Technology. M. Xiong’s research was supported by the Hong Kong Research Grants Council under Project No. 606211 and DAG11SC02.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Cunsheng Ding

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Keqin Feng

  3. The School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Rongquan Feng

  4. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Maosheng Xiong

  5. The School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Aixian Zhang

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  1. Cunsheng Ding
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  2. Keqin Feng
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Correspondence to Cunsheng Ding.

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Ding, C., Feng, K., Feng, R. et al. Unit time-phase signal sets: Bounds and constructions. Cryptogr. Commun. 5, 209–227 (2013). https://doi.org/10.1007/s12095-013-0085-y

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