Skip to main content
SpringerLink
Log in

The Diagonal Reduction Algorithm Using Fast Givens

  • Conference paper
  • First Online:
Computer Mathematics

  • 718 Accesses

Abstract

Recently, a new lattice basis reduction notion, called diagonal reduction, was proposed for lattice-reduction-aided detection (LRAD) of multiinput multioutput (MIMO) systems. In this paper, we improve the efficiency of the diagonal reduction algorithm by using the fast Givens transformations. The technique of the fast Givens is applicable to a family of LLL-type lattice reduction methods to improve efficiency. Also, in this paper, we investigate dual diagonal reduction and derive an upper bound of the proximity factors for a family of dual reduction aided successive interference cancelation (SIC) decoding. Our upper bound not only extends an existing bound for dual LLL reduction to a family of dual reduction methods, but also improves the existing bound.

973 Program Project Under Grant 2010CB327900.

National Natural Science Foundation of China Under Grant 11271084.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Flop count: addition/multiplication/division/max/rounding, 1 flop.

References

  1. Agrell, E., Eriksson, T., Vardy, A., Zeger, K.: Closest point search in lattices. IEEE Trans. Inf. Theory 48, 2201–2214 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, S., Babai, L., Stern, J.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54, 317–331 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babai, L.: On Lovász’s lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Golub, G.H., Van Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  5. Hassibi, B., Vikalo, H.: On the sphere-decoding algorithm I: Expected complexity. IEEE Trans. Signal Process. 53, 2806–2818 (2005)

    Article  MathSciNet  Google Scholar 

  6. Jaldén, J., Ottersen, B.: On the complexity of sphere decoding in digital communications. IEEE Trans. Signal Process. 53, 1474–1484 (2005)

    Article  MathSciNet  Google Scholar 

  7. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factorizing polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ling, C.: Towards characterizing the performance of approximate lattice decoding. Proc. Int. Symp. Turbo Codes/Int. Conf. Source Channel Coding ’06. Munich, Germany (2006).

    Google Scholar 

  9. Ling, C.: On the proximity factors of lattice reduction-aided decoding. IEEE Trans. Signal Process. 59, 2795–2808 (2011)

    Article  MathSciNet  Google Scholar 

  10. Ling, C., Howgrave-Graham, N.: Effective LLL reduction for lattice decoding. In: Proceedings IEEE Int. Symp. Inf. Theory (ISIT). Nice, France (2007).

    Google Scholar 

  11. Ling, C., Mow, W.H., Gan, L.: Dual-lattice ordering and partial lattice reduction for SIC-based MIMO detection. IEEE J. Sel. Topics Signal Process. 3, 975–985 (2009)

    Article  Google Scholar 

  12. Mow, W.H.: Universal lattice decoding: Principle and recent advances. Wireless Wirel. Commun. Mobile Comput. 3, 553–569 (2003). (Special Issue on Coding and Its Applications Wireless CDMA System)

    Article  Google Scholar 

  13. Nguyen, P.Q., Vallée, B.: The LLL Algorithm: Survey and Applications. Springer, Berlin (2009)

    Google Scholar 

  14. Schnorr, C.P., Euchner, M.: Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Math. Progr. 66, 181–199 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Taherzadeh, M., Mobasher, A., Khandani, A.K.: LLL reduction achieves the receive diversity in MIMO decoding. IEEE Trans. Inf. Theory 53, 4801–4805 (2007)

    Article  MathSciNet  Google Scholar 

  16. Wübben, D., Seethaler, D., Jaldén, J., Marz, G.: Lattice reduction: a survey with applications in wireless communications. IEEE Signal Process. Mag. 28, 70–91 (2011)

    Article  Google Scholar 

  17. Xie, X., Chang, X.W., Borno, M.A.: Partial LLL reduction. In: Proceedings of IEEE GLOBECOM (2011)

    Google Scholar 

  18. Zhang, W., Qiao, S., Wei, Y.: HKZ and Minkowski reduction algorithms for lattice-reduction- aided MIMO detection. IEEE Trans. Signal Process. 60, 5963–5976 (2012)

    Article  MathSciNet  Google Scholar 

  19. Zhang, W., Qiao, S., Wei, Y.: A diagonal lattice reduction algorithm for MIMO detection. IEEE Signal Process. Lett. 19, 311–314 (2012)

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank Professor Lihong Zhi and referees for their useful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266000, People’s Republic of China

    Wen Zhang

  2. Shanghai Key Laboratory of Contemporary Applied Mathematics, Department of Computing and Software, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Sanzheng Qiao

  3. Shanghai Key Laboratory of Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

Authors
  1. Wen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanzheng Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yimin Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yimin Wei .

Editor information

Editors and Affiliations

  1. Academy of Mathematics & System Science, Beijing, China

    Ruyong Feng

  2. Department of Mathematics and Computer S, University of Antwerp, Antwerp, Antwerpen, Belgium

    Wen-shin Lee

  3. Department of Mathematical Information S, Tokyo University of Science, Tokyo, Tokyo, Japan

    Yosuke Sato

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zhang, W., Qiao, S., Wei, Y. (2014). The Diagonal Reduction Algorithm Using Fast Givens. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_30

Download citation

Publish with us

Policies and ethics

Search

Navigation