Skip to main content

Geometric Properties of Grassmannian Frames for and

Abstract

Grassmannian frames are frames satisfying a min-max correlation criterion. We translate a geometrically intuitive approach for two- and three-dimensional Euclidean space ( and) into a new analytic method which is used to classify many Grassmannian frames in this setting. The method and associated algorithm decrease the maximum frame correlation, and hence give rise to the construction of specific examples of Grassmannian frames. Many of the results are known by other techniques, and even more generally, so that this paper can be viewed as tutorial. However, our analytic method is presented with the goal of developing it to address unresovled problems in-dimensional Hilbert spaces which serve as a setting for spherical codes, erasure channel modeling, and other aspects of communications theory.

References

  1. 1.

    Christensen O: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston, Mass, USA; 2002.

    Google Scholar 

  2. 2.

    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-X

    MathSciNet  Article  Google Scholar 

  3. 3.

    Benedetto JJ, Fickus M: Finite normalized tight frames. Advances in Computational Mathematics 2003, 18(2–4):357–385.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Conway JH, Hardin RH, Sloane NJA: Packing lines, planes, etc.: packings in Grassmannian spaces. Experimental Mathematics 1996, 5(2):139–159.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Eldar YC, Bölcskei H: Geometrically uniform frames. IEEE Transactions on Informormation Theory 2003, 49(4):993–1006. 10.1109/TIT.2003.809602

    MathSciNet  Article  Google Scholar 

  6. 6.

    Tropp JA, Dhillon IS, Heath RW Jr., Strohmer T: Designing structured tight frames via an alternating projection method. IEEE Transactions on Informormation Theory 2005, 51(1):188–209.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Goyal VK, Kovačević J, Kelner JA: Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis 2001, 10(3):203–233. 10.1006/acha.2000.0340

    MathSciNet  Article  Google Scholar 

  8. 8.

    Casazza PG, Kovačević J: Equal-norm tight frames with erasures. Advances in Computational Mathematics 2003, 18(2–4):387–430.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Holmes R, Paulsen V: Optimal frames for erasures. Linear Algebra and its Applications 2004, 377(1):31–51.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fejes Tóth L: Distribution of points in the elliptic plane. Acta Mathematica Academiae Scientiarum Hungaricae 1965, 16: 437–440. 10.1007/BF01904849

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lay DC: Linear Algebra and Its Applications. Addison-Wesley, Boston, Mass, USA; 2003.

    Google Scholar 

  12. 12.

    Duffin RJ, Schaeffer AC: A class of nonharmonic Fourier series. Transactions of the American Mathematical Society 1952, 72(2):341–366. 10.1090/S0002-9947-1952-0047179-6

    MathSciNet  Article  Google Scholar 

  13. 13.

    Benedetto JJ, Frazier MW (Eds): Wavelets: Mathematics and Applications. CRC Press, Boca Raton, Fla, USA; 1994.

    Google Scholar 

  14. 14.

    Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.

    Google Scholar 

  15. 15.

    Rosenfeld M: In praise of the Gram matrix. In The Mathematics of Paul Erdős, II, Algorithms and Combinatorics. Volume 14. Springer, Berlin, Germany; 1997:318–323.

    Google Scholar 

  16. 16.

    Welch LR: Lower bounds on the maximum cross-correlation of signals. IEEE Transactions on Informormation Theory 1974, 20(3):397–399. 10.1109/TIT.1974.1055219

    Article  Google Scholar 

  17. 17.

    Panik MJ: Fundamentals of Convex Analysis: Duality, Separation, Representation, and Resolution. Kluwer Academic, Dordrecht, the Netherlands; 1993.

    Google Scholar 

  18. 18.

    Lay SR: Convex Sets and Their Applications. John Wiley & Sons, New York, NY, USA; 1982.

    Google Scholar 

  19. 19.

    Webster R: Convexity. Oxford University Press, New York, NY, USA; 1994.

    Google Scholar 

  20. 20.

    Krein M, Milman D: On extreme points of regular convex sets. Studia Mathematica 1940, 9: 133–138.

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lemmens PWH, Seidel JJ: Equiangular lines. Journal of Algebra 1973, 24(3):494–512. 10.1016/0021-8693(73)90123-3

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to John J Benedetto.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benedetto, J.J., Kolesar, J.D. Geometric Properties of Grassmannian Frames for and. EURASIP J. Adv. Signal Process. 2006, 049850 (2006). https://doi.org/10.1155/ASP/2006/49850

Download citation

Keywords

  • Hilbert Space
  • Information Technology
  • Euclidean Space
  • Geometric Property
  • Quantum Information