Skip to main content

Equiangular tight frames

A survey on equiangular tight frames in the space \( \mathbb{R}^n \) is presented. Several equivalent definitions of a tight frame are given. The construction of the Mercedes–Benz frame, the well-known example of a tight frame on the plane, is generalized to the space \( \mathbb{R}^n \). The existence problems for the Mercedes–Benz systems and other more general equiangular tight frames are discussed. It is shown that the Welch inequality becomes the equality only on equiangular tight frames (if they exist). Necessary and sufficient conditions for the existence of an equiangular tight (n,m)-frame are formulated in terms of the so-called signature matrices. All the main results are completely proved. Bibliography: 37 titles. Illustrations: 3 figures.

References

  1. J. Kovačevič and A. Chebira, “Life beyond bases: the advent of frames. Part 1,” IEEE Signal Process. Magazine, July 2007, 86–114.

  2. J. Kovačevič and A. Chebira, “Life beyond bases: the advent of frames. Part 2,” IEEE Signal Process. Magazine, September 2007, 115–125.

  3. P. G. Casazza, D. Redmond, and J. C. Tremain, “Real equiangular frames.” (http://www.math.missouri.edu/pete)

  4. P. Casazza and J. Kovačevič, “Equal-norm tight frames with erasures,” Adv. Comp. Math. 18, No. 2–4, 387–430 (2003)

    MATH  Article  Google Scholar 

  5. V. K. Goyal, J. Kovačevič, and J. A. Kelner, “Quantized frame expansions with erasures,” Appl. Comput. Harmonic Anal. 10, No. 3, 203–233 (2001).

    MATH  Article  Google Scholar 

  6. J. J. Benedetto and M. Fickus, “Finite normalized tight frames. Frame potentials,” Adv. Comput. Math. 18, No. 2–4, 357–385 (2003).

    MATH  Article  MathSciNet  Google Scholar 

  7. P. G. Casazza, “Custom building finite frames,” Contemporary Math. 345, 61–86 (2004).

    MathSciNet  Google Scholar 

  8. I. Daubechies, Ten Lectures on Wavelets, Philadelphia (1992).

  9. P. Delsarte, J. M. Goetals, and J. J. Seidel, “Bounds for systems of lines and Jacobi polynomials,” Philips Res. Repts. 30, No. 3, 91–105 (1975).

    MATH  Google Scholar 

  10. P. W. H. Lemmens and J. J. Seidel, “Equiangular lines,” J. Algebra 24, No. 3, 494–512 (1973).

    MATH  Article  MathSciNet  Google Scholar 

  11. L. R. Welch, “Lower bounds on the maximum cross-correlation of signals,” IEEE Trans. Inf. Theory 20, 397–399 (1974).

    MATH  Article  MathSciNet  Google Scholar 

  12. T. Strohmer and R. W. Heath, “Grassmannian frames with applications to coding and communication,” Appl. Comput. Harmonic Anal. 14, No. 3, 257–275 (2003).

    MATH  Article  MathSciNet  Google Scholar 

  13. R. B. Holmes and V. I. Paulsen, “Optimal frames for erasures,” Linear Algebra Appl. 377, 31–51 (2004).

    MATH  Article  MathSciNet  Google Scholar 

  14. A. M. Duryagin and A. B. Pevnyi, “Grassmannian frames” [in Russian], Workshop DHA & CAGD. Selected Topics. 4 November 2007. (http://dha.spb.ru/reps07.shtml#1104).

  15. V. V. Maximenko and A. B. Pevnyi, “Existence of equiangular tight frames” [in Russian], Workshop DHA & CAGD. Selected Topics. 26 March 2008. (http://dha.spb.ru/reps08.shtml#0326).

  16. V. N. Malozemov and A. B. Pevnyi, “Mercedes–Benz Systems and tight frames” [in Russian], Workshop DHA & CAGD. Selected Topics. 28 February 2007. (http://dha.spb.ru/reps07.shtml#0228).

  17. V. N. Malozemov and A. B. Pevnyi, “Equiangular systems of vectors and tight frames” [in Russian], Workshop DHA & CAGD. Selected Topics. 18 September 2007. (http://dha.spb.ru/reps07.shtml#0918).

  18. V. N. Malozemov and A. B. Pevnyi, “The fourth definition of a tight frame” [in Russian], Workshop DHA & CAGD. Selected Topics. 30 May 2007. (http://dha.spb.ru/reps07.shtml#0530).

  19. V. N. Malozemov and N. A. Solov’eva, “On unitary matrices and singular decompositions” [in Russian], Workshop DHA & CAGD. Selected Topics. 30 January 2008. (http://dha.spb.ru/reps08.shtml#0130).

  20. V. N. Malozemov and A. B. Pevnyi, “There are no Mercedes–Benz Systems for m > n  + 1” [in Russian], Workshop DHA & CAGD. Selected Topics. 13 February 2008. (http://dha.spb.ru/reps08.shtml#0213).

  21. A. B. Pevnyi, “Frames in finite-dimensional spaces and minimization of the frame potential” [in Russian], Workshop DHA & CAGD. Selected Topics. 28 March 2006. (http://dha.spb.ru/reps06.shtml#0328)

  22. A. N. Sabaev, “Note on Mercedes–Benz systems” [in Russian], Workshop DHA & CAGD. Selected Topics. 16 September 2008. (http://dha.spb.ru/reps08.shtml#0916)

  23. M. N. Istomina and A. B. Pevnyi, “The Mercedes–Benz frame in the n-dimensional space” [in Russian], Vestnik Syktyvkar. Gos. Univ. Ser. 1 No. 6, 219–222 (2006).

  24. M. N. Istomina and A. B. Pevnyi, “On location of points on a sphere and the Mercedes–Benz frame” [in Russian], Mat. Prosveschenie. Ser. 3 No. 11, 105–112 (2007).

  25. V. N. Malozemov and A. B. Pevnyi, “Mercedes–Benz systems and tight frames” [in Russian], Vestnik Syktyvkar. Gos. Univ. Ser. 1 No. 7, 141–154 (2007).

  26. N. N. Andreev and V. A. Yudin, “Extremal location of points on a sphere” [in Russian], Mat. Prosveschenie. Ser. 3 No. 1, 115–121 (1997).

  27. B. Reznick, “Sums of even powers of real linear forms,” Mem. Am. Math. Soc. 96, No. 463, 1–155 (1992).

    MathSciNet  Google Scholar 

  28. J. J. Benedetto and J. D. Kolesar, “Geometric properties of Grassmannian frames for \( \mathbb{R}^2 \) and \( \mathbb{R}^3 \),” EURASIP J. Appl. Signal Process. (2006), ID 49850, 1–17.

  29. M. A. Sustik, J. A. Tropp, I. S. Dhillon, and R. W. Heath, “On the existence of equiangular tight frames,” Linear Algebra Appl. 426, 619–635 (2007).

    MATH  Article  MathSciNet  Google Scholar 

  30. J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, “Designing structured tight frames via an alternating projection method,” IEEE Trans. Inf. Theory 51, 188–209 (2005).

    Article  MathSciNet  Google Scholar 

  31. D. Han and D. R. Larson, “Frames, bases and group representation,” Mem. Am. Math. Soc. 147, No. 697, 1–94 (2000).

    MathSciNet  Google Scholar 

  32. R. Vale and S. Waldron, “Tight frames and their symmetries,” Const. Approx. 21, No. 1, 83–112 (2005).

    MATH  MathSciNet  Google Scholar 

  33. B. Venkov, “Réseaux et designs sphériques,” In: Réseaux Euclidiens, Designs sphériques et Formes Modulaires, Enseign. Math., Gèneve (2001), pp. 10–86.

    Google Scholar 

  34. S. Waldron, “Generalised Welch bound equality sequences are tight frames,” IEEE Trans. Inf. Theory 49, No. 9, 2307–2009 (2003).

    Article  MathSciNet  Google Scholar 

  35. S. Waldron and N. Hay, “On computing all harmonic frames of n vectors in \( \mathbb{C}^d \),” Appl. Comput. Harmonic Anal. 21, 168–181 (2006).

    Google Scholar 

  36. P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the Welch bound with difference sets,” IEEE Trans. Inf. Theory 51, No. 5, 1900–1907 (2005).

    Article  MathSciNet  Google Scholar 

  37. V. V. Voevodin and Vl. V. Voevodin, Handbook on Linear Algebra. Electronic System LINEAL [in Russian], St.-Petersburg (2006).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. N. Malozemov.

Additional information

Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 3–25.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Malozemov, V.N., Pevnyi, A.B. Equiangular tight frames. J Math Sci 157, 789–815 (2009). https://doi.org/10.1007/s10958-009-9366-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-009-9366-6

Keywords

  • Benz
  • Orthogonal Matrix
  • Normed System
  • Nonzero Vector
  • Tight Frame