Analysis of a Class of Infinite Dimensional Frames

  • Cishen Zhang
  • Jingxin Zhang
  • Xiaofang Chen
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 174)


Frames are a mathematical tool which can represent redundancies in many application problems. In this article, a class of infinite dimensional and bi-directional frames are studied. It is shown that the infinite dimensional and bi-directional frames can be represented by milti-input, multi-output state space equations. Such a state space representation can enable the application of powerful linear system methods and numerical tools to the performance analysis and evaluation of frames.


Frames Linear systems Mixed causal-anticausal realizations State space equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Duffin, R.J., Schaeffer, A.C.: A Class of Nonharmonic Fourier Series. Trans. of the American Mathematical Society 72(2), 341–366 (1952)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Daubechies, I., Grossman, A., Meyer, Y.: Painless Nonorthogonal Expansions. Journal of Mathematical Physcics 27(5), 1271–1283 (1986)MATHCrossRefGoogle Scholar
  3. 3.
    Heil, C., Walnut, D.: Continuous and Discrete Wavelet Transforms. SIAM Review 31(4), 628–666 (1989)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Peter, G.C.: Modern Tools for Weyl-Heisenberg (Gabor) Frame Theory. Advances in Imaging and Electron Physics 115, 1–127 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Daubechies, I.: The Wavelet Transform, Time-frequency Localization and Signal Analysis. IEEE Trans. on Information Theory 36(5), 961–1005 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (1992)Google Scholar
  7. 7.
    Peter, J.B., Edward, H.A.: The Lapacian Pyramid as a Compact Image Code. IEEE Trans. on Communications 31(4), 532–540 (1983)CrossRefGoogle Scholar
  8. 8.
    Benedetto, J.J., Powell, A.M., Yilmaz, Ö.: Sigma-delta Quantization and Finite Frames. IEEE Trans. on Information Theory 52(5), 1990–2005 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dragotti, P.L., Velisavljevic, V., Vetterli, M., Beferull-Lozano, B.: Discrete Directinoal Wavelet Bases and Frames for Image Compression and Denoising. In: Proc.SPIE Conf. Wavelet Applications Signal Image Processing, pp. 1287–1295 (2003)Google Scholar
  10. 10.
    Riccardo, B., Roberto, R.: Bounds on Error Amplification in Oversampled Filter Banks for Robust Transmission. IEEE Trans. on Signal Processing 54(4), 1399–1411 (2006)CrossRefGoogle Scholar
  11. 11.
    Kovacevic, J., Chebira, A.: Life beyond bases: The advent of frames (Part I). IEEE Signal Processing Mag. 24(4), 86–104 (2007)CrossRefGoogle Scholar
  12. 12.
    Kovacevic, J., Chebira, A.: Life beyond bases: The advent of frames (Part II). IEEE Signal Processing Mag. 24(5), 115–125 (2007)CrossRefGoogle Scholar
  13. 13.
    Martin, V., Zoran, C.: Oversampled FIR Filter Banks and Frames in l 2(Z). In: IEEE Interational Conference on Acoustic, Speech and Signal Processing Conference Proceddings, pp. 1530–1533 (1996)Google Scholar
  14. 14.
    Zoran, C., Martin, V.: Oversampled Filter Banks. IEEE Trans. on Signal Processing 46(5), 1245–1255 (1998)CrossRefGoogle Scholar
  15. 15.
    Helmut, B., Franz, H., Hans, G.F.: Frame-theoretic Analysis of Oversampled Filter Banks. IEEE Trans. on Signal Processing 46(12), 3256–3268 (1998)CrossRefGoogle Scholar
  16. 16.
    Helmut, B., Franz, H.: Noise Reduction in Oversampled Filter Banks Using Predictive Quantization. IEEE Trans. on Information Theory 47(1), 155–172 (2001)MATHCrossRefGoogle Scholar
  17. 17.
    Alfred, M.: Frame Analysis for Biothogonal Cosine-modulated Filterbanks. IEEE Trans. on Signal Processing 51(1), 172–181 (2003)CrossRefGoogle Scholar
  18. 18.
    Li, C., Jingxin, Z., Cishen, Z.: Frame-Theory-Based Analysis and Design of Oversampled Filter Banks: Direct Computational Method. IEEE Trans. on Signal Processing 55(2), 507–519 (2007)CrossRefGoogle Scholar
  19. 19.
    David, S., Yehoshua, Y.Z.: Frame Analysis of Wavelet-type Filter Banks. Signal Processing 67(2), 125–139 (1998)MATHCrossRefGoogle Scholar
  20. 20.
    Ilker, B., Ivan, W.S.: On the Frame Bounds of Iterated Filter Banks. Applied and Computational Harmonic Analysis 27(2), 255–262 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhauser (2003)Google Scholar
  22. 22.
    Huang, S., Tongwen, C.: On Causality and Anticausality of Cascaded Discrete-time Systems. IEEE Trans. on Circuit and Systems I: Fundamental Theory and Applications 43(3), 240–242 (1996)CrossRefGoogle Scholar
  23. 23.
    Li, C., Jingxin, Z., Cishen, Z., Edoardo, M.: Efficient Computation of Frame Bounds Using LMI-Based Optimization. IEEE Trans. on Signal Processing 56(7), 3029–3033 (2008)MATHCrossRefGoogle Scholar
  24. 24.
    Anders, R.: On the Kalman-Yakubovich-Popov Lemma. Systems and Control Letters 28(1), 7–10 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Cishen Zhang
    • 1
  • Jingxin Zhang
    • 2
  • Xiaofang Chen
    • 3
  1. 1.Faculty of Engineering and Industrial SciencesSwinburne University of TechnologyMelbourneAustralia
  2. 2.Department of Electrical and Computer Systems EngineeringMonash UniversityClaytonAustralia
  3. 3.School of Electrical Electronic EngineeringNanyang Technological UniversityNanyangSingapore

Personalised recommendations