Open Access

A Unified Transform for LTI Systems—Presented as a (Generalized) Frame

  • Arie Feuer1,
  • Paul M.J. Van den Hof2 and
  • Peter S.C. Heuberger2
EURASIP Journal on Advances in Signal Processing20062006:091604

Received: 19 August 2004

Accepted: 31 May 2005

Published: 20 February 2006


We present a set of functions in and show it to be a (tight) generalized frame (as presented by G. Kaiser (1994)). The analysis side of the frame operation is called the continuous unified transform. We show that some of the well-known transforms (such as Laplace, Laguerre, Kautz, and Hambo) result by creating different sampling patterns in the transform domain (or, equivalently, choosing a number of subsets of the original frame). Some of these resulting sets turn out to be generalized (tight) frames as well. The work reported here enhances the understanding of the interrelationships between the above-mentioned transforms. Furthermore, the impulse response of every stable finite-dimensional LTI system has a finite representation using the frame we introduce here, with obvious benefits in identification problems.


Authors’ Affiliations

Department of Electrical Engineering, Technion-Israel Institute of Technology
Delft Center for Systems and Control, Delft University of Technology


  1. Heuberger PSC, de Hoog TJ, Van den Hof PMJ, Wahlberg B: Orthonormal basis functions in time and frequency domain: Hambo transform theory. SIAM Journal on Control and Optimization 2003, 42(4):1347-1373. 10.1137/S0363012902405340MathSciNetView ArticleMATHGoogle Scholar
  2. Oppenheim AV, Willsky AS, Young IT: Signals and Systems. Prentice-Hall, Englewood Cliffs, NJ, USA; 1983.MATHGoogle Scholar
  3. Vetterli M, Kovacevic J: Wavelets and Subband Coding. Prentice-Hall, Englewood Cliffs, NJ, USA; 1995.MATHGoogle Scholar
  4. Mallat S: A Wavelet Tour of Signal Processing. 2nd edition. Academic Press, San Diego, Calif, USA; 1999.MATHGoogle Scholar
  5. Daubechies I: Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, Pa, USA; 1992.Google Scholar
  6. Chui CK: An Introduction to Wavelets, Wavelet Analysis and Its Applications. Academic Press, Boston, Mass, USA; 1992.Google Scholar
  7. Christensen O: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston, Mass, USA; 2003.View ArticleMATHGoogle Scholar
  8. Kaiser G: A Friendly Guide to Wavelets. Birkhäuser, Boston, Mass, USA; 1994.MATHGoogle Scholar
  9. Ali ST, Antoine JP, Gazeau JP: Continuous frames in Hilbert space. Annals of Physics 1993, 222(1):1-37. 10.1006/aphy.1993.1016MathSciNetView ArticleMATHGoogle Scholar
  10. Viscito E, Allebach JP: The analysis and design of multidimensional FIR perfect reconstruction filter banks for arbitrary sampling lattices. IEEE Transactions on Circuits And Systems—Part I : Fundamental Theory and Applications 1991, 38(1):29-41.View ArticleGoogle Scholar
  11. Apostol TM: Mathematical Analysis. 2nd edition. Addison-Wesley, Sydney, Australia; 1974.MATHGoogle Scholar
  12. Paley REAC, Wiener N: Fourier Transforms in the Complex Domain. American Mathematical Society, New York, NY, USA; 1934.MATHGoogle Scholar
  13. de Hoog TJ: Rational orthonormal bases and related transforms in linear system modeling, M.S. thesis. Delft University of Technology, Delft, The Netherlands; 2001.Google Scholar
  14. Donoho DL, Stark PB: Uncertainty principles and signal recovery. SIAM Journal on Applied Mathematics 1989, 49(3):906-931. 10.1137/0149053MathSciNetView ArticleMATHGoogle Scholar
  15. Donoho DL, Elad M:Optimally sparse representation in general (nonorthogonal) dictionaries via minimization. Proceedings of the National Academy of Sciences of the United States of America 2003, 100(5):2197-2202. 10.1073/pnas.0437847100MathSciNetView ArticleMATHGoogle Scholar


© Feuer et al. 2006