Skip to main content
SpringerLink
Log in

Discrete Multidimensional Systems

  • Chapter
  • First Online:
Multidimensional Signals and Systems

  • 234 Accesses

Abstract

Linear discrete multidimensional systems are classified in the same way as one-dimensional systems: there are finite impulse response systems (FIR-systems) and infinite impulse response systems (IIR-systems). FIR-systems are widely applied in image processing and in processing of sampled volume data. IIR-systems in multiple dimensions are more complex to handle than in one dimension because they cannot be conveniently described by pole-zero diagrams. A different class of discrete multidimensional systems arises from the discretization of differential equations. They require the solution of potentially large systems of linear equations, which constitutes a problem of multidimensional signals and systems of its own. Finally the relations between multidimensional systems and multi-input-multi-output systems are discussed.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 84.99
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

References

  1. Rabenstein, R., Schäfer, M.: Multidimensional Signals and Systems: Applications. Springer Nature, Heidelberg, Berlin (to appear)

    Google Scholar 

  2. Bachelier, O., Cluzeau, T., David, R., Àlvarez, F.J.S., Yeganefar, N., Yeganefar, N.: Structural stability, asymptotic stability and exponential stability for linear multidimensional systems: the good, the bad and the ugly. International Journal of Control 91(12), 2714–2725 (2018). https://doi.org/10.1080/00207179.2017.1390258

    Article  MathSciNet  MATH  Google Scholar 

  3. Bracewell, R.N.: Fourier Analysis and Imaging. Kluwer Academic/Plenum Publishers, New York (2003)

    Book  MATH  Google Scholar 

  4. Brandt, A., Livne, O.E.: Multigrid techniques. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2011)

    Google Scholar 

  5. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2 edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2002)

    Google Scholar 

  6. Bronshtein, I., Semendyayev, K., Musiol, G., Mühlig, H.: Handbook of Mathematics. Springer-Verlag, Berlin (2015)

    Book  MATH  Google Scholar 

  7. Dudgeon, D.E., Mersereau, R.M.: Multidimensional Digital Signal Processing. Prentice-Hall, Englewood Cliffs, NJ (1984)

    MATH  Google Scholar 

  8. Fettweis, A.: Wave digital filters: Theory and practice. Proceedings of the IEEE 74(2), 270–327 (1986)

    Article  Google Scholar 

  9. Fettweis, A.: Robust numerical integration using wave-digital concepts. Multidimensional Systems and Signal Processing 17, 7–25 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fettweis, A., Nitsche, G.: Transformation approach to numerically integrating PDEs by means of WDF principles. Multidimensional Systems and Signal Processing 2, 127–159 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fornasini, E., Marchesini, G.: E.fornasini, g.marchesini, “doubly indexed dynamical systems: state space models and structural properties”, math. sys. theory, vol.12, pp.59–72, 1978. Mathematical Systems Theory (1978)

    Google Scholar 

  12. Franke, D.: Systeme mit örtlich verteilten Parametern. Eine Einführung in die Modellbildung, Analyse und Regelung. Hochschultext. Springer, Berlin u.a. (1987)

    Google Scholar 

  13. Galkowski, K.: State-space Realisations of Linear 2-D Systems with Extensions to the General ND (n > 2) Case. Lecture Notes in Control and Information Sciences. Springer (2001)

    MATH  Google Scholar 

  14. Girod, B., Rabenstein, R., Stenger, A.: Signals and Systems. Wiley, Chichester (2001)

    Google Scholar 

  15. Golub, G.H., Loan, C.F.V.: Matrix Computations, 4 edn. Johns Hopkins University Press, Baltimore, USA (2012)

    MATH  Google Scholar 

  16. Großmann, C., Roos, H.G.: Numerische Behandlung partieller Differentialgleichungen. Vieweg+Teubner Verlag, Wiesbaden (2005)

    Google Scholar 

  17. Hackbush, W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)

    Book  Google Scholar 

  18. Hjelmstad, K.D.: Fundamentals of Structural Dynamics. Springer Nature Switzerland (2022)

    Google Scholar 

  19. Jazar, R.N.: Perturbation Methods in Science and Engineering. Springer Nature Switzerland (2021)

    Google Scholar 

  20. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, UK (1995)

    Google Scholar 

  21. Kaczorek, T.: Two-Dimensional Linear Systems. Springer, Berlin (1985)

    MATH  Google Scholar 

  22. Kahl, K., Kintscher, N.: Automated local Fourier analysis (aLFA). BIT Numerical Mathematics 60(3), 651–686 (2020). https://doi.org/10.1007/s10543-019-00797-w. https://www.scopus.com/inward/record.uri?eid=2-s2.0-85078421494&doi=10.1007%2fs10543-019-00797-w&partnerID=40&md5=12ea31c1134446029db07b9c263ac636. All Open Access, Green Open Access, Hybrid Gold Open Access

  23. MIT BCS Perceptual Science Group: Three frames of original flower garden sequence (MPEG suite). http://persci.mit.edu/demos/jwang/garden-layer/orig-seq.html. Accessed on Apr. 25, 2022

  24. Öchsner, A.: Weighted residual methods for finite elements. In: H. Altenbach, A. Öchsner (eds.) Encyclopedia of Continuum Mechanics, pp. 2771–2786. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_20

    Chapter  Google Scholar 

  25. Quarteroni, A.: Numerical Models for Differential Problems. Springer-Verlag Milan (2009)

    Google Scholar 

  26. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2 edn. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  27. Roesser, R.: A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control 20(1), 1–10 (1975). doi: 10.1109/TAC.1975.1100844

  28. Rogers, E., Galkowski, K., Owens, D.H.: Control Systems Theory and Applications for Linear Repetitive Processes, vol. 349. Springer (2007). https://eprints.soton.ac.uk/263634/

  29. Rogers, E., Galkowski, K., Paszke, W., Moore, K.L., Bauer, P.H., Hladowski, L., Dabkowski, P.: Multidimensional control systems: case studies in design and evaluation. Multidimensional Systems and Signal Processing 26(4), 895–939 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2 edn. Society for Industrial and Applied Mathematic (SIAM), Philadelphia, USA (2003)

    Google Scholar 

  31. Schäfer, M.: Computational Engineering – Introduction to Numerical Methods, 2 edn. Springer Nature Switzerland (2022)

    Google Scholar 

  32. Schroeder, H., Blume, H.: One- and Multidimensional Signal Processing. Algorithms and Applications in Image Processing. Wiley, Chichester (2000)

    Google Scholar 

  33. Strauss, W.A.: Partial Differential Equations. John Wiley & Sons (2008)

    Google Scholar 

  34. Traub, J.: Iterative Methods for the Solution of Equations, 2 edn. AMS/Chelsea Publication, Providence, RI, USA (1982)

    MATH  Google Scholar 

  35. Trottenberg, U., Oosterlee, C., Schuller, A.: Multigrid. Academic Press, San Diego (2001)

    MATH  Google Scholar 

  36. Tveito, A., Winther, R.: Introduction to Partial Differential Equations – A Computational Approach. Springer-Verlag, Berlin Heidelberg (2005)

    MATH  Google Scholar 

  37. Varga, R.S.: Matrix Iterative Analysis, 2 edn. Springer-Verlag, Berlin (2000)

    Book  MATH  Google Scholar 

  38. Wang, J., Adelson, E.: Representing moving images with layers. IEEE Transactions on Image Processing 3(5), 625–638 (1994). https://doi.org/10.1109/83.334981

    Article  Google Scholar 

  39. Wienands, R., Joppich, W.: Practical Fourier Analysis for Multigrid Methods. Chapman & Hall/CRC, London, UK (2011)

    Google Scholar 

  40. Woods, J.: Multidimensional Signal, Image, and Video Processing and Coding, 2 edn. Elsevier, Amsterdam (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Multimedia Communications and Signal Processing, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany

    Rudolf Rabenstein

  2. Digital Communications, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany

    Maximilian Schäfer

Authors
  1. Rudolf Rabenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maximilian Schäfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2023 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Rabenstein, R., Schäfer, M. (2023). Discrete Multidimensional Systems. In: Multidimensional Signals and Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-26514-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-26514-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-26513-6

  • Online ISBN: 978-3-031-26514-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation