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Connections Between Linear Systems and Convolutional Codes

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Codes, Systems, and Graphical Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 123))

Abstract

The article reviews different definitions for a convolutional code which can be found in the literature. The algebraic differences between the definitions are worked out in detail. It is shown that bi-infinite support systems are dual to finite-support systems under Pontryagin duality. In this duality the dual of a controllable system is observable and vice versa. Uncontrollability can occur only if there are bi-infinite support trajectories in the behavior, so finite and half-infinite-support systems must be controllable. Unobservability can occur only if there are finite support trajectories in the behavior, so bi-infinite and half-infinite-support systems must be observable. It is shown that the different definitions for convolutional codes are equivalent if one restricts attention to controllable and observable codes.

This work was supported in part by NSF grant DMS-96-10389. This research has been carried out while the author was a guest professor at EPFL in Switzerland. The author would like to thank EPFL for its support and hospitality.

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References

  1. S. Eilenberg. Automata, languages, and machines. Vol. A. Academic Press, New York, 1974. Pure and Applied Mathematics, vol. 59.

    Google Scholar 

  2. P. Elias. Coding for noisy channels. IRE Conv. Rec., 4:37–46, 1955.

    Google Scholar 

  3. F. Fagnani and S. Zampieri. Minimal syndrome formers for group codes. IEEE Trans. Inform. Theory, 45(1): 3–31, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Fornasini and M.E. Valcher. Observability and extendability offinite support nD behaviors. In Proc. of the 34th IEEE Conference on Decision and Control, pp. 3277–3282, New Orleans, Louisiana, 1995.

    Google Scholar 

  5. E. Fornasini and M.E. Valcher. Multidimensional systems with finite support behaviors: Signal structure, generation, and detection. SIAM J. Control Optim., 36(2):760–779, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  6. G.D. Forney. Convolutional codes I: Algebraic structure. IEEE Trans. Inform. Theory, IT-16(5):720–738, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  7. G.D. Forney. Structural analysis of convolutional codes via dual codes. IEEE Trans. Inform. Theory, IT-19(5):512–518, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  8. G.D. Forney. Minimal bases of rational vector spaces, with applications to multivariable linear systems. SIAM J. Control, 13(3):493–520, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  9. G.D. Forney. Group codes and behaviors. In G. Picci and D.S. Gilliam, editors, Dynamical Systems, Contral, Coding, Computer Vision: New Trends, Interlaces, and Interplay, pp. 301–320. Birkäuser, Boston-Basel-Berlin, 1999.

    Google Scholar 

  10. G.D. Forney, B. Marcus, N.T. Sindhushayana, and M. Trott. A multilingual dietionary: System theory, coding theory, symbolic dynamics and automata theory. In Different Aspects of Coding Theory, Proceedings of Symposia in Applied Mathematics number 50, pp. 109–138. American Mathematical Society, 1995.

    MathSciNet  Google Scholar 

  11. G.D. Forney and M.D. Trott. Controllability, observability, and duality in behavioral group systems. In Proc. of the 34th IEEE Conlerence on Decision and Contral, pp. 3259–3264, New Orleans, Louisiana, 1995.

    Google Scholar 

  12. G.D. Forney and M.D. Trott. The dynamies of group codes: Dual abelian group codes and systems. Preprint, August 1997; submitted to IEEE Trans. Inform. Theory, January 2000.

    Google Scholar 

  13. P.A. Fuhrmann. Aigebraic system theory: An analyst’s point of view. J. Franklin Inst., 301:521–540, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  14. P.A. Fuhrmann. Duality in polynomial models with some applications to geometric control theory. IEEE Trans. Automat. Control, 26(1):284–295, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  15. P.A. FUHRMANN. A Polynomial Approach to Linear Algebra. Universitext. Springer-Verlag, New York, 1996.

    Book  MATH  Google Scholar 

  16. H. Gluesing-LuErssen, J. Rosenthal, and P.A. Weiner. Duality between multidimensional convolutional codes and systems. E-print math.OC/9905046, May 1999.

    Google Scholar 

  17. M.L.J. Hautus and M. Heymann. Linear feedback—an algebraic approach. SIAM J. Control, 16:83–105, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Hinrichsen and D. PrÄtzel-Wolters. Solution modules and system equivalence. Internat. J. Contral, 32:777–802, 1980.

    Article  MATH  Google Scholar 

  19. D. Hinrichsen and D. PrÄtzel-Wolters. Generalized Hermite matrices and complete invariants of strict system equivalence. SIAM J. Control Optim., 21:289–305, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Johannesson and Z. Wan. A linear algebra approach to minimal convolutional encoders. IEEE Trans. Inform. Theory, IT-39(4):1219–1233, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Johannesson and K. Sh. Zigangirov. Fundamentals of Convolutional Coding. IEEE Press, New York, 1999.

    Book  Google Scholar 

  22. R. E. Kalman. Aigebraic structure of linear dynamical systems. I. The module of Σ. Proc. Nat. Acad. Sci. U.S.A., 54:1503–1508, 1965.

    MathSciNet  MATH  Google Scholar 

  23. R.E. Kalman, P.L. Falb, and M.A. Arbib. Topics in Mathematical System Theory. McGraw-Hill, New York, 1969.

    MATH  Google Scholar 

  24. B. Kitchens. Symbolic dynamics and convolutional codes. Preprint, February 2000.

    Google Scholar 

  25. G. KÖthe. Topological Vector Spaces I. Springer Verlag, 1969.

    Google Scholar 

  26. M. Kuijper. First-Order Representations 01 Linear Systems. Birkhäuser, Boston, 1994.

    Book  Google Scholar 

  27. M. Kuijper and J.M. Schumacher. Realization of autoregressive equations in pencil and descriptor form. SIAM J. Control Optim., 28(5):1162–1189, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Lin and D.J. Costello. Error Contral Coding: Fundamentals and Applications. Prentice-Hall, Englewood Cliffs, NJ, 1983.

    Google Scholar 

  29. D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1995.

    Google Scholar 

  30. H.A. Loeliger and T. Mittelholzer. Convolutional codes over groups. IEEE Trans. Inform. Theory, 42(6):1660–1686, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Lomadze. Finite-dimensional time-invariant linear dynamical systems: Algebraic theory. Acta Appl. Math, 19:149–201, 1990.

    MathSciNet  MATH  Google Scholar 

  32. B. Marcus. Symbolic dynamics and connections to coding theory, automata theory and system theory. In Different aspects of coding theory (San Francisco, CA, 1995), vol. 50 of Proc. Sympos. Appl. Math., pp. 95–108. Amer. Math. Soc., Providence, RI, 1995.

    Google Scholar 

  33. J.L. Massey, D.J. Costello, and J. Justesen. Polynomial weights and code constructions. IEEE Trans. Inform. Theory, IT-19(1):101–110, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  34. J.L. Massey and M.K. Sain. Codes, automata, and continuous systems: Explicit interconnections. IEEE Trans. Automat. Contr., AC-12(6):644–650, 1967.

    Article  Google Scholar 

  35. R.J. McEliece. The algebraic theory of convolutional codes. In V. Pless and W.C. Huffman, editors, Handbook of Coding Theory, volume 1, pages 1065–1138. Elsevier Science Publishers, Amsterdam, The Netherlands, 1998.

    Google Scholar 

  36. H. Nussbaumer. Computer Communication Systems, vol. 1. John Wiley & Sons, Chichester, New York, 1990. Translated by John C.C. Nelson.

    Google Scholar 

  37. U. Oberst. Multidimensional constant linear systems. Acta Appl. Math, 20:1–175, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  38. PH. Piret. Convolutional Codes, an Algebraic Approach. MIT Press, Cambridge, MA, 1988.

    MATH  Google Scholar 

  39. M.S. Ravi and J. Rosenthal. A smooth compactification of the space of transfer functions with fixed McMillan degree. Acta Appl. Math, 34:329–352, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  40. M.S. Ravi and J. Rosenthal. A general realization theory for higher order linear differential equations. Systems & Control Letters, 25(5):351–360, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Rosenthal and J.M. Schumacher. Realization by inspection. IEEE Trans. Automat. Contr., AC-42(9):1257–1263, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Rosenthal, J.M. Schumacher, and E.V. York. On behaviors and convolutional codes. IEEE Trans. Inform. Theory, 42(6), Part I):1881–1891, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Rosenthal and R. Smarandache. Maximum distance separable convolutional codes. Appl. Algebra Engrg. Comm. Comput., 10(1):15–32, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Rosenthal and E.V. York. BCH convolutional codes. IEEE Trans. Inform. Theory, 45(6):1833–1844, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  45. M.K. Sain and J.L. MASSEY. Invertibility of linear time-invariant dynamical systems. IEEE Trans. Automat. Contr., AC-14:141–149, 1969.

    Article  MathSciNet  Google Scholar 

  46. R. Smarandache, H. Gluesing-LuErssen, and J. Rosenthal. Constructions of MDS-convolutional codes. Submitted to IEEE Trans. Inform. Theory, August 1999.

    Google Scholar 

  47. L. Staiger. Subspaces of GF(q) W and convolutional codes. Information and Contral, 59:148–183, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  48. M.E. Valcher and E. Fornasini. On 2D finite support convolutional codes: An algebraic approach. Multidim. Sys. and Sign. Proc., 5:231–243, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Weiner. Multidimensional Convolutional Codes. PhD thesis, University of Notre Dame, 1998. Available at http://www.nd.edurrosen/preprints.html.

    Google Scholar 

  50. J.C. Willems. From time series to linear system. Part I: Finite dimensional linear time invariant systems. Automatica, 22:561–580, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  51. J.C. Willems. Models for dynamics. In U. Kirchgraber and H.O. Walther, editors, Dynamics Reported, volume 2, pp. 171–269. John Wiley & Sons Ltd, 1989.

    Google Scholar 

  52. J.C. Willems. Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control, AC-36(3):259–294, 1991.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, 46556-5683, UK

    Joachim Rosenthal

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  1. Joachim Rosenthal
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Editors and Affiliations

  1. IBM Almaden Research Center, K65-802, 650 Harry Rd., San Jose, CA, 95120, USA

    Brian Marcus

  2. Department of Mathematics, University of Notre Dame, NotreDame, IN, 46556-5683, USA

    Joachim Rosenthal

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Rosenthal, J. (2001). Connections Between Linear Systems and Convolutional Codes. In: Marcus, B., Rosenthal, J. (eds) Codes, Systems, and Graphical Models. The IMA Volumes in Mathematics and its Applications, vol 123. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0165-3_2

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  • DOI: https://doi.org/10.1007/978-1-4613-0165-3_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95173-7

  • Online ISBN: 978-1-4613-0165-3

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