Circuits, Systems and Signal Processing

, Volume 29, Issue 6, pp 1141–1165 | Cite as

Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series

  • John M. DavisEmail author
  • Ian A. Gravagne
  • Robert J. MarksII


The convergence of Laplace transforms on time scales is generalized to the bilateral case. The bilateral Laplace transform of a signal on a time scale subsumes the continuous time bilateral Laplace transform, and the discrete time bilateral z-transform as special cases. As in the unilateral case, the regions of convergence (ROCs) time scale Laplace transforms are determined by the time scale’s graininess. ROCs for the bilateral Laplace transforms of double sided time scale exponentials are determined by two modified Hilger circles. The ROC is the intersection of points external to modified Hilger circle determined by behavior for positive time and the points internal to the second modified Hilger circle determined by negative time. Since graininess lies between zero and infinity, there can exist conservative ROCs applicable for all time scales. For continuous time (ℝ) bilateral transforms, the circle radii become infinite and results in the familiar ROC between two lines parallel to the imaginary z axis. Likewise, on ℤ, the ROC is an annulus. For signals on time scales bounded by double sided exponentials, the ROCs are at least that of the double sided exponential. The Laplace transform is used to define the box minus shift through which time scale convolution can be defined. Generalizations of familiar properties of signals on ℝ and ℤ include identification of the identity convolution operator, the derivative theorem, and characterizations of wide sense stationary stochastic processes for an arbitrary time scales including autocorrelation and power spectral density expressions.


Time scales Laplace transform z-transforms Region of convergence Hilger circle Stationarity Autocorrelation Power spectral density Hilger delta 


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  1. 1.
    M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001) zbMATHGoogle Scholar
  2. 2.
    C.J. Chyan, J.M. Davis, J. Henderson, W.K.C. Yin, Eigenvalue comparisons for differential equations on a measure chain. Electron. J. Differ. Equ. 1998, 35 (1998), 7 pp. MathSciNetGoogle Scholar
  3. 3.
    J.M. Davis, J. Henderson, K.R. Prasad, Upper and lower bounds for the solution of the general matrix Riccati differential equation on a time scale. J. Comput. Appl. Math. 141, 133–145 (2002) CrossRefMathSciNetGoogle Scholar
  4. 4.
    J.M. Davis, I.A. Gravagne, B.J. Jackson, R.J. Marks II, A.A. Ramos, The Laplace transform on time scales revisited. J. Math. Anal. Appl. 332, 1291–1307 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J.M. Davis, I.A. Gravagne, R.J. Marks II, Convergence of causal Laplace transforms on time scales. Circuits Syst. Signal Process (2010). doi: 10.1007/s00034-010-9182-8
  6. 6.
    I.A. Gravagne, J.M. Davis, J. Dacunha, R.J. Marks II, Bandwidth sharing for controller area networks using adaptive sampling, in Proc. Int. Conf. Robotics and Automation (ICRA), New Orleans, LA, April 2004, pp. 5250–5255 Google Scholar
  7. 7.
    S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg (1988) Google Scholar
  8. 8.
    R.J. Marks II, Introduction to Shannon Sampling and Interpolation Theory (Springer, New York, 1991) zbMATHGoogle Scholar
  9. 9.
    R.J. Marks II (ed.), Advanced Topics in Shannon Sampling and Interpolation Theory (Springer, Berlin, 1993) Google Scholar
  10. 10.
    R.J. Marks II, Handbook of Fourier Analysis and Its Applications (Oxford University Press, London, 2009) zbMATHGoogle Scholar
  11. 11.
    R.J. Marks II, I. Gravagne, J.M. Davis, J.J. DaCunha, Nonregressivity in switched linear circuits and mechanical systems. Math. Comput. Model. 43, 1383–1392 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R.J. Marks II, I.A. Gravagne, J.M. Davis, A generalized Fourier transform and convolution on time scales. J. Math. Anal. Appl. 340(2), 901–919 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1968) Google Scholar
  14. 14.
    A. Papoulis, Generalized sampling expansion. IEEE Trans. Circuits Syst. CAS-24, 652–654 (1977) CrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977) zbMATHGoogle Scholar
  16. 16.
    K. Yao, J.B. Thomas, On some stability and interpolating properties of nonuniform sampling expansions. IEEE Trans. Circuit Theory CT-14, 404–408 (1967) Google Scholar
  17. 17.
    K. Yao, J.B. Thomas, On a class of nonuniform sampling representation, in Symp. Signal Transactions Processing (Columbia University Press, New York, 1968) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • John M. Davis
    • 1
    Email author
  • Ian A. Gravagne
    • 2
  • Robert J. MarksII
    • 2
  1. 1.Department of MathematicsBaylor UniversityWacoUSA
  2. 2.Department of Electrical and Computer EngineeringBaylor UniversityWacoUSA

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