A time scale is any closed subset of the real line. Continuous time and discrete time are special cases. The unilateral Laplace transform of a signal on a time scale subsumes the continuous-time unilateral Laplace transform, and the discrete-time unilateral z-transform as special cases. The regions of convergence (ROCs) time scale Laplace transforms are determined by the time scale’s graininess. For signals with finite area, the ROC for the Laplace transform resides outside of a Hilger circle determined by the time scales’s smallest graininess. For transcendental functions associated with the solution of linear time-invariant differential equations, the ROCs are determined by function parameters (e.g., sinusoid frequency) and the largest and smallest graininess values in the time scale. Since graininess always lies between zero and infinity, there are ROCs applicable to a specified signal on any time scale. All ROCs reduce to the familiar half-plane ROCs encountered in the continuous-time unilateral Laplace transform and circle ROCs for the unilateral z-transform. If a time scale unilateral Laplace transform converges at some point in the transform plane, a region of additional points can be identified as also belonging to the larger ROC.
Time scales Laplace transform z-transforms Region of convergence Hilger circle