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

Abstract

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.