Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

C.1 LAPLACE TRANSFORM

Definition: Laplace Transform X(s) The Laplace transform of x(t) for is

(C.1)

where is a complex variable with units of s−1. The real part σ is neper frequency (nepers/second), and the imaginary part ω is radian frequency (radians/s). The region of convergence (ROC) specifies the range of σ for which X(s) exists.

Equation (C.1) is the bilateral Laplace transform. For the unilateral Laplace transform, the integral has a finite lower limit, usually zero:

(C.2)

The unilateral and bilateral Laplace transforms are identical when x(t) is nonzero only for , which can be emphasized by writing x(t)u(t), where u(t) is the unit-step function. The inverse Laplace transform is

(C.3)

where the integration is performed along a line parallel to the axis, and σ lies in the ROC. ...