Random Processes: Filtering, Estimation, and Detection

APPENDIX AThe Bilateral Laplace Transform

The unilateral Laplace transform, F(p), of a given positive time function, f(t), is defined as

The integral has meaning only if the integral converges. The set of values of p where the integral converges is called the region of convergence (ROC) for the transform of f(t). The unilateral inverse transform of F(p) gives the f(t) and can be written as the following line integral:

where the σ − j∞ to σ + j∞ line lies within the region of convergence for the transform F(p). The unilateral Laplace transform works only with the positive time part of a given f(t), and thus the inverse is unique for positive time functions only.

When f(t) contains both a positive and negative time part, the bilateral Laplace transform is useful. The bilateral Laplace transform of f(t) is defined as

and it exists for certain p, whose set forms the region of convergence for the transform F(p). This region of convergence is usually a strip in the complex p-plane as shown in Figure A.1.

To obtain f(t) from the F(p), it can be shown that inverse bilateral Laplace transform is

As this is an integral in the complex p-plane, it is referred to as the complex inversion integral ...