ROCs of Unilateral and Bilateral Laplace Transforms
For causal signals, performing unilateral Laplace transform and bilateral Laplace transform will get the same result. This equivalence occurs because all values of a causal signal are zero for negative time, which means the integration from negative infinity to zero (present in the bilateral transform) contributes nothing to the result.
In this case, the radius of convergence is always in the form of (the constant is often referred to as the abscissa of convergence).
Example: Find the Laplace transform of
Note that will only converge if , which implies that .
Example: Find the Laplace transform of
These equations converge if and , thus .
For non-casual signals, we must use bilateral Laplace transform, and the ROC can be more complicated.
Laplace transforms are often represented as rational functions:
The roots of the numerator are the zeros of the Laplace transform.
The roots of the denominator are the poles of the Laplace transform
Poles play a crucial role in determining the convergence of Laplace transforms. When is at a a pole, approaches , so approaches infinity. Thus, the Laplace transform never converge at its poles. And as a result, the ROC of a Laplace transform never includes any of its poles!
Further, poles segregate the s-plane into multiple sections. If a rational Laplace transform has distinct poles, there can be up to different possible regions of convergence. As a result, there are possible inverse Laplace transform, with each ROC as a strip or region in the complex frequency domain bounded by poles and/or extending to infinity.