In this lesson, we cover the Bilateral
Laplace Transform.
We will see that for the Bilateral Laplace Transform we must specify Region
of Convergence (ROC) because multiple signals have the same Laplace
Transform. The Bilateral Laplace Transform is defined as:

Example 1

Example 2

Notice that the Laplace Transforms for
Examples 1 and 2 were the same, except that they had different
Regions of Convergence. Therefore, we must always specify the
ROC when working with the Bilateral Laplace Transform so that we
can determine whether the time signal is left-sided or right-sided.

Region of Convergence

We
state without proof here that
the ROC of the Laplace Transform of the sum of multiple time functions
is the INTERSECTION of the individuals ROCs. Use this fact to solve the
next two Examples.

Example 3

Example 4

Inverse Bilateral Laplace Transform

To compute the Inverse Bilateral Laplace
Transform, we'll again use Partial Fraction Expansion.
We will ignore the case of multiple
poles. Write the Bilateral Laplace Transform H(s) as:

Here, s_{k} are the poles, r_{k}
are the residues, and
b_{N} is nonzero if the order of the
numerator B(s) is greater than the order of the
denominator A(s).

Example 5

Example 6

Example 7

Remember,
you can always check your results of taking the inverse
transform by taking the transform and comparing.