The Bilateral Laplace Transform of a signal x(t) is defined as:
The complex variable s = σ + jω, where ω is the frequency variable of the Fourier Transform (simply set σ = 0). The Laplace Transform converges for more functions than the Fourier Transform since it could converge off of the jω axis. Here is a plot of the s-plane:
The Inverse Bilateral Laplace Transform of X(s) is:
Notice that to compute the inverse Laplace Transform, it requires a contour integral. (When taking the inverse transform, the value of c for the contour integral must be in the region where the integral exists.) Fortunately, we will see more convenient ways (namely, Partial Fraction Expansion) to take the inverse transform so you are not required to know how to do contour integration.
If we define x(t) to be 0 for t < 0, this gives us the unilateral Laplace transform:
As we'll see, an important difference between the bilateral and unilateral Laplace Transforms is that you need to specify the region of convergence (ROC) for the bilateral case.
We point out (without proof) several features of ROCs:
Taking the Laplace Transform is clearly a linear operation:
L[ax1(t) + bx2(t)] = aX1(s) + bX2(s)
where X1(s) is the Laplace Transform of x1(t) and X2(s) is the Laplace Transform of x2(t).
In Example 1, we needed to specify that Re(s) > 0. If this is not the case, the integral would have not converged at the upper limit of infinity.
The general Laplace Transform for an exponential function is: