4.1.1. The Laplace Transform

Let \(x(t)\) be a continous time signal. Its one-sided Laplace transform \(X(s)\) is defined with:

\[X(s) = \int_{0}^{\infty} x(t) e^{-st} dt\]

The two-sided or bilateral Laplace transform integrates from \(-\infty\) to \(\infty\). Using the one-sided Laplace transform is equivalent with transforming causal signals and systems, i.e. a signal such that \(x(t)=0\) for \(x<0\). And most often we assume that \(x(0)=0\).

Just like for the Z-transform we have to specify the ROC for the Laplace transform. Existence of the integral defining the Laplace transform in general is not be guaranteed for all points in the complex plane.

A signal \(x(t)\) and its Laplace transform \(X(s)\) are denoted as the transform pair:

\(\newcommand{\ltarrow}{\stackrel{\op L}{\longrightarrow}}\) \(\newcommand{\op}[1]{\mathsf #1}\)

\[x(t) \ltarrow X(s)\]

For completeness we give the inverse Laplace transform:

\[x(t) = \frac{1}{2\pi j} \lim_{T\rightarrow\infty}\int_{\gamma-jT}^{\gamma+jT} e^{st} X(s) ds\]

where \(\gamma\) is a real number so that the contour path of integration is in the region of convergence of \(X(s)\).

The inverse transform is not much used in practice. Most often the inverse transform is deduced from known transform pairs and the properties of the Laplace transform.