Introduction to Laplace Transform Analysis

The one-sided Laplace transform of a signal $x(t)$ is defined by

$$X(s) \isdef {\cal L}_s\{x\} \isdef \int_0^\infty x(t) e^{-st}dt$$

where $t$ is real and $s=\sigma + j\omega$ is a complex variable. The one-sided Laplace transform is also called the unilateral Laplace transform. There is also a two-sided, or bilateral, Laplace transform obtained by setting the lower integration limit to $-\infty$ instead of 0. Since we will be analyzing only causal^B.1 linear systems using the Laplace transform, we can use either. However, it is customary in engineering treatments to use the one-sided definition.

When evaluated along the $s=j\omega$ axis (i.e., $\sigma=0$), the Laplace transform reduces to the unilateral Fourier transform:

$$X(j\omega) = \int_0^\infty x(t) e^{-j\omega t}dt.$$

The Fourier transform is normally defined bilaterally ( $0\leftarrow -\infty$ above), but for causal signals $x(t)$, there is no difference. We see that the Laplace transform can be viewed as a generalization of the Fourier transform from the real line (a simple frequency axis) to the entire complex plane. We say that the Fourier transform is obtained by evaluating the Laplace transform along the $j\omega$ axis in the complex $s$ plane.

An advantage of the Laplace transform is the ability to transform signals which have no Fourier transform. To see this, we can write the Laplace transform as

$$X(s) = \int_0^\infty x(t) e^{-(\sigma + j\omega)t} dt = \int_0^\infty \left[x(t)e^{-\sigma t}\right] e^{-j\omega t} dt .$$

Thus, the Laplace transform can be seen as the Fourier transform of an exponentially windowed input signal. For $\sigma>0$ (the so-called ``strict right-half plane'' (RHP)), this exponential weighting forces the Fourier-transformed signal toward zero as $t\to\infty$. As long as the signal $x(t)$ does not increase faster than $\exp(Bt)$ for some $B$, its Laplace transform will exist for all $\sigma>B$. We make this more precise in the next section.