The Laplace Transform

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

$$X(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}{\cal L}_s\{x\} \mathrel{\stackrel{\mathrm{\Delta}}{=}}\int_0^\infty x(t) e^{-st} dt$$

• $t =$ time in seconds
• $s=\sigma + j\omega$ is a complex variable
• Appropriate for causal signals

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

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

Thus, the Laplace transform generalizes the Fourier transform from the real line (the frequency axis) to the entire complex plane.

$\parbox{0.8\textwidth}{The Fourier transform equals the Laplace transform evaluated along the $j\omega$\ axis in the complex $s$\ plane}$

The Laplace Transform can also be seen as the Fourier transform of an exponentially windowed causal signal $x(t)$