Laplace transform
Laplace transforms turn time-domain functions, where $t$ is the variable (time), into frequency-domain functions, where $s$ is the variable (complex frequency).
Formal definition
$$F(s) = \int_{0}^{\infty} f(t)e^{-st} dt$$
Inverse Laplace transform
Similar to the Z-transform, we usually calculate the inverse Laplace transform by reorganizing the Laplace representation into a form we recognize with partial fractions and then pattern matching. Again, the time-domain representation depends on the desired region of convergence - the same Laplace domain representation can result in different time-domain representations, depending on the RoC.
| Laplace domain representation | Region of convergence | Time-domain representation |
|---|---|---|
| $H(s)=\frac{1}{s-p}$ | $s > p$ | $h(t)=e^{pt}u(t)$ |
| $H(s)=\frac{1}{s-p}$ | $s < p$ | $h(t)=-e^{pt}u(-t)$ |
| $H(s) = 1 $ | All | $ h(t) = \delta(t) $ |
Why Laplace transforms are cool
Integration in the time domain becomes division by $s$ in the Laplace domain, and differentiation in the time domain becomes multiplication by $s$ in the Laplace domain. This is useful for block diagrams.