Laplace Transform Table
A comprehensive reference for Control Systems and Signal Processing. Convert functions from the time domain \(f(t)\) to the s-domain \(F(s)\).
Definition
The **bilateral Laplace transform** converts time-domain signals into the complex s-domain, making solving differential equations much easier (turning them into algebraic equations).
\[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t) e^{-st} dt \]
Common Transform Pairs
| Signal Name | Time Domain \(f(t)\) | Laplace Domain \(F(s)\) |
|---|---|---|
| Unit Impulse | \( \delta(t) \) | \( 1 \) |
| Unit Step | \( u(t) \) | \( \frac{1}{s} \) |
| Ramp | \( t \) | \( \frac{1}{s^2} \) |
| Exponential Decay | \( e^{-at} \) | \( \frac{1}{s+a} \) |
| Sine Wave | \( \sin(\omega t) \) | \( \frac{\omega}{s^2 + \omega^2} \) |
| Cosine Wave | \( \cos(\omega t) \) | \( \frac{s}{s^2 + \omega^2} \) |
| Damped Sine | \( e^{-at}\sin(\omega t) \) | \( \frac{\omega}{(s+a)^2 + \omega^2} \) |
| Damped Cosine | \( e^{-at}\cos(\omega t) \) | \( \frac{s+a}{(s+a)^2 + \omega^2} \) |
Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | \( af(t) + bg(t) \) | \( aF(s) + bG(s) \) |
| Time Shifting | \( f(t-a)u(t-a) \) | \( e^{-as}F(s) \) |
| Frequency Shifting | \( e^{-at}f(t) \) | \( F(s+a) \) |
| Time Scaling | \( f(at) \) | \( \frac{1}{a}F(\frac{s}{a}) \) |
| Differentiation | \( f'(t) \) | \( sF(s) - f(0) \) |
| Integration | \( \int_{0}^{t} f(\tau) d\tau \) | \( \frac{1}{s}F(s) \) |