Laplace Time scaling Property

Scaling the time variable compresses or expands the frequency spectrum:

$$\mathcal{L}\{f(at)\} = \frac{1}{a} F\left(\frac{s}{a}\right) \quad \text{for } a > 0$$

Derivation:

$$\begin{align*} \mathcal{L}\{f(t)\} &= \int_{0}^{\infty} e^{-st}f(t)\, dt = F(s)\\ \mathcal{L}\{f(at)\} &= \int_{0}^{\infty} e^{-st}f(at)\, dt \\ \text{take,}\, u &= at \\ \frac{u}{a} &= t \\ ({t \to 0, u \to 0}) \\ ({t \to \infty, u \to \infty \, \text{if a>0}}) \\ \frac{1}{a}\cdot du &= dt \\ \\ \mathcal{L}\{f(at)\} &= \int_{0}^{\infty} e^{-s\frac{u}{a}}f(u)\frac{1}{a}\cdot du \\ \mathcal{L}\{f(at)\} &= \frac{1}{a}\int_{0}^{\infty} e^{-\left(\frac{s}{a}\right)\cdot u}f(u)\, du \\ \mathcal{L}\{f(at)\} &= \frac{1}{a} \cdot F\left(\frac{s}{a}\right) \end{align*}$$

• $a > 0$ criteria is for the bounds to be preserved from $0$ to $\infty$
• In real world, $t$ must be positive since time cannot be reversed.