11.3: Properties of the Laplace Transform
- Page ID
- 22909
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| Property | Signal | Laplace Transform | Region of Convergence |
|---|---|---|---|
| Linearity | \(\alpha x_{1}(t)+\beta x_{2}(t)\) | \(\alpha X_{1}(s)+\beta X_{2}(s)\) | At least \(\mathrm{ROC}_{1} \cap \mathrm{ROC}_{2}\) |
| Time Shifting | \(x(t−\tau)\) | \(e^{-(s \tau)} X(s)\) | \(\mathrm{ROC}\) |
| Frequency Shifting (modulation) | \(e^{\eta t} x(t)\) | \(X(s-\eta)\) | Shifted \(\mathrm{ROC}\) (\(s-\eta\) must be in the region of convergence) |
| Time Scaling | \(x(\alpha t)\) | \((1-|\alpha|) X(s-\alpha)\) | Scaled \(\mathrm{ROC}\) (\(s-\alpha\) must be in the region of convergence) |
| Conjugation | \(x^*(t)\) | \(X^*(s^*)\) | \(\mathrm{ROC}\) |
| Convolution | \(x_{1}(t) * x_{2}(t)\) | \(X_{1}(t) X_{2}(t)\) | At least \(\mathrm{ROC}_{1} \cap \mathrm{ROC}_{2}\) |
| Time Differentiation | \(\frac{d}{d t} x(t)\) | \(sX(s)\) | At least \(\mathrm{ROC}\) |
| Frequency Differentiation | \((-t)x(t)\) | \(\frac{d}{d s} X(s)\) | \(\mathrm{ROC}\) |
| Integration in Time | \(\int_{-\infty}^{t} x(\tau) d \tau\) | \((1-s) X(s)\) | At least \(\operatorname{ROC} \cap(\operatorname{Re}(s)>0)\) |