This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.
The properties, theorems and transforms of the two-sided Laplace transform as derived in the previous sections are summarized in the following. The corresponding tables serve as a reference for the application of the Laplace transform in the theory of signals and systems. Please refer to the respective sections for details.
The two-sided Laplace transform and its inverse are defined as
\begin{align} X(s) &= \int_{-\infty}^{\infty} x(t) \, e^{- s t} \; dt \\ x(t) &= \frac{1}{2 \pi j} \int_{\sigma - j \infty}^{\sigma + j \infty} X(s) \, e^{s t} \; ds \end{align}where $s \in \text{ROC} \{ x(t) \}$.
The properties and theorems of the two-sided Laplace transform are given as
| | $x(t)$ | $X(s) = \mathcal{L} \{ x(t) \}$ | ROC |
|---|---|---|---|
| Linearity | $A \, x_1(t) + B \, x_2(t)$ | $A \, X_1(s) + B \, X_2(s)$ | $\supseteq \text{ROC}\{x_1(t)\} \cap \text{ROC}\{x_2(t)\}$ |
| Real-valued signal | $x(t) = x^*(t)$ | $X(s) = X^*(s^*)$ | |
| Scaling | $x(a t)$ | $\frac{1}{\lvert a \rvert} X\left( \frac{s}{a} \right)$ | $s: \frac{s}{a} \in \text{ROC}\{x(t)\}$ |
| Convolution | $x(t) * h(t)$ | $X(s) \cdot H(s)$ | $\supseteq \text{ROC}\{x(t)\} \cap \text{ROC}\{h(t)\}$ |
| Shift | $x(t - \tau)$ | $e^{-s \tau} \cdot X(s)$ | $\text{ROC}\{x(t)\}$ |
| Differentiation (causal signal) | $\frac{d}{dt} x(t)$ | $s \cdot X(s) - x(0+)$ | $\supseteq \text{ROC}\{x(t)\}$ |
| Integration | $\int_{-\infty}^{t} x(t) \; dt$ | $\frac{1}{s} \cdot X(s)$ | $\supseteq \text{ROC}\{x(t)\} \cap \{s: \Re \{s\} > 0 \}$ |
| Modulation | $e^{s_0 t}\cdot x(t)$ | $X(s - s_0)$ | $s: s - \Re \{s_0\} \in \text{ROC}\{x(t)\}$ |
where $A, B, s_0 \in \mathbb{C}$, $a \in \mathbb{R} \setminus \{0\}$ and $\tau \in \mathbb{R}$.
Two-sided Laplace transforms which are frequently used are given as
| $x(t)$ | $X(s) = \mathcal{L} \{ x(t) \}$ | ROC |
|---|---|---|
| $\delta(t)$ | $1$ | $\mathbb{C}$ |
| $\epsilon(t)$ | $\frac{1}{s}$ | $\Re \{s\} > 0$ |
| $t \epsilon(t)$ | $\frac{1}{s^2}$ | $\Re \{s\} > 0$ |
| $e^{- s_0 t} \epsilon(t)$ | $\frac{1}{s + s_0}$ | $\Re \{s\} > \text{Re}\{-s_0\}$ |
| $\sin(\omega_0 t) \epsilon(t)$ | $\frac{\omega_0}{s^2 + \omega_0^2}$ | $\Re \{s\} > 0$ |
| $\cos(\omega_0 t) \epsilon(t)$ | $\frac{s}{s^2 + \omega_0^2}$ | $\Re \{s\} > 0$ |
| $t^n e^{-s_0 t} \epsilon(t)$ | $\frac{n!}{(s+s_0)^{n+1}}$ | $\Re \{s\} > \text{Re}\{-s_0\}$ |
| $e^{-s_0 t} \cos(\omega_0 t) \epsilon(t)$ | $\frac{s + s_0}{(s+s_0)^2 + \omega_0^2}$ | $\Re \{s\} > \Re \{-s_0\}$ |
| $e^{-s_0 t} \sin(\omega_0 t) \epsilon(t)$ | $\frac{\omega_0}{(s+s_0)^2 + \omega_0^2}$ | $\Re \{s\} > \Re \{-s_0\}$ |
| $t \cos(\omega_0 t) \epsilon(t)$ | $\frac{s^2 - \omega_0^2}{(s^2 + \omega_0^2)^2}$ | $\Re \{s\} > 0$ |
| $t \sin(\omega_0 t) \epsilon(t)$ | $\frac{2 \omega_0 s}{(s^2 + \omega_0^2)^2}$ | $\Re \{s\} > 0$ |
where $s_0 \in \mathbb{C}$, $\omega_0 \in \mathbb{R}$ and $n \in \mathbb{N}$. More one- and two-sided transforms may be found in the literature or online.
Copyright
The notebooks are provided as Open Educational Resource. Feel free to use the notebooks for your own educational purposes. The text is licensed under Creative Commons Attribution 4.0, the code of the IPython examples under the MIT license. Please attribute the work as follows: Lecture Notes on Signals and Systems by Sascha Spors.