The Laplace Transform

This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Communications Engineering, Universit├Ąt Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.

Summary of Properties, Theorems and Transforms

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.

Definition

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 $s \in \text{ROC} \{ x(t) \}$.

Properties and Theorems

The properties and theorems of the two-sided Laplace transform are given as

 $x(t)$ $X(s) = \mathcal{L} \{ x(t) \}$ $\text{ROC}$
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 $\tau \in \mathbb{R}$.

Selected Transforms

Two-sided Laplace transforms which are frequently used are given as

$x(t)$ $X(s) = \mathcal{L} \{ x(t) \}$ $\text{ROC}$
$\delta(t)$ $1$ $\mathbb{C}$
$\epsilon(t)$ $\frac{1}{s}$ $\Re \{s\} > 0$
$t \epsilon(t)$ $\frac{1}{s^2}$ $\Re \{s\} > 0$
$e^{- s_0 t} \epsilon(t)$ $\frac{1}{s + s_0}$ $\Re \{s\} > \text{Re}\{-s_0\}$
$\sin(\omega_0 t) \epsilon(t)$ $\frac{\omega_0}{s^2 + \omega_0^2}$ $\Re \{s\} > 0$
$\cos(\omega_0 t) \epsilon(t)$ $\frac{s}{s^2 + \omega_0^2}$ $\Re \{s\} > 0$
$t^n e^{-s_0 t} \epsilon(t)$ $\frac{n!}{(s+s_0)^{n+1}}$ $\Re \{s\} > \text{Re}\{-s_0\}$
$e^{-s_0 t} \cos(\omega_0 t) \epsilon(t)$ $\frac{s + s_0}{(s+s_0)^2 + \omega_0^2}$ $\Re \{s\} > \Re \{-s_0\}$
$e^{-s_0 t} \sin(\omega_0 t) \epsilon(t)$ $\frac{\omega_0}{(s+s_0)^2 + \omega_0^2}$ $\Re \{s\} > \Re \{-s_0\}$
$t \cos(\omega_0 t) \epsilon(t)$ $\frac{s^2 - \omega_0^2}{(s^2 + \omega_0^2)^2}$ $\Re \{s\} > 0$
$t \sin(\omega_0 t) \epsilon(t)$ $\frac{2 \omega_0 s}{(s^2 + \omega_0^2)^2}$ $\Re \{s\} > 0$

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

This notebook is provided as Open Educational Resource. Feel free to use the notebook for your own 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: Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples.