Transforms of Discontinuous and Periodic Functions

Transforms of Discontinuous and Periodic FunctionsExpected Educational ResultsTransforms of Discontinuous and Periodic FunctionsRectangular Window FunctionDefinition: Rectangular Window FunctionGraphs of the Rectangular Window FunctionInvestigation 05Investigation 06Laplace Transform of the Rectangular Window FunctionInvestigation 07Investigation 08CC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Section 7.6 – Transforms of Discontinuous and Periodic Functions

Expected Educational Results

Objective 23–1: I can sketch step and window functions.
Objective 23–2: I can evaluate the Laplace transform of step and window functions.
Objective 23–3: I can sketch periodic functions.
Objective 23–4: I can evaluate the Laplace transform of periodic functions.
Objective 23–5: I can evaluate the Laplace transform of the Gamma function.

Transforms of Discontinuous and Periodic Functions

Rectangular Window Function

Definition: Rectangular Window Function

rectangular window function $\Pi_{a,b}(t)$ is defined by

$\Pi_{a,b}(t)=u(t-a)-u(t-b)=\begin{cases} \hfill 0, \hfill & t<a \\ \hfill 1, \hfill & a<t<b \\ \hfill 0, \hfill & t>b \\ \end{cases}$

Graphs of the Rectangular Window Function

Investigation 05

Sketch the following functions. Explain.

$f(t)=\Pi_{1,3}(t)$
$f(t)=\Pi_{-2,1}(t)$

Investigation 06

Sketch the following functions. Explain.

$f(t)=t^2\,\left[u(t)-u(t-9)\right]$
$f(t)=\cos{(t)}\,\Pi_{\frac{\pi}{2},\frac{5\pi}{2}}(t)$
$f(t)=t^2\,\Pi_{2,3}(t)$

Laplace Transform of the Rectangular Window Function

Investigation 07

Evaluate the following Laplace transforms.

$\mathcal{L}\left\{\Pi_{1,3}(t)\right\}(s)$
$\mathcal{L}\left\{e^t\,\Pi_{1,2}(t)\right\}(s)$
$\mathcal{L}\left\{\sin{(t)}\,\Pi_{0,\frac{\pi}{4}}(t)\right\}(s)$
$\mathcal{L}\left\{\Pi_{a,b}(t)\right\}(s)$ $a<b$

Investigation 08

Find the Laplace transforms of the following functions.

$f(t)=\begin{cases} \hfill t^2, \hfill & 0<t<\pi \\ \hfill \sin{(t)}, \hfill & t>\pi \\ \end{cases}$
$f(t)=\begin{cases} \hfill t, \hfill & 0<t<2 \\ \hfill 1+t, \hfill & 2<t<3 \\ \hfill e^t, \hfill & 3<t<5 \\ \hfill t^2, \hfill & t>5 \\ \end{cases}$

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Sunday, 8 November 2020 22:02 EDT