Chapter 2, Problem 13PS

Pulse Function For positive numbers $a<b$, the pulse function is defined as

$P_{a,b}\left(x\right)=H\left(x-a\right)-H\left(x-b\right)=\{\begin{array}{l}\begin{array}{cc}0,& x<a\end{array}\\ \begin{array}{cc}1,& a\le x<b\end{array}\\ \begin{array}{cc}0,& x\ge b\end{array}\end{array}$

where $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$ is the Heaviside function.

(a) Sketch the graph of the pulse function.

(b) Find the following limits:

(i) $\underset{x\to a^{+}}{\lim }{P}_{a,b}\left(x\right)$ (ii) $\underset{x\to a^{-}}{\lim }{P}_{a,b}\left(x\right)$

(iii) $\underset{x\to b^{+}}{\lim }{P}_{a,b}\left(x\right)$ (iv) $\underset{x\to b^{-}}{\lim }{P}_{a,b}\left(x\right)$

(c) Discuss the continuity of the pulse function.

(d) Why is $U\left(x\right)=\frac{1}{b-a}{P}_{a,b}\left(x\right)$ called the unit pulse function?