Chapter 1, Problem 13PS

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

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

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

(a) Sketch the graph of the pulse function.

(b) Find the following limits:

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

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

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

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

(c) Discuss the continuity of the pulse function.

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