How to get a rectangular pulse function over the entire domain of $\mathbb{R}$

Question

I am reading this article:

https://lpsa.swarthmore.edu/Fourier/Series/ExFS.html#EvenPulse

The following plot is depicted there.

This is called a rectangular pulse function.

The article states then

$$x_T(t) = \left\{ {\matrix{ {A,\quad |t| \le {{{T_p}} \over 2}} \cr {0,\quad |t| \gt {{{T_p}} \over 2}} \cr } } \right.,\quad \quad - {T \over 2} < t \le {T \over 2}$$

However, I want this function to extend over the entire domain of $\mathbb{R}$, just as $\sin(t)$ and $\cos(t)$ do.

How can I extend this function so it covers the entire domain of $\mathbb{R}$?

Answer 1

I have a proposed answer, although I'm sure someone can probably come up with a better one.

Let $h(t) = \left(\left(t-\frac{T}{2}\right)\text{ mod } T\right)-\frac{T}{2}$

$$x_T(t) = \left\{ {\matrix{ {A,\quad |h(t)| \le {{{T_p}} \over 2}} \cr {0,\quad |h(t)| \gt {{{T_p}} \over 2}} \cr } } \right.$$

I think this should create a pulse function whose domain is $\mathbb{R}$.

Assume $A=5$, $T=10$, $T_p = 5$. Here's a graph: