Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as

$${\displaystyle \operatorname {rect} \left({\frac {t}{T}}\right)=\Pi \left({\frac {t}{T}}\right)=\left\{{\begin{array}{rl}0,&{\text{if }}|t|>{\frac {T}{2}}\\{\frac {1}{2}},&{\text{if }}|t|={\frac {T}{2}}\\1,&{\text{if }}|t|<{\frac {T}{2}}.\end{array}}\right.}$$

Alternative definitions of the function define ${\textstyle \operatorname {rect} \left(t=\pm {\frac {T}{2}}\right)}$ to be 0, 1, or undefined. The area under the curve does not change for the different definitions of the functions at ${\textstyle t=\pm {\frac {T}{2}}}$.

The rectangular function can be used as the basis for a rectangular wave.