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.

The rect function has been introduced 1953 by Woodward in "Probability and Information Theory, with Applications to Radar" as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively.

The rectangular function is a special case of the more general boxcar function:

$${\displaystyle \operatorname {rect} \left({\frac {t-X}{Y}}\right)=H(t-(X-Y/2))-H(t-(X+Y/2))=H(t-X+Y/2)-H(t-X-Y/2)}$$

where ${\displaystyle H(x)}$ is the Heaviside step function; the function is centered at ${\displaystyle X}$ and has duration ${\displaystyle Y}$, from ${\displaystyle X-Y/2}$ to ${\displaystyle X+Y/2.}$