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

rect ⁡ ( t T ) = Π ( t T ) = { 0 , if | t | > T 2 1 2 , if | t | = T 2 1 , if | t | < T 2 . {\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 rect ⁡ ( t = ± T 2 ) {\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 t = ± T 2 {\textstyle t=\pm {\frac {T}{2}}} . The rectangular function can be used as the basis for a rectangular wave.

History 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.

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

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

Fourier transform of the rectangular function

The unitary Fourier transforms of the rectangular function are

∫ − ∞ ∞ rect ⁡ ( t ) ⋅ e − i 2 π f t d t = sin ⁡ ( π f ) π f = sinc ⁡ ( π f ) = sinc π ⁡ ( f ) , {\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} (\pi f)=\operatorname {sinc} _{\pi }(f),}

using ordinary frequency f, where sinc π {\displaystyle \operatorname {sinc} _{\pi }} is the normalized form of the sinc function and

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