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The rectangular function (also known as the rectangle function, rect

function, Pi function, Heaviside Pi function,cite-ref-1[1] gate function, unit

pulse, or the normalized boxcar function) is defined ascite-ref-wolfram-2-0[2]

rect ⁡ ( t a ) = Π ( t a ) = { 0 , if | t | > a 2 1 2 , if | t | = a 2 1

, if | t | < a 2 . {\displaystyle \operatorname {rect} \left({\frac

{t}{a}}\right)=\Pi \left({\frac

{t}{a}}\right)=\left\{{\begin{array}{rl}0,&{\text{if }}|t|>{\frac

{a}{2}}\\{\frac {1}{2}},&{\text{if }}|t|={\frac

{a}{2}}\\1,&{\text{if }}|t|<{\frac {a}{2}}.\end{array}}\right.}

Alternative definitions of the function define rect ⁡ ( ± 1 2 )

{\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}

to be 0,cite-ref-3[3] 1,cite-ref-4[4]cite-ref-5[5] or undefined.

Its periodic version is called a rectangular wave.

• History

• Relation to the boxcar function

• Fourier transform of the rectangular function

• Relation to the triangular function

• Use in probability

• Rational approximation

• Demonstration of validity

• Dirac delta function

• See also

• References

The rect function has been introduced 1953 by Woodwardcite-ref-6[6] in

"Probability and Information Theory, with Applications to Radar"cite-ref-7[7] as

an ideal cutout operator, together with the sinc functioncite-ref-8[8]cite-ref-9[9] 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:

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.}

The unitary Fourier transforms of the rectangular function arecite-ref-wolfram-2-1[2] ∫ − ∞

∞ 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

formcite-ref-10[10] of the sinc function and 1 2 π ∫ − ∞ ∞ rect ⁡ ( t ) ⋅ e − i ω t

d t = 1 2 π ⋅ sin ⁡ ( ω / 2 ) ω / 2 = 1 2 π ⋅ sinc ⁡ ( ω / 2 ) ,

{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty

}\operatorname {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt

{2\pi }}}\cdot {\frac {\sin \left(\omega /2\right)}{\omega

/2}}={\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {sinc}

\left(\omega /2\right),} using angular frequency ω {\displaystyle

\omega } , where sinc {\displaystyle \operatorname {sinc} } is the

unnormalized form of the sinc function.

For rect ⁡ ( x / a ) {\displaystyle \operatorname {rect} (x/a)} , its

Fourier transform is ∫ − ∞ ∞ rect ⁡ ( t a ) ⋅ e − i 2 π f t d t = a sin

⁡ ( π a f ) π a f = a sinc π ⁡ ( a f ) . {\displaystyle \int

_{-\infty }^{\infty }\operatorname {rect} \left({\frac

{t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=a{\frac {\sin(\pi af)}{\pi

af}}=a\ \operatorname {sinc} _{\pi }{(af)}.}

We can define the triangular function as the convolution of two

rectangular functions:

t r i ( t / T ) = r e c t ( 2 t / T ) ∗ r e c t ( 2 t / T ) .

{\displaystyle \operatorname {tri(t/T)} =\operatorname {rect(2t/T)}

*\operatorname {rect(2t/T)} .\,}

Viewing the rectangular function as a probability density function, it

is a special case of the continuous uniform distribution with a = − 1 /

2 , b = 1 / 2. {\displaystyle a=-1/2,b=1/2.} The

characteristic function is

φ ( k ) = sin ⁡ ( k / 2 ) k / 2 , {\displaystyle \varphi (k)={\frac

{\sin(k/2)}{k/2}},}

and its moment-generating function is

M ( k ) = sinh ⁡ ( k / 2 ) k / 2 , {\displaystyle M(k)={\frac

{\sinh(k/2)}{k/2}},}

where sinh ⁡ ( t ) {\displaystyle \sinh(t)} is the hyperbolic sine

function.

The pulse function may also be expressed as a limit of a

rational function:

Π ( t ) = lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 . {\displaystyle \Pi

(t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac

{1}{(2t)^{2n}+1}}.}

First, we consider the case where | t | < 1 2 . {\textstyle |t|<{\frac

{1}{2}}.} Notice that the term ( 2 t ) 2 n {\textstyle (2t)^{2n}} is

always positive for integer n . {\displaystyle n.} However, 2 t < 1

{\displaystyle 2t<1} and hence ( 2 t ) 2 n {\textstyle (2t)^{2n}}

approaches zero for large n . {\displaystyle n.}

It follows that: lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 0 + 1 = 1 ,

| t | < 1 2 . {\displaystyle \lim _{n\rightarrow \infty ,n\in

\mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac

{1}{0+1}}=1,|t|<{\tfrac {1}{2}}.}

Second, we consider the case where | t | > 1 2 . {\textstyle

|t|>{\frac {1}{2}}.} Notice that the term ( 2 t ) 2 n {\textstyle

(2t)^{2n}} is always positive for integer n . {\displaystyle n.}

However, 2 t > 1 {\displaystyle 2t>1} and hence ( 2 t ) 2 n

{\textstyle (2t)^{2n}} grows very large for large n . {\displaystyle

n.}

It follows that: lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 + ∞ + 1 = 0

, | t | > 1 2 . {\displaystyle \lim _{n\rightarrow \infty ,n\in

\mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{+\infty

+1}}=0,|t|>{\tfrac {1}{2}}.}

Third, we consider the case where | t | = 1 2 . {\textstyle |t|={\frac

{1}{2}}.} We may simply substitute in our equation:

lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = lim n → ∞ , n ∈ ( Z ) 1 1 2 n

+ 1 = 1 1 + 1 = 1 2 . {\displaystyle \lim _{n\rightarrow \infty

,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}=\lim _{n\rightarrow

\infty ,n\in \mathbb {(} Z)}{\frac {1}{1^{2n}+1}}={\frac

{1}{1+1}}={\tfrac {1}{2}}.}

We see that it satisfies the definition of the pulse function.

Therefore,

rect ⁡ ( t ) = Π ( t ) = lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = { 0

if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 . {\displaystyle

\operatorname {rect} (t)=\Pi (t)=\lim _{n\rightarrow \infty ,n\in

\mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\begin{cases}0&{\mbox{if

}}|t|>{\frac {1}{2}}\\{\frac {1}{2}}&{\mbox{if }}|t|={\frac

{1}{2}}\\1&{\mbox{if }}|t|<{\frac {1}{2}}.\\\end{cases}}}

The rectangle function can be used to represent the Dirac delta function

δ ( x ) {\displaystyle \delta (x)} .cite-ref-0-11-0[11] Specifically, δ ( x ) = lim a

→ 0 1 a rect ⁡ ( x a ) . {\displaystyle \delta (x)=\lim _{a\to

0}{\frac {1}{a}}\operatorname {rect} \left({\frac {x}{a}}\right).}

For a function g ( x ) {\displaystyle g(x)} , its average over the

width a {\displaystyle a} around 0 in the function domain is

calculated as,

g a v g ( 0 ) = 1 a ∫ − ∞ ∞ d x g ( x ) rect ⁡ ( x a ) . {\displaystyle

g_{avg}(0)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\

g(x)\operatorname {rect} \left({\frac {x}{a}}\right).} To obtain g (

0 ) {\displaystyle g(0)} , the following limit is applied,

g ( 0 ) = lim a → 0 1 a ∫ − ∞ ∞ d x g ( x ) rect ⁡ ( x a )

{\displaystyle g(0)=\lim _{a\to 0}{\frac {1}{a}}\int \limits

_{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac

{x}{a}}\right)} and this can be written in terms of the Dirac delta

function as, g ( 0 ) = ∫ − ∞ ∞ d x g ( x ) δ ( x ) . {\displaystyle

g(0)=\int \limits _{-\infty }^{\infty }dx\ g(x)\delta (x).} The

Fourier transform of the Dirac delta function δ ( t ) {\displaystyle

\delta (t)} is

δ ( f ) = ∫ − ∞ ∞ δ ( t ) ⋅ e − i 2 π f t d t = lim a → 0 1 a ∫ − ∞ ∞

rect ⁡ ( t a ) ⋅ e − i 2 π f t d t = lim a → 0 sinc ⁡ ( a f ) .

{\displaystyle \delta (f)=\int _{-\infty }^{\infty }\delta

(t)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}{\frac {1}{a}}\int

_{-\infty }^{\infty }\operatorname {rect} \left({\frac

{t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}\operatorname

{sinc} {(af)}.} where the sinc function here is the normalized sinc

function. Because the first zero of the sinc function is at f = 1 / a

{\displaystyle f=1/a} and a {\displaystyle a} goes to infinity, the

Fourier transform of δ ( t ) {\displaystyle \delta (t)} is

δ ( f ) = 1 , {\displaystyle \delta (f)=1,} means that the frequency

spectrum of the Dirac delta function is infinitely broad. As a pulse is

shorten in time, it is larger in spectrum.

• Fourier transform

• Square wave

• Step function

• Top-hat filter

• Boxcar function

cite-note-11. ↑ citerefwolfram-research2008Wolfram Research (2008). "HeavisidePi, Wolfram Language function". Retrieved October 11, 2022.

cite-note-wolfram-22. ↑ reference-mathworld-rectangle-functionciterefweissteinWeisstein, Eric W. "Rectangle Function". MathWorld.

cite-note-33. ↑ citerefwang2012Wang, Ruye (2012). Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis. Cambridge University Press. pp. 135–136. ISBN 9780521516884.

cite-note-44. ↑ citereftang2007Tang, K. T. (2007). Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models. Springer. p. 85. ISBN 9783540446958.

cite-note-55. ↑ citerefkumar2011Kumar, A. Anand (2011). Signals and Systems. PHI Learning Pvt. Ltd. pp. 258–260. ISBN 9788120343108.

cite-note-66. ↑ citerefklauder1960Klauder, John R (1960). "The Theory and Design of Chirp Radars". Bell System Technical Journal. 39 (4): 745–808. doi:10.1002/j.1538-7305.1960.tb03942.x.

cite-note-77. ↑ citerefwoodward1953Woodward, Philipp M (1953). Probability and Information Theory, with Applications to Radar. Pergamon Press. p. 29.

cite-note-88. ↑ citerefhiggins1996Higgins, John Rowland (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press Inc. p. 4. ISBN 0198596995.

cite-note-99. ↑ citerefzayed1996Zayed, Ahmed I (1996). Handbook of Function and Generalized Function Transformations. CRC Press. p. 507. ISBN 9780849380761.

cite-note-1010. ↑ Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html

cite-note-0-1111. ↑ citerefkharebutolarajora2023Khare, Kedar; Butola, Mansi; Rajora, Sunaina (2023). "Chapter 2.4 Sampling by Averaging, Distributions and Delta Function". Fourier Optics and Computational Imaging (2nd ed.). Springer. pp. 15–16. doi:10.1007/978-3-031-18353-9. ISBN 978-3-031-18353-9.