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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 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 π f t d t = sin ⁡ ⁡ ( π π f ) π π f = sinc ⁡ ⁡ ( π π f ) = sinc π 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 π ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t ) ⋅ ⋅ e − − i ω ω t d t = 1 2 π π ⋅ ⋅ sin ⁡ ⁡ ( ω ω / 2 ) ω ω / 2 = 1 2 π π ⋅ ⋅ sinc ⁡ ⁡ ( ω 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 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 π f t d t = a sin ⁡ ⁡ ( π π a f ) π π a f = a sinc π 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 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 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 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 δ ( x ) {\displaystyle \delta (x)} .cite-ref-0-11-0[11] Specifically, δ 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 δ ( x ) . {\displaystyle g(0)=\int \limits _{-\infty }^{\infty }dx\ g(x)\delta (x).} The Fourier transform of the Dirac delta function δ δ ( t ) {\displaystyle \delta (t)} is

δ δ ( f ) = ∫ ∫ − − ∞ ∞ ∞ ∞ δ δ ( t ) ⋅ ⋅ e − − i 2 π π f t d t = lim a → → 0 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t a ) ⋅ ⋅ e − − i 2 π 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 δ ( t ) {\displaystyle \delta (t)} is

δ 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.