Therectangular function (also known as therectangle function,rect function,Pi function,Heaviside Pi function,^[1]gate function,unit pulse, or thenormalizedboxcar function) is defined as^[2]

Alternative definitions of the function define$\mathrm{rect}\left(\pm \frac{1}{2}\right)$ to be 0,^[3] 1,^[4][5] or undefined.

Its periodic version is called arectangular wave.

Therect function has been introduced 1953 byWoodward^[6] in "Probability and Information Theory, with Applications to Radar"^[7] as an idealcutout operator, together with thesinc function^[8][9] as an idealinterpolation operator, and their counter operations which aresampling (comb operator) andreplicating (rep operator), respectively.

The rectangular function is a special case of the more generalboxcar function:

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

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

Theunitary Fourier transforms of the rectangular function are^[2]$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{rect}(t)\cdot e^{-i2\pi ft}dt=\frac{\sin (\pi f)}{\pi f}=\mathrm{sinc}(\pi f)={\mathrm{sinc}}_{\pi }(f),}$$using ordinary frequencyf, where${\displaystyle {\mathrm{sinc}}_{\pi }}$ is the normalized form^[10] of thesinc function and$${\displaystyle \frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{rect}(t)\cdot e^{-i\omega t}dt=\frac{1}{\sqrt{2\pi }}\cdot \frac{\sin \left(\omega /2\right)}{\omega /2}=\frac{1}{\sqrt{2\pi }}\cdot \mathrm{sinc}\left(\omega /2\right),}$$using angular frequency${\displaystyle \omega }$, where${\displaystyle \mathrm{sinc}}$ is the unnormalized form of thesinc function.

For${\displaystyle \mathrm{rect}(x/a)}$, its Fourier transform is$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{rect}\left(\frac{t}{a}\right)\cdot e^{-i2\pi ft}dt=a\frac{\sin (\pi af)}{\pi af}=a{\mathrm{sinc}}_{\pi }(af).}$$

We can define thetriangular function as theconvolution of two rectangular functions:

$${\displaystyle \mathrm{t}\mathrm{r}\mathrm{i}(\mathrm{t}/\mathrm{T})=\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(2\mathrm{t}/\mathrm{T})\ast \mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(2\mathrm{t}/\mathrm{T}).}$$

Viewing the rectangular function as aprobability density function, it is a special case of thecontinuous uniform distribution with${\displaystyle a=-1/2,b=1/2.}$ Thecharacteristic function is

$${\displaystyle \phi (k)=\frac{\sin (k/2)}{k/2},}$$

and itsmoment-generating function is

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

where${\displaystyle \sinh (t)}$ is thehyperbolic sine function.

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

$${\displaystyle \mathrm{\Pi }(t)=\underset{n\to \mathrm{\infty },n\in \mathbb{(}Z)}{lim}\frac{1}{(2t{)}^{2n}+1}.}$$

First, we consider the case where Notice that the term$(2t{)}^{2n}$ is always positive for integer${\displaystyle n.}$ However, and hence$(2t{)}^{2n}$ approaches zero for large${\displaystyle n.}$

It follows that:

Second, we consider the case where$|t|>\frac{1}{2}.$ Notice that the term$(2t{)}^{2n}$ is always positive for integer${\displaystyle n.}$ However,${\displaystyle 2t>1}$ and hence$(2t{)}^{2n}$ grows very large for large${\displaystyle n.}$

It follows that:$${\displaystyle \underset{n\to \mathrm{\infty },n\in \mathbb{(}Z)}{lim}\frac{1}{(2t{)}^{2n}+1}=\frac{1}{+\mathrm{\infty }+1}=0,|t|>\frac{1}{2}.}$$

Third, we consider the case where$|t|=\frac{1}{2}.$ We may simply substitute in our equation:

$${\displaystyle \underset{n\to \mathrm{\infty },n\in \mathbb{(}Z)}{lim}\frac{1}{(2t{)}^{2n}+1}=\underset{n\to \mathrm{\infty },n\in \mathbb{(}Z)}{lim}\frac{1}{1^{2n}+1}=\frac{1}{1+1}=\frac{1}{2}.}$$

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

The rectangle function can be used to represent theDirac delta function${\displaystyle \delta (x)}$.^[11] Specifically,$${\displaystyle \delta (x)=\underset{a\to 0}{lim}\frac{1}{a}\mathrm{rect}\left(\frac{x}{a}\right).}$$For a function${\displaystyle g(x)}$, its average over the width${\displaystyle a}$ around 0 in the function domain is calculated as,

$${\displaystyle g_{avg}(0)=\frac{1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\mathrm{rect}\left(\frac{x}{a}\right).}$$To obtain${\displaystyle g(0)}$, the following limit is applied,

$${\displaystyle g(0)=\underset{a\to 0}{lim}\frac{1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\mathrm{rect}\left(\frac{x}{a}\right)}$$and this can be written in terms of the Dirac delta function as,$${\displaystyle g(0)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\delta (x).}$$The Fourier transform of the Dirac delta function${\displaystyle \delta (t)}$ is

$${\displaystyle \delta (f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)\cdot e^{-i2\pi ft}dt=\underset{a\to 0}{lim}\frac{1}{a}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{rect}\left(\frac{t}{a}\right)\cdot e^{-i2\pi ft}dt=\underset{a\to 0}{lim}\mathrm{sinc}(af).}$$where thesinc function here is the normalized sinc function. Because the first zero of the sinc function is at${\displaystyle f=1/a}$ and${\displaystyle a}$ goes to infinity, the Fourier transform of${\displaystyle \delta (t)}$ is

$${\displaystyle \delta (f)=1,}$$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

• Special functions

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