Rectangular function

File:Rectangular function.svg

Rectangular function with a = 1

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

$$\mathrm{rect}\left(\frac{t}{a}\right)=\mathrm{\Pi }\left(\frac{t}{a}\right)=\{\begin{array}{rl}0,& \text{if }|t|>\frac{a}{2}\\ \frac{1}{2},& \text{if }|t|=\frac{a}{2}\\ 1,& \text{if }|t|<\frac{a}{2}.\end{array}$$

Alternative definitions of the function define $\mathrm{rect}(\pm \frac{1}{2})$ to be 0, 1, or undefined.

Its periodic version is called 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:

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

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

Fourier transform of the rectangular function

File:Sinc function (normalized).svg

Plot of normalized $\mathrm{sinc}(x)$ function (i.e. $\mathrm{sinc}(\pi x)$) with its spectral frequency components.

The unitary Fourier transforms of the rectangular function are $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{rect}(t)\cdot e^{-i2\pi ft}dt=\frac{\sin (\pi f)}{\pi f}=\mathrm{sinc}(\pi f)={\mathrm{sinc}}_{\pi }(f),$$ using ordinary frequency Template:Mvar, where ${\mathrm{sinc}}_{\pi }$ is the normalized form of the sinc function and $$\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{rect}(t)\cdot e^{-i\omega t}dt=\frac{1}{\sqrt{2\pi }}\cdot \frac{\sin (\omega /2)}{\omega /2}=\frac{1}{\sqrt{2\pi }}\cdot \mathrm{sinc}(\omega /2),$$ using angular frequency $\omega$, where $\mathrm{sinc}$ is the unnormalized form of the sinc function.

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

Relation to the triangular function

We can define the triangular function as the convolution of two rectangular functions:

$$\mathrm{tri(t/T)}=\mathrm{rect(2t/T)}*\mathrm{rect(2t/T)}.$$

Use in probability

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with $a=-1/2,b=1/2.$ The characteristic function is

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

and its moment-generating function is

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

where $\sinh (t)$ is the hyperbolic sine function.

Rational approximation

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

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

Demonstration of validity

First, we consider the case where $|t|<\frac{1}{2}.$ Notice that the term $(2t{)}^{2n}$ is always positive for integer $n.$ However, $2t<1$ and hence $(2t{)}^{2n}$ approaches zero for large $n.$

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

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

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

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

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

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

$$\mathrm{rect}(t)=\mathrm{\Pi }(t)=\underset{n\to \mathrm{\infty },n\in \mathbb{(}Z)}{\lim }\frac{1}{(2t{)}^{2n}+1}=\{\begin{array}{ll}0& \text{if }|t|>\frac{1}{2}\\ \frac{1}{2}& \text{if }|t|=\frac{1}{2}\\ 1& \text{if }|t|<\frac{1}{2}.\\ \end{array}$$

Dirac delta function

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

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

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

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

$$\delta (f)=1,$$ 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.

See also

• Fourier transform
• Square wave
• Step function
• Top-hat filter
• Boxcar function