rectangular function

The rect function has been introduced 1953 by Woodward{{Cite journal |last=Klauder |first=John R |title=The Theory and Design of Chirp Radars |pages=745–808 |journal=Bell System Technical Journal |year=1960 |volume=39 |issue=4 |doi=10.1002/j.1538-7305.1960.tb03942.x |url=https://ieeexplore.ieee.org/document/6773600 |url-access=subscription }} in "Probability and Information Theory, with Applications to Radar"{{Cite book |last=Woodward |first=Philipp M |title=Probability and Information Theory, with Applications to Radar |publisher=Pergamon Press |pages=29 |year=1953 }} as an ideal cutout operator, together with the sinc function{{Cite book |last=Higgins |first=John Rowland |title=Sampling Theory in Fourier and Signal Analysis: Foundations |pages=4 |publisher=Oxford University Press Inc. |year=1996 |isbn=0198596995 }}{{Cite book |last=Zayed |first=Ahmed I |title=Handbook of Function and Generalized Function Transformations |pages=507 |publisher=CRC Press |year=1996 |isbn=9780849380761 }} 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:

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

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

File:Sinc_function_(normalized).svg

The unitary Fourier transforms of the rectangular function are

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{rect\}(t)\backslash cdot\; e^\{-i\; 2\backslash pi\; f\; t\}\; \backslash ,\; dt$$

=\frac{\sin(\pi f)}{\pi f} = \operatorname{sinc}(\pi f) =\operatorname{sinc}_\pi(f),

using ordinary frequency {{mvar|f}}, where $\backslash operatorname\{sinc\}\_\backslash pi$ is the normalized formWolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html of the sinc function and

$$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{rect\}(t)\backslash cdot\; e^\{-i\; \backslash omega\; t\}\; \backslash ,\; dt$$

=\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 $\backslash omega$, where $\backslash operatorname\{sinc\}$ is the unnormalized form of the sinc function.

For $\backslash operatorname\{rect\}\; (x/a)$, its Fourier transform is$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{rect\}\backslash left(\backslash frac\{t\}\{a\}\backslash right)\backslash cdot\; e^\{-i\; 2\backslash pi\; f\; t\}\; \backslash ,\; dt$$

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

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

$$\backslash operatorname\{tri(t/T)\}\; =\; \backslash operatorname\{rect(2t/T)\}\; *\; \backslash operatorname\{rect(2t/T)\}.\backslash ,$$

{{Main |Uniform distribution (continuous)}}

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

$$\backslash varphi(k)\; =\; \backslash frac\{\backslash sin(k/2)\}\{k/2\},$$

and its moment-generating function is

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

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

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

$$\backslash Pi(t)\; =\; \backslash lim\_\{n\backslash rightarrow\; \backslash infty,\; n\backslash in\; \backslash mathbb(Z)\}\; \backslash frac\{1\}\{(2t)^\{2n\}+1\}.$$

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

It follows that:

Second, we consider the case where $|t|>\backslash frac\{1\}\{2\}.$ 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:

$$\backslash lim\_\{n\backslash rightarrow\; \backslash infty,\; n\backslash in\; \backslash mathbb(Z)\}\; \backslash frac\{1\}\{(2t)^\{2n\}+1\}\; =\; \backslash frac\{1\}\{+\backslash infty+1\}\; =\; 0,\; |t|>\backslash tfrac\{1\}\{2\}.$$

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

$$\backslash lim\_\{n\backslash rightarrow\; \backslash infty,\; n\backslash in\; \backslash mathbb(Z)\}\; \backslash frac\{1\}\{(2t)^\{2n\}+1\}\; =\; \backslash lim\_\{n\backslash rightarrow\; \backslash infty,\; n\backslash in\; \backslash mathbb(Z)\}\; \backslash frac\{1\}\{1^\{2n\}+1\}\; =\; \backslash frac\{1\}\{1+1\}\; =\; \backslash tfrac\{1\}\{2\}.$$

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

$$\backslash operatorname\{rect\}(t)\; =\; \backslash Pi(t)\; =\; \backslash lim\_\{n\backslash rightarrow\; \backslash infty,\; n\backslash in\; \backslash mathbb(Z)\}\; \backslash frac\{1\}\{(2t)^\{2n\}+1\}\; =\; \backslash begin\{cases\}$$

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 $\backslash delta\; (x)$.{{Cite book |last1=Khare |first1=Kedar |title=Fourier Optics and Computational Imaging |last2=Butola |first2=Mansi |last3=Rajora |first3=Sunaina |publisher=Springer |year=2023 |isbn=978-3-031-18353-9 |edition=2nd |pages=15–16 |chapter=Chapter 2.4 Sampling by Averaging, Distributions and Delta Function |doi=10.1007/978-3-031-18353-9}} Specifically,$$\backslash delta\; (x)\; =\; \backslash lim\_\{a\; \backslash to\; 0\}\; \backslash frac\{1\}\{a\}\backslash operatorname\{rect\}\backslash left(\backslash frac\{x\}\{a\}\backslash right).$$For a function $g(x)$, its average over the width $a$ around 0 in the function domain is calculated as,

$$g\_\{avg\}(0)\; =\; \backslash frac\{1\}\{a\}\; \backslash int\backslash limits\_\{-\; \backslash infty\}^\{\backslash infty\}\; dx\backslash \; g(x)\; \backslash operatorname\{rect\}\backslash left(\backslash frac\{x\}\{a\}\backslash right).$$

To obtain $g(0)$, the following limit is applied,

$$g(0)\; =\; \backslash lim\_\{a\; \backslash to\; 0\}\; \backslash frac\{1\}\{a\}\; \backslash int\backslash limits\_\{-\; \backslash infty\}^\{\backslash infty\}\; dx\backslash \; g(x)\; \backslash operatorname\{rect\}\backslash left(\backslash frac\{x\}\{a\}\backslash right)$$

and this can be written in terms of the Dirac delta function as,

$$g(0)\; =\; \backslash int\backslash limits\_\{-\; \backslash infty\}^\{\backslash infty\}\; dx\backslash \; g(x)\; \backslash delta\; (x).$$The Fourier transform of the Dirac delta function $\backslash delta\; (t)$ is

$$\backslash delta\; (f)$$

= \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt

= \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \operatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\pi f t} \, dt

= \lim_{a \to 0} \operatorname{sinc}{(a f)}.

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 $\backslash delta\; (t)$ is

$$\backslash delta\; (f)\; =\; 1,$$

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

{{Reflist}}

{{DEFAULTSORT:Rectangular Function}}

Category:Special functions