# Rectangular function

Rectangular function

The rectangular function , also called rect function, is a discontinuous mathematical function with the following definition:

${\ displaystyle \ operatorname {rect} (t) = \ sqcap (t) = {\ begin {cases} 0 & {\ text {if}} | t |> {\ frac {1} {2}} \\ [3pt ] {\ frac {1} {2}} & {\ mbox {if}} | t | = {\ frac {1} {2}} \\ [3pt] 1 & {\ text {if}} | t | < {\ frac {1} {2}}. \ end {cases}}}$

Alternative definitions, which are particularly common in the field of signal processing , define the rectangular function in simplified form as:

${\ displaystyle \ operatorname {rect_ {d}} (t) = {\ begin {cases} 1 & {\ text {if}} | t | \ leq {\ frac {1} {2}} \\ [3pt] 0 & {\ text {if}} | t |> {\ frac {1} {2}}. \ end {cases}}}$

## General

The rectangle function can also be expressed using the Heaviside function as: ${\ displaystyle \ Theta (x)}$

${\ displaystyle \ operatorname {rect} (t) = \ Theta \ left (t + {\ frac {1} {2}} \ right) \ cdot \ Theta \ left ({\ frac {1} {2}} - t \ right) = \ Theta \ left (t + {\ frac {1} {2}} \ right) - \ Theta \ left (t - {\ frac {1} {2}} \ right)}$.

It is set. ${\ displaystyle \ Theta (0) = {\ tfrac {1} {2}}}$

The Fourier transformation of the rectangular function results in the sinc function : ${\ displaystyle \ operatorname {sinc} (x) = \ sin (\ pi x) / (\ pi x)}$

${\ displaystyle {\ mathcal {F}} \ {\ operatorname {rect} (t) \} = \ operatorname {sinc} (f)}$

This also applies to . The opposite is true ${\ displaystyle \ operatorname {rect_ {d}} (t)}$

${\ displaystyle {\ mathcal {F}} \ {\ operatorname {sinc} (t) \} = \ operatorname {rect} (f)}$.

Here it is important to use the first definition of the rectangular function, for the last equation is wrong. ${\ displaystyle \ operatorname {rect_ {d}}}$

## Shifting and scaling

A rectangular function centered at and having a duration of is expressed by ${\ displaystyle t_ {0}}$${\ displaystyle T}$

${\ displaystyle \ operatorname {rect} \ left ({\ frac {t-t_ {0}} {T}} \ right) \ ,.}$

## Derivation

As a discontinuous function, the rectangular function is neither differentiable in the classical sense nor is it weakly differentiable . However, it is possible to derive a distribution through the Dirac delta distribution : ${\ displaystyle \ delta}$

${\ displaystyle \ operatorname {rect} '(t) = \ delta \ left (t + {\ frac {1} {2}} \ right) - \ delta \ left (t - {\ frac {1} {2}} \ right)}$

## Other connections

The convolution of two equal rectangular functions results in the triangle function , the integration a ramp function . The Rademacher functions are a form with a periodic continuation of the rectangular function .

The multiple folding with folds ${\ displaystyle m}$

${\ displaystyle \ operatorname {rect} (t) * \ operatorname {rect} (t) * \ operatorname {rect} (t) * \ ldots}$

with a suitable scaling results in the Gaussian bell curve . ${\ displaystyle m \ to \ infty}$