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Rectangle Function

The rectangle function $\Pi(x)$ is a function which is 0 outside the interval $[-1/2, 1/2]$ and unity inside it. It is also called the Gate Function, Pulse Function, or Window Function, and is defined by

\Pi(x) \equiv \cases{
0 & for $\vert x\vert > {\textstyle{1...
... 2}}$\cr
1 & for $\vert x\vert < {\textstyle{1\over 2}}$.\cr}
\end{displaymath} (1)

The function $f(x)=h\Pi((x-c)/b)$ has height $h$, center $c$, and full-width $b$. Identities satisfied by the rectangle function include
$\displaystyle \Pi(x)$ $\textstyle =$ $\displaystyle H(x+{\textstyle{1\over 2}})-H(x-{\textstyle{1\over 2}})$ (2)
  $\textstyle =$ $\displaystyle H({\textstyle{1\over 2}}+x)+H({\textstyle{1\over 2}}-x)-1$ (3)
  $\textstyle =$ $\displaystyle H({\textstyle{1\over 4}}-x^2)$ (4)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\mathop{\rm sgn}\nolimits (x+{\textstyle{1\over 2}})-\mathop{\rm sgn}\nolimits (x-{\textstyle{1\over 2}})],$ (5)

where $H(x)$ is the Heaviside Step Function. The Fourier Transform of the rectangle function is given by
{\mathcal F}[\Pi(x)]=\int_{-\infty}^\infty e^{-2\pi ikx}\Pi(x)\,dx=\mathop{\rm sinc}\nolimits (\pi k),
\end{displaymath} (6)

where $\mathop{\rm sinc}\nolimits (x)$ is the Sinc Function.

See also Fourier Transform--Rectangle Function, Heaviside Step Function, Ramp Function

© 1996-9 Eric W. Weisstein