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

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

$$\Pi(x) = H(x+{\textstyle{1\over 2}})-H(x-{\textstyle{1\over 2}})$$ (2)

$$= H({\textstyle{1\over 2}}+x)+H({\textstyle{1\over 2}}-x)-1$$ (3)

$$= H({\textstyle{1\over 4}}-x^2)$$ (4)

$$= {\textstyle{1\over 2}}[\mathop{\rm sgn}\nolimits (x+{\textstyle{1\over 2}})-\mathop{\rm sgn}\nolimits (x-{\textstyle{1\over 2}})],$$ (5)

where $H(x)$ is the Heaviside Step Function. The Fourier Transform of the rectangle function is given by

$${\mathcal F}[\Pi(x)]=\int_{-\infty}^\infty e^{-2\pi ikx}\Pi(x)\,dx=\mathop{\rm sinc}\nolimits (\pi k),$$ (6)

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

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