Rectangular function: Difference between revisions

Revision as of 20:00, 3 July 2006

Revision as of 18:40, 5 August 2006

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[[Image:Rectangular function.svg|thumb|right|Rectangular function]]

The '''rectangular function''' (also known as the '''rectangle function''', '''rect function''' or the normalized '''[[boxcar function]]''') is defined as,

:<math>\mathrm{rect}(xt) = \sqcap(xt) = \begin{cases}

0 & \mbox{if } |xt| > \frac{1}{2} \\[3pt]

\frac{1}{2} & \mbox{if } |xt| = \frac{1}{2} \\[3pt]

1 & \mbox{if } |xt| < \frac{1}{2}

\end{cases} </math>

Alternate definitions of the function define <math>\mathrm{rect}(\pm 1/2)</math> to be 0, 1, or undefined. We can also express the rectangular function in terms of the [[Heaviside step function]], ''<math>u(t)''</math>:

:<math>\mathrm{rect}(xt) = u \left( xt + \frac{1}{2} \right) - u \left( xt - \frac{1}{2} \right) </math>

or, alternatively:

:<math>\mathrm{rect}(xt) = u \left( xt + \frac{1}{2} \right) \cdot u \left( \frac{1}{2} - xt \right) </math>

The rectangular function is normalized:

:<math>\int_{-\infty}^\infty \mathrm{rect}(xt)\,dxdt=1</math>

The [[Continuous_Fourier_transform#Table_of_important_Fourier_transforms|unitary Fourier transforms]] of the rectangular function are''':''',

:<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \mathrm{rect}(t)\cdot e^{-i \omega t} \, dt

=\frac{1}{\sqrt{2\pi}}\cdot \mathrm{sinc}\left(\frac{\omega}{2\pi}\right)</math>,

and, in terms of the normalized [[sinc function]]. ,

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= \mathrm{sinc}(f)</math>

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

:<math>\mathrm{tri}(t) = \mathrm{rect}(t) * \mathrm{rect}(t) </math>

Viewing the rectangular function as a [[probability distribution]] function, its [[characteristic function (probability theory)|characteristic function]] is therefore written,

:<math>\varphi(k) = \frac{\sin(k/2)}{k/2}\,</math>

and its [[moment generating function]] is:,

:<math>M(k)=\frac{\mathrm{sinh}(k/2)}{k/2}\,</math>

where "<math>\mathrm{sinh"}(t)</math> is the [[hyperbolic sine]] function.

==See also==