Rectangular function: Difference between revisions

Revision as of 15:52, 4 May 2006

Revision as of 16:40, 12 May 2006

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

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

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

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

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

\end{cases} </math>

or in terms of the [[Heaviside step function]], ''u(t)'':

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

or, alternatively:

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

The rectangular function is normalized:

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The [[continuous Fourier transform|Fourier transform]] of the rectangular function is

:<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrm{rect}(x)e^{-i \omega xt} \, dxdt

=\frac{\textrm{sinc}(\omega/2)}{\sqrt{2\pi}}</math>