Rectangular function: Difference between revisions

Revision as of 15:55, 2 July 2005

Revision as of 07:16, 13 December 2005

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\end{matrix} \right.</math>

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

:<math>\mathbf{rect}(t) = u(t + \frac{1/}{2}) - u(t - \frac{1/}{2})</math>

or, alternatively:

:<math>\mathbf{rect}(t) = u(t + \frac{1}{2}) u(\frac{1}{2} - t) </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^{-ikxi \omega x}\,dx

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

where "sinc" is the [[sinc function]]. Viewing the rectangular function as a [[probability distribution]] function, its [[characteristic function]] is therefore written