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Concept
Fonction porte
Résumé
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
Alternative definitions of the function define to be 0, 1, or undefined.
Its periodic version is called a rectangular wave.
The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively.
The rectangular function is a special case of the more general boxcar function:
where is the Heaviside step function; the function is centered at and has duration , from to
The unitary Fourier transforms of the rectangular function are
using ordinary frequency f, where is the normalized form of the sinc function and
using angular frequency , where is the unnormalized form of the sinc function.
For , its Fourier transform isNote that as long as the definition of the pulse function is only motivated by its behavior in the time-domain experience, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans. However, some aspects of the theoretical result may be understood intuitively, as finiteness in time domain corresponds to an infinite frequency response. (Vice versa, a finite Fourier transform will correspond to infinite time domain response.)
We can define the triangular function as the convolution of two rectangular functions:
Uniform distribution (continuous)
Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with The characteristic function is
and its moment-generating function is
where is the hyperbolic sine function.
Source officielle
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