Rectangular function

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Rectangular function

The rectangular function , also called rect function, is a discontinuous mathematical function with the following definition:

Alternative definitions, which are particularly common in the field of signal processing , define the rectangular function in simplified form as:


The rectangle function can also be expressed using the Heaviside function as:


It is set.

The Fourier transformation of the rectangular function results in the sinc function :

This also applies to . The opposite is true


Here it is important to use the first definition of the rectangular function, for the last equation is wrong.

Shifting and scaling

A rectangular function centered at and having a duration of is expressed by


As a discontinuous function, the rectangular function is neither differentiable in the classical sense nor is it weakly differentiable . However, it is possible to derive a distribution through the Dirac delta distribution :

Other connections

The convolution of two equal rectangular functions results in the triangle function , the integration a ramp function . The Rademacher functions are a form with a periodic continuation of the rectangular function .

The multiple folding with folds

with a suitable scaling results in the Gaussian bell curve .

See also

Individual evidence

  1. ^ Hans Dieter Lüke: Signal transmission. Basics of digital and analog communication systems . 6th, revised and expanded edition. Springer, Berlin et al. 1995, ISBN 3-540-58753-5 , pp. 2 .