Triangular function

The triangular function, also tri - function triangle - function or tent function is a mathematical function with the following definition:

It can be defined equivalent to using the convolution with the rectangular function rect, as it is clearly shown in the adjacent diagram:

By a parameter a ≠ 0, the triangular function can be scaled:

The delta functions is found mainly in the area of ​​signal processing for the representation of idealized waveforms application. It is used, alongside a Gaussian function, the Heaviside function and the rectangular function for the description of elementary signals. Technical applications are in the range of matched filters or window functions such as the Bartlett window.

The Fourier transform of the delta function gives the squared sinc function:

General form

In general, one would like to scale the triangle function. Of interest here are the stretching in the x- direction, and the height at the top. For the stretch T is half the period, so the distance from the beginning of the triangular function to the center T0. The height at the point t0 by

Given.

Derivation

The derivation of the delta function is a sum of two rectangle functions rect represents:

Which can be represented ε as the sum of three step functions:

Where 2T representing the period of time t0 the center and a is the height of the triangular function. Therefore, the pre-factor occurs as the slope of the triangular function in the derivative.

Source

  • Hans Dieter Lüke: signal transmission. 6th edition. Springer Verlag, 1995, ISBN 3-540-54824-6.
  • Mathematical function
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