Triangular function

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Triangular function

The triangular function (also known as the triangle function, hat function, or tent function) is defined either as:

or, equivalently, as the convolution of two identical unit rectangular functions:

The triangular function can also be represented as the product of the rectangular and absolute value functions:

The function is useful in signal processing and communication systems engineering as a representation of an idealized signal, and as a prototype or kernel from which more realistic signals can be derived. It also has applications in pulse code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also equivalent to the triangular window sometimes called the Bartlett window.


For any parameter,  :

Fourier transform[edit]

The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function:

where is the normalized sinc function.

Extended version to repeat tent in all R domain[edit]

|(x mod 2)-1|

Alternative definition[edit]

Triangle function (alternate)

Note that in some cases the triangle function may be defined to have a base of length 1 instead of length 2:

See also[edit]