Triangular function

Triangular function

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

: $egin{align}operatorname{tri}(t) = and (t) quad &overset{underset{mathrm{def{{=} max(1 - |t|, 0) \&= egin{cases}1 - |t|, & |t| < 1 \0, & mbox{otherwise} end{cases}end{align}$

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

: $egin{align}operatorname{tri}(t) = operatorname{rect}(t) * operatorname{rect}(t) quad&overset{underset{mathrm{def{{=} int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}(t- au) d au\&= int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}( au-t) d au .end{align}$

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.

Scaling

For any parameter, $a
e 0,$ :

: $egin{align}operatorname{tri}(t/a) &= int_{-infty}^infty mathrm{rect}( au) cdot mathrm{rect}( au - t/a) d au \&= egin{cases}1 - |t/a|, & |t| < |a| \0, & mbox{otherwise} .end{cases}end{align}$

Fourier transform

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

: $egin{align}mathcal{F}{operatorname{tri}(t)} &= mathcal{F}{operatorname{rect}(t) * operatorname{rect}(t)}\&= mathcal{F}{operatorname{rect}(t)}cdot mathcal{F}{operatorname{rect}(t)}\&= mathcal{F}{operatorname{rect}(t)}^2\&= mathrm{sinc}^2(f) .end{align}$

ee also

*Tent map

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