Triangular function

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The triangular function $\scriptstyle \mathrm {tri} (t),\,t\,\in \,\mathbb {R} \,$, (also known as the triangle function, hat function, or tent function) is defined as

$\mathrm {tri} (t)=\land (t)\equiv {\begin{cases}1-|t|,&|t|<1,\\0,&{\mbox{otherwise}}.\end{cases}}\,$

Equivalently

$\mathrm {tri} (t)=\max(1-|t|,\,0).\,$

Formulae

It is the convolution of two identical unit rectangular functions

$\mathrm {tri} (t)=\mathrm {rect} (t)*\mathrm {rect} (t)\equiv \int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (t-\tau )\ d\tau =\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (\tau -t)\ d\tau .\,$

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

$\mathrm {tri} (t)=\mathrm {rect} ({\tfrac {t}{2}})\left(1-\left|t\right|\right)\,$

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, $\scriptstyle a\,\neq \,0\,$

${\begin{aligned}\mathrm {tri} ({\tfrac {t}{a}})&=\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (\tau -{\tfrac {t}{a}})\ d\tau \\&={\begin{cases}1-|{\tfrac {t}{a}}|,&|t|<|a|,\\0,&{\mbox{otherwise}}.\end{cases}}\end{aligned}}\,$

Fourier transform

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

${\begin{aligned}{\mathcal {F}}\{\mathrm {tri} (t)\}&={\mathcal {F}}\{\mathrm {rect} (t)*\mathrm {rect} (t)\}\\&={\mathcal {F}}\{\mathrm {rect} (t)\}\cdot {\mathcal {F}}\{\mathrm {rect} (t)\}\\&={\mathcal {F}}\{\mathrm {rect} (t)\}^{2}\\&=\mathrm {sinc} ^{2}(f).\end{aligned}}\,$

See also

• Tent map
• Triangular distribution