Triangular function

Triangular function

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

${\begin{aligned}\operatorname {tri} (t)=\land (t)\quad &{\overset {\underset {\mathrm {def} }{}}{=}}\ \max(1-|t|,0)\\&={\begin{cases}1-|t|,&|t|<1\\0,&{\mbox{otherwise}}\end{cases}}\end{aligned}}$

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

${\begin{aligned}\operatorname {tri} (t)=\operatorname {rect} (t)*\operatorname {rect} (t)\quad &{\overset {\underset {\mathrm {def} }{}}{=}}\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (t-\tau )\ d\tau \\&=\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (\tau -t)\ d\tau .\end{aligned}}$

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

$\operatorname {tri} (t)=\operatorname {rect} (t/2)\left(1-\left|t\right|\right)$

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, $a\neq 0\,$ :

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

Fourier transform

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

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

where $\operatorname {sinc} (x)=\sin(\pi x)/(\pi x)$ is the normalized sinc function.

Alternative definition

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:

${\begin{aligned}\operatorname {tri} (t)=\land (t)\quad &{\overset {\underset {\mathrm {def} }{}}{=}}\ \max(1-|2t|,0)\\&={\begin{cases}1-|2t|,&|2t|<1\\0,&{\mbox{otherwise}}\end{cases}}\end{aligned}}$

See also

• Tent map
• Triangular distribution
• Triangle wave, a piecewise linear periodic function