Triangular function

A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an

Definitions

The most common definition is as a piecewise function:

${\displaystyle {\begin{aligned}\operatorname {tri} (x)=\Lambda (x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-|x|,0{\big )}\\&={\begin{cases}1-|x|,&|x|<1;\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}}$

Equivalently, it may be defined as the convolution of two identical unit rectangular functions:

${\displaystyle {\begin{aligned}\operatorname {tri} (x)&=\operatorname {rect} (x)*\operatorname {rect} (x)\\&=\int _{-\infty }^{\infty }\operatorname {rect} (x-\tau )\cdot \operatorname {rect} (\tau )\,d\tau .\\\end{aligned}}}$

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

${\displaystyle \operatorname {tri} (x)=\operatorname {rect} (x/2){\big (}1-|x|{\big )}.}$

Note that some authors instead define the triangle function to have a base of width 1 instead of width 2:

${\displaystyle {\begin{aligned}\operatorname {tri} (2x)=\Lambda (2x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-2|x|,0{\big )}\\&={\begin{cases}1-2|x|,&|x|<{\tfrac {1}{2}};\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}}$

In its most general form a triangular function is any linear B-spline:^[1]

${\displaystyle \operatorname {tri} _{j}(x)={\begin{cases}(x-x_{j-1})/(x_{j}-x_{j-1}),&x_{j-1}\leq x<x_{j};\\(x_{j+1}-x)/(x_{j+1}-x_{j}),&x_{j}\leq x<x_{j+1};\\0&{\text{otherwise}}.\end{cases}}}$

Whereas the definition at the top is a special case

${\displaystyle \Lambda (x)=\operatorname {tri} _{j}(x),}$

where ${\displaystyle x_{j-1}=-1}$, ${\displaystyle x_{j}=0}$, and ${\displaystyle x_{j+1}=1}$.

A linear B-spline is the same as a continuous piecewise linear function ${\displaystyle f(x)}$, and this general triangle function is useful to formally define ${\displaystyle f(x)}$ as

${\displaystyle f(x)=\sum _{j}y_{j}\cdot \operatorname {tri} _{j}(x),}$

where ${\displaystyle x_{j}<x_{j+1}}$ for all integer ${\displaystyle j}$. The piecewise linear function passes through every point expressed as coordinates with ordered pair ${\displaystyle (x_{j},y_{j})}$, that is,

${\displaystyle f(x_{j})=y_{j}}$.

Scaling

For any parameter ${\displaystyle a\neq 0}$:

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

Fourier transform

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

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

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

See also

• Källén function, also known as triangle function
• Tent map
• Triangular distribution
• Triangle wave, a piecewise linear periodic function
• Trigonometric functions

References