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[[Image:Triangular function.svg|thumb|right|Triangular function]]
[[File:Triangle function.png|thumb|right|Triangle function (alternate)]]

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

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

or, equivalently, as the [[convolution]] of two identical unit [[rectangular function]]s''':'''

:<math>
\begin{align}
\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{align}
</math>

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

:<math> \operatorname{tri}(t) = \operatorname{rect}(t/2) \left ( 1 - \left |t \right | \right ) </math>

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 signal]]s and as a [[matched filter]] for receiving the signals.  It is also equivalent to the '''triangular window''' sometimes called the [[Bartlett window]].

Note that in some cases the triangle function may be defined to have a base of length <math>t/2</math> instead of <math>t</math>:

<math>
\begin{align}
\operatorname{tri}(t) = \and (t) \quad 
&\overset{\underset{\mathrm{def}}{}}{=} \ \max(1 - |2t|, 0) \\
&= 
\begin{cases}
1 - |2t|, & |2t| < 1 \\
0, & \mbox{otherwise} 
\end{cases}
\end{align}
</math>

==Scaling==
For any parameter, <math>a \ne 0\,</math>&nbsp;''':'''

:<math>
\begin{align}
\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{align}
</math>

==Fourier transform==
The transform is easily determined using the [[Fourier_transform#Convolution_theorem|convolution property of Fourier transforms]] and the [[Fourier_transform#Square-integrable_functions|Fourier transform of the rectangular function]]''':'''

:<math>
\begin{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}
</math>

==See also==
*[[Tent map]]
*[[Triangular distribution]]
*[[Triangle wave]], a piecewise linear periodic function

[[Category:Special functions]]

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