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

- $$

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}

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

- $$

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) dtau
&= int_{-infty}^infty mathrm{rect}(tau) cdot mathrm{rect}(tau-t) dtau .
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, $a\; ne\; 0,$ :

- $$

begin{align}
operatorname{tri}(t/a) &= int_{-infty}^infty mathrm{rect}(tau) cdot mathrm{rect}(tau - t/a) dtau
&=
begin{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:

- $$

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}

## See also