Unit triangle function

The unit triangle function is a triangle of height 1 and base 2, centered at the origin:

$\mathrm{tri}(t)=\{\begin{array}{ll}1-|t|,& |t|<1\\ 0,& |t|\ge 1.\end{array}$

Image: Triangular function, CC BY-SA 3.0

It rises linearly from 0 at $t=-1$ to a peak of 1 at $t=0$, then falls linearly back to 0 at $t=+1$.

As a self-convolution

The most useful fact about the triangle is that it’s the convolution of a unit rectangle with itself:

$\mathrm{rect}(t)\ast \mathrm{rect}(t)=\mathrm{tri}(t).$

The derivation, by flip-and-slide: $\mathrm{rect}(\tau )$ is $1$ on $(-1/2,1/2)$, and $\mathrm{rect}(t-\tau )$ is $1$ on $(t-1/2,t+1/2)$. The integrand of the convolution is the indicator of the intersection of these intervals. The width of the intersection depends on $t$:

• $|t|\ge 1$: no overlap. $\mathrm{tri}(t)=0$.
• $0\le t<1$: overlap is $(t-1/2,1/2)$, width $1-t$.
• $-1<t\le 0$: overlap is $(-1/2,t+1/2)$, width $1+t=1-|t|$.

So $\mathrm{tri}(t)=1-|t|$ for $|t|<1$, zero outside. Two rectangles slid past each other smooth into a triangle.

Sanity check via the area property of convolution: area of $\mathrm{rect}$ is 1, so area of $\mathrm{rect}\ast \mathrm{rect}$ should be $1\times 1=1$. The triangle has base 2 and height 1, area $=(1/2)(2)(1)=1$. ✓

Fourier transform

By the convolution theorem, convolution in time becomes multiplication in frequency. So

$\mathrm{tri}(t)\stackrel{\mathcal{F}}{\leftrightarrow }{\mathrm{sinc}}^2(f).$

The triangle’s spectrum is the sinc squared. This pair is the second most-used in the course (after rect/sinc itself), and it shows up in Fourier series examples for periodic triangle waves, in pulse-shaping for digital communications, and in any context where two rectangular gates are cascaded or chained.

Scaling

The general scaled triangle $\mathrm{tri}(t/w)$ is a triangle of height 1 and base $2w$, centered at the origin. By the time-scaling property of the Fourier transform, its spectrum is $w{\mathrm{sinc}}^2(wf)$.