Triangular function

Definitions

The most common definition is as a piecewise function: \begin{align} \operatorname{tri}(x) = \Lambda(x) \ &\overset{\underset{\text{def}}{}}{=} \ \max\big(1 - |x|, 0\big) \ &= \begin{cases} 1 - |x|, & |x| 0 & \text{otherwise}. \ \end{cases} \end{align}

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

\begin{align} \operatorname{tri}(x) &= \operatorname{rect}(x) * \operatorname{rect}(x) \ &= \int_{-\infty}^\infty \operatorname{rect}(x - \tau) \cdot \operatorname{rect}(\tau) ,d\tau. \ \end{align}

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

\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:

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

In its most general form a triangular function is any linear B-spline:

\operatorname{tri}j(x) = \begin{cases} (x - x{j-1})/(x_j - x_{j-1}), & x_{j-1} \le x (x_{j+1} - x)/(x_{j+1} - x_j), & x_j \le x 0 & \text{otherwise}. \end{cases}

Whereas the definition at the top is a special case

\Lambda(x) = \operatorname{tri}_j(x),

where x_{j-1} = -1, x_j = 0, and x_{j+1} = 1.

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

f(x) = \sum_j y_j \cdot \operatorname{tri}_j(x),

where x_j for all integer j. The piecewise linear function passes through every point expressed as coordinates with ordered pair (x_j, y_j), that is,

f(x_j) = y_j.

Scaling

For any parameter a \ne 0:

\begin{align} \operatorname{tri}\left(\tfrac{t}{a}\right) &= \tfrac{1}{\sqrt{a}} \operatorname{rect}\left(\tfrac{t}{a}\right)* \tfrac{1}{\sqrt{a}} \operatorname{rect}\left(\tfrac{t}{a}\right) \[1ex] &= \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| 0 & \text{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} where \operatorname{sinc}(x) = \sin(\pi x) / (\pi x) is the normalized sinc function.

For the general form, we have:

\begin{align} \mathcal{F}{\operatorname{tri}\left(\tfrac{t}{a}\right)} &= \mathcal{F}\left{\tfrac{1}{\sqrt{a}} \operatorname{rect}\left(\tfrac{t}{a}\right) * \tfrac{1}{\sqrt{a}}\operatorname{rect}\left(\tfrac{t}{a}\right)\right} \ &= \tfrac{1}{a} \ \mathcal{F}\left{\operatorname{rect}\left(\tfrac{t}{a}\right)\right} \cdot \mathcal{F}\left{\operatorname{rect}\left(\tfrac{t}{a}\right)\right}\ &= \tfrac{1}{a} \ \mathcal{F}\left{\operatorname{rect}\left(\tfrac{t}{a}\right)\right}^2 \ &= \tfrac{1}{a} \ {a}^2 \ \mathrm{sinc}^2(a \cdot f) = {a} \ \mathrm{sinc}^2(a \cdot f). \end{align}

References

References

1. "INF-MAT5340 Lecture Notes".