Triangle Function — Definition, Formula & Examples

The triangle function (also called the triangular function) is a piecewise linear function that rises linearly from 0 to 1, then falls linearly back to 0, forming a triangle shape. It appears frequently in signal processing and Fourier analysis as a basic waveform.

The triangle function $\operatorname{tri}(t)$ is defined as $\operatorname{tri}(t) = \max(1 - |t|,\, 0)$ , which equals $1 - |t|$ for $|t| \leq 1$ and $0$ otherwise. Equivalently, it is the convolution of the rectangular function with itself.

Key Formula

\operatorname{tri}(t) = \begin{cases} 1 - |t|, & |t| \leq 1 \\ 0, & |t| > 1 \end{cases}

Where:

$t$ = The input variable (often representing time or a normalized coordinate)
$|t|$ = The absolute value of t, ensuring symmetry about the origin

How It Works

The triangle function peaks at

t = 0

with a value of 1 and decreases linearly to 0 at

t = -1

and

t = 1

. Outside the interval

[-1, 1]

, it is identically zero. You can scale and shift it:

\operatorname{tri}\!\left(\frac{t - t_0}{\tau}\right)

centers the peak at

t_0

and stretches the base to width

2\tau

. In Fourier analysis, the triangle function's transform is

\operatorname{sinc}^2(f)

, making it a natural window function for smoothing spectral data.

Worked Example

Problem: Evaluate tri(t) at t = −0.5, t = 0, and t = 1.5.

At t = −0.5: Since |−0.5| = 0.5 ≤ 1, use the formula 1 − |t|.

\operatorname{tri}(-0.5) = 1 - 0.5 = 0.5

At t = 0: Since |0| = 0 ≤ 1, substitute directly.

\operatorname{tri}(0) = 1 - 0 = 1

At t = 1.5: Since |1.5| = 1.5 > 1, the function equals zero.

\operatorname{tri}(1.5) = 0

Answer: tri(−0.5) = 0.5, tri(0) = 1, and tri(1.5) = 0.

Why It Matters

The triangle function serves as a standard window and interpolation kernel in digital signal processing courses. Engineers use it when designing filters, performing linear interpolation between sampled data points, and analyzing bandwidth through its Fourier transform

\operatorname{sinc}^2(f)

Common Mistakes

Mistake: Confusing the triangle function with a periodic triangle wave.

Correction: The triangle function tri(t) is a single, finite-duration pulse defined on [−1, 1]. A triangle wave is a periodic repetition of triangular segments extending over all t. They are related but distinct objects.