# Triangle wave

A bandlimited triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).

A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics, demonstrating odd symmetry. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

## Harmonics

Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See Fourier Analysis for a mathematical description.

It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4n−1)th harmonic by −1 (or changing its phase by π)[citation needed], and rolling off the harmonics by the inverse square of their relative frequency to the fundamental.

This infinite Fourier series converges to the triangle wave with cycle frequency f over time t:

{\displaystyle {\begin{aligned}x_{\mathrm {triangle} }(t)&{}={\frac {8}{\pi ^{2}}}\sum _{k=0}^{\infty }(-1)^{k}\,{\frac {\sin \left(2\pi (2k+1)ft\right)}{(2k+1)^{2}}}\\&{}={\frac {8}{\pi ^{2}}}\left(\sin(2\pi ft)-{1 \over 9}\sin(6\pi ft)+{1 \over 25}\sin(10\pi ft)-\cdots \right)\end{aligned}}}

## Definitions

Sine, square, triangle, and sawtooth waveforms

Another definition of the triangle wave, with range from -1 to 1 and period 2a is:

${\displaystyle x(t)={\frac {2}{a}}\left(t-a\left\lfloor {\frac {t}{a}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {t}{a}}+{\frac {1}{2}}\right\rfloor }}$
where the symbol ${\displaystyle \scriptstyle \lfloor n\rfloor }$ represent the floor function of n.

Also, the triangle wave can be the absolute value of the sawtooth wave:

${\displaystyle x(t)=\left|2\left({t \over a}-\left\lfloor {t \over a}+{1 \over 2}\right\rfloor \right)\right|}$

or, for a range from -1 to +1:

${\displaystyle x(t)=2\left|2\left({t \over a}-\left\lfloor {t \over a}+{1 \over 2}\right\rfloor \right)\right|-1}$

The triangle wave can also be expressed as the integral of the square wave:

${\displaystyle \int \operatorname {sgn}(\sin(x))\,dx\,}$

A simple equation with a period of 4, with ${\displaystyle y(0)=1}$. As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power:

${\displaystyle y(x)=|x\,{\bmod {\,}}4-2|-1}$

or, a more complex and complete version of the above equation with a period of ${\displaystyle p}$, amplitude ${\displaystyle a}$, and starting with ${\displaystyle y(0)=a/2}$:

${\displaystyle y(x)={\frac {2a}{p}}{\Biggl (}{\biggl |}\left(x{\bmod {p}}\right)-{\frac {p}{2}}{\biggr |}-{\frac {p}{4}}{\Biggr )}}$

The function (1) is a specialization of (2), with a=2 and p=4:

${\displaystyle y(x)={\frac {2\times 2}{4}}{\Biggl (}{\biggl |}\left(x{\bmod {4}}\right)-{\frac {4}{2}}{\biggr |}-{\frac {4}{4}}{\Biggr )}\Leftrightarrow }$
${\displaystyle y(x)={\Biggl (}{\biggl |}\left(x{\bmod {4}}\right)-2{\biggr |}-1{\Biggr )}}$

An odd version of the function (1) can be made, just shifting by one the input value, which will change the phase of the original function:

${\displaystyle y(x)=|(x-1)\,{\bmod {\,}}4-2|-1}$

Generalizing the formula (3) to make the function odd for any period and amplitude gives:

${\displaystyle y(x)={\frac {4a}{p}}{\Biggl (}{\biggl |}\left((x-{\frac {p}{4}}){\bmod {p}}\right)-{\frac {p}{2}}{\biggr |}-{\frac {p}{4}}{\Biggr )}}$

In terms of sine and arcsine with period p and amplitude a:

${\displaystyle y(x)={\frac {2a}{\pi }}\arcsin \left(\sin \left({\frac {2\pi }{p}}x\right)\right)}$

Note: sin y = cos x

## Arc Length

The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p":

${\displaystyle s={\sqrt {(4a)^{2}+p^{2}}}}$