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. 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 summing odd harmonics of the fundamental while multiplying every other odd harmonic by -1 (or, equivalently, changing its phase by ?) and multiplying the amplitude of the harmonics by one over the square of their mode number, n, (which is equivalent to one over the square of their relative frequency to the fundamental).

The above can be summarised mathematically as follows:

{\displaystyle {\begin{aligned}x_{\mathrm {triangle} }(t)&{}=\sum _{i=0}^{N}(-1)^{i}n^{-2}\left(\sin[nt]\right)\end{aligned}}}

where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), and i is the harmonic label which is related to its mode number by ${\displaystyle n=2i+1}$.

This infinite Fourier series converges to the triangle wave as N tends to infinity, as shown in the animation.

## Definitions

Sine, square, triangle, and sawtooth waveforms

Another definition of the triangle wave, with range from -1 to 1 and period ${\displaystyle p,}$ is:

${\displaystyle x(t)={\frac {4}{p}}\left(t-{\frac {p}{2}}\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor }}$
where the symbol ${\displaystyle \lfloor n\rfloor }$ represent the floor function of n.

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

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

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

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

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, generalizing the above equation for 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 former function is a specialization of the latter for 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 first function 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 this 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:

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

## Arc Length

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

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

## References

• Weisstein, Eric W. "Fourier Series - Triangle Wave". MathWorld.

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