# Triangle wave

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

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

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).

## Definitions

Sine, square, triangle, and sawtooth waveforms

### Trigonometric functions

A triangle wave with period p and amplitude a can be expressed in terms of sine and arcsine (whose value ranges from -π/2 to π/2):

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

### 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 π) 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)&{}={\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin \left(2\pi f_{0}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), ${\displaystyle f_{0}}$  is the fundamental frequency, 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.

### Floor function

Another definition of the triangle wave, with range from -1 to 1 and period 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 ${\displaystyle \lfloor \,\ \rfloor }$  is the floor function.

### Sawtooth wave

Also, the triangle wave is 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:

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

### Square wave

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

${\displaystyle x(t)=\int _{0}^{t}\operatorname {sgn}(\sin(u))\,du.}$

### Modulo operation

Here is a simple equation with a period of 4 and initial value ${\displaystyle y(0)=1}$ :

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

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. The previous equation can be generalized for a period of ${\displaystyle p,}$  amplitude ${\displaystyle a,}$  and initial value ${\displaystyle y(0)=a/2}$ :

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

The former function is a specialization of the latter for a = 2 and p = 4:

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

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:

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

## Arc length

The arc length per period for a triangle wave, denoted by s, is given in terms of the amplitude a and period length p by

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