# Triangle wave

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Short description: Non-sinusoidal waveform

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.

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

### Definition

A triangle wave of period p that spans the range [0,1] is defined as: $\displaystyle{ x(t)= 2 \left| \frac{t}{p} - \left \lfloor \frac{t}{p} + \frac{1}{2} \right \rfloor \right| }$ where $\displaystyle{ \lfloor\,\ \rfloor }$ is the floor function. This can be seen to be the absolute value of a shifted sawtooth wave.

For a triangle wave spanning the range [−1,1] the expression becomes: $\displaystyle{ x(t)= 2 \left | 2 \left ( \frac{t}{p} - \left \lfloor {t \over p} + {1 \over 2} \right \rfloor \right) \right | - 1. }$

A more general equation for a triangle wave with amplitude $\displaystyle{ a }$ and period $\displaystyle{ p }$ using the modulo operation and absolute value is:

Triangle wave with amplitude=5, period=4

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

For example, for a triangle wave with amplitude 5 and period 4: $\displaystyle{ y(x) = 5 \left | \bigl( (x - 1) \bmod 4 \bigr) - 2 \right | - 5. }$

A phase shift can be obtained by altering the value of the $\displaystyle{ - p/4 }$ term, and the vertical offset can be adjusted by altering the value of the $\displaystyle{ - a }$ term.

As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.

Note that in many programming languages, the % operator is a remainder operator (with result the same sign as the dividend), not a modulo operator; the modulo operation can be obtained by using ((x % p) + p) % p in place of x % p. In e.g. JavaScript, this results in an equation of the form 4*a/p * Math.abs((((x-p/4)%p)+p)%p - p/2) - a.

### Relation to the square wave

The triangle wave can also be expressed as the integral of the square wave: $\displaystyle{ x(t) = \int_0^t \sgn\left(\sin\frac{u}{p}\right)\,du. }$

### Expression in 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): $\displaystyle{ y(x) = \frac{2a}{\pi} \arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right). }$ The identity $\displaystyle{ \cos{x} = \sin\left(\frac{p}{4}-x\right) }$ can be used to convert from a triangle "sine" wave to a triangular "cosine" wave. This phase-shifted triangle wave can also be expressed with cosine and arccosine: $\displaystyle{ y(x) = a - \frac{2a}{\pi} \arccos\left(\cos\left(\frac{2\pi}{p}x\right)\right). }$

### Expressed as alternating linear functions

Another definition of the triangle wave, with range from −1 to 1 and period p, is: $\displaystyle{ x(t) = \frac{4}{p} \left (t-\frac{p}{2} \left \lfloor\frac{2 t}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 t}{p} + \frac{1}{2} \right \rfloor }$

### 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{align} x_\mathrm{triangle}(t) & {} = \frac8{\pi^2}\sum_{i=0}^{N - 1} (-1)^i n^{-2} \sin\left(2\pi f_0 n t\right) \end{align} } 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 quickly to the triangle wave as N tends to infinity, as shown in the animation.

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