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Triangle wave

## Summary

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.

Triangle wave
A bandlimited triangle wave[1] pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
General information
General definition${\displaystyle x(t)=4\left\vert t-\left\lfloor t+3/4\right\rfloor +1/4\right\vert -1}$
Fields of applicationElectronics, synthesizers
Domain, codomain and image
Domain${\displaystyle \mathbb {R} }$
Codomain${\displaystyle \left[-1,1\right]}$
Basic features
ParityOdd
Period1
Specific features
Root${\displaystyle \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }$
DerivativeSquare wave
Fourier series${\displaystyle x(t)=-{\frac {8}{{\pi }^{2}}}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{\left(2k-1\right)^{2}}}\sin \left(2\pi \left(2k-1\right)t\right)}$

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

### 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

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

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

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

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:

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

### 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 ${\textstyle \cos {x}=\sin \left({\frac {p}{4}}-x\right)}$  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

${\displaystyle x(t)={\frac {4}{p}}\left(t-{\frac {p}{2}}\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor }.}$

### Harmonics

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:

${\displaystyle x_{\text{triangle}}(t)={\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin(2\pi f_{0}nt),}$

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

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