Gabriel Lamé

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

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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, due to its 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), 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:

$\begin{array}{rl}{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}(t)& =\frac{8}{{\pi }^2}\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k\frac{\sin ((2k+1)t)}{(2k+1{)}^2}\\ & =\frac{8}{{\pi }^2}(\sin (t)-\frac{1}{9}\sin (3t)+\frac{1}{25}\sin (5t)-\cdots )\end{array}$

Definitions

Sine, square, triangle, and sawtooth waveforms

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

$x(t)=\frac{2}{a}(t-a\lfloor \frac{t}{a}+\frac{1}{2}\rfloor )(-1{)}^{\lfloor \frac{t}{a}+\frac{1}{2}\rfloor }$

where the symbol $\lfloor n\rfloor$ represent the floor function of n.

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

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

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

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

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

$\int \mathrm{sgn}(\sin (x))dx$

A simple equation with a period of 4, with $y(0)=1$. 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:

$y(x)=|xmod4-2|-1$

or, a more complex and complete version of the above equation with a period of 2π and starting with $y(0)=0$:

$y(x)=|4\left((\frac{x}{2\pi }-0.25)mod1\right)-2|-1$

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

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

See also

• List of periodic functions
• Triangle function
• Waves
• Sound
• Zigzag

References

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