Triangle wave

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

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

Triangle wave with amplitude=5, period=4

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

For example, for a triangle wave with amplitude 5 and period 4: $$y(x)=5\left|\right((x-1)mod4)-2|-5.$$

A phase shift can be obtained by altering the value of the $-p/4$ term, and the vertical offset can be adjusted by altering the value of the $-a$ 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: $$x(t)=\underset{0}{\overset{t}{\int }}\mathrm{sgn}(\sin \frac{u}{p})du.$$

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): $$y(x)=\frac{2a}{\pi }\arcsin (\sin \left(\frac{2\pi }{p}x\right)).$$ The identity $\cos x=\sin (\frac{p}{4}-x)$ 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: $$y(x)=a-\frac{2a}{\pi }\arccos (\cos \left(\frac{2\pi }{p}x\right)).$$

Expressed as alternating linear functions

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

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: $$\begin{array}{rl}{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}(t)& =\frac{8}{{\pi }^2}\sum _{i=0}^{N-1}(-1{)}^i{n}^{-2}\sin \left(2\pi f_{0}nt\right)\end{array}$$ where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), $f_0$ is the fundamental frequency, and i is the harmonic label which is related to its mode number by $n=2i+1$.

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

See also

• List of periodic functions
• Sine wave
• Square wave
• Sawtooth wave
• Pulse wave
• Sound
• Triangle function
• Wave
• Zigzag

References

• Weisstein, Eric W.. "Fourier Series - Triangle Wave". http://mathworld.wolfram.com/FourierSeriesTriangleWave.html.