Triangle wave
5 seconds of triangle wave at 220 Hz


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A triangle wave is a nonsinusoidal 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).
Contents
 Harmonics 1
 Definitions 2
 See also 3
 References 4
Harmonics
It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4nā1)th harmonic by ā1 (or changing its phase by Ļ), 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{align} x_\mathrm{triangle}(t) & {} = \frac {8}{\pi^2} \sum_{k=0}^\infty (1)^k \, \frac{ \sin \left( (2k+1) t \right)}{(2k+1)^2} \\ & {} = \frac{8}{\pi^2} \left( \sin ( t){1 \over 9} \sin (3 t)+{1 \over 25} \sin (5 t)  \cdots \right) \end{align}
Definitions
Another definition of the triangle wave, with range from 1 to 1 and period 2a is:
 x(t)=\frac{2}{a} \left (ta \left \lfloor\frac{t}{a}+\frac{1}{2} \right \rfloor \right )(1)^\left \lfloor\frac{t}{a}+\frac{1}{2} \right \rfloor
 where the symbol \scriptstyle \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 \left ( {t \over a}  \left \lfloor {t \over a} + {1 \over 2} \right \rfloor \right) \right 
or, for a range from 1 to +1:
 x(t)= 2 \left  2 \left ( {t \over a}  \left \lfloor {t \over a} + {1 \over 2} \right \rfloor \right) \right   1
The triangle wave can also be expressed as the integral of the square wave:
 \int\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) = x\,\bmod\,4  21
or, a more complex and complete version of the above equation with a period of 2Ļ and starting with y(0) = 0:
 y(x) = \left4\left(\left(\frac{x}{2\pi}  0.25\right)\,\bmod\,1\right)  2\right1
In terms of sine and arcsine with period p and amplitude a:
 y(x) = \frac{2a}{\pi}\arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right)
See also
References
 Weisstein, Eric W., "Fourier Series  Triangle Wave", MathWorld.